diff --git "a/CMPhysBench.json" "b/CMPhysBench.json" new file mode 100644--- /dev/null +++ "b/CMPhysBench.json" @@ -0,0 +1,12308 @@ +[ + { + "id": 1, + "topic": "Theoretical Foundations", + "question": "For an oscillator with charge $q$, its energy operator without an external field is\n\n\\begin{equation*}\nH_{0}=\\frac{p^{2}}{2 m}+\\frac{1}{2} m \\omega^{2} x^{2} \n\\end{equation*}\n\nIf a uniform electric field $\\mathscr{E}$ is applied, causing an additional force on the oscillator $f= q \\mathscr{E}$, the total energy operator becomes\n\n\\begin{equation*}\nH=\\frac{p^{2}}{2 m}+\\frac{1}{2} m \\omega^{2} x^{2}-q \\mathscr{E} x \n\\end{equation*}\n\nFind the expression for the new energy levels $E_{n}$.", + "final_answer": [ + "E_{n} =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{q^{2} \\mathscr{E}^{2}}{2 m \\omega^{2}}" + ], + "answer_type": "Expression", + "answer": "In $H_{0}$ and $H$, $p$ is the momentum operator,\n\n$p=-\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} x}$\n\n\nThe potential energy term in equation (2) can be rewritten as\n\n$\\frac{1}{2} m \\omega^{2} x^{2}-q \\mathscr{E} x=\\frac{1}{2} m \\omega^{2}[(x-x_{0})^{2}-x_{0}^{2}]$\n\nwhere\n\n\\begin{equation*}\nx_{0}=\\frac{q \\mathscr{E}}{m \\omega^{2}} \\tag{3}\n\\end{equation*}\n\n\nBy performing a coordinate shift, let\n\n\\begin{equation*}\nx^{\\prime}=x-x_{0} \\tag{4}\n\\end{equation*}\n\n\nBecause\n\n\\begin{equation*}\np=-\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} x}=-\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} x}=p^{\\prime} \\tag{5}\n\\end{equation*}\n\n$H$ can be expressed as\n\n\\begin{equation*}\nH=\\frac{p^{\\prime 2}}{2 m}+\\frac{1}{2} m \\omega^{2} x^{\\prime 2}-\\frac{1}{2} m \\omega^{2} x_{0}^{2} \\tag{6}\n\\end{equation*}\n\n\nComparing equations (1) and (6), it is evident that the difference between $H$ and $H_{0}$ is that the variable changes from $x$ to $x^{\\prime}$, with an added constant term $(-\\frac{1}{2} m \\omega^{2} x_{0}^{2})$. Hence, we get\n\n\\begin{gather*}\nE_{n}=E_{n}^{(0)}-\\frac{1}{2} m \\omega^{2} x_{0}^{2} \\tag{7}\\\\\n\\varphi_{n}(x)=\\psi_{n}(x^{\\prime})=\\psi_{n}(x-x_{0}) \\tag{8}\n\\end{gather*}\n\n\nIt is well-known that the energy levels of the free oscillator are\n\n\\begin{equation*}\nE_{n}^{(0)}=(n+\\frac{1}{2}) \\hbar \\omega, \\quad n=0,1,2, \\cdots \\tag{9}\n\\end{equation*}\n\n\nThus,\n\n\\begin{align*}\nE_{n} & =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{1}{2} m \\omega^{2} x_{0}^{2} \\\\\n& =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{q^{2} \\mathscr{E}^{2}}{2 m \\omega^{2}} \\tag{10}\n\\end{align*}\n\n\nIntroducing the coordinate shift operator\n\n\\begin{equation*}\nD_{x}(x_{0})=\\mathrm{e}^{-\\mathrm{i} x_{0} p / \\hbar}=\\mathrm{e}^{-x_{0} \\frac{d}{d x}} \\tag{11}\n\\end{equation*}\n\n\nIts effect on the wave function is\n\n\\begin{equation*}\nD_{x}(x_{0}) \\psi(x)=\\psi(x-x_{0}) \\tag{11'}\n\\end{equation*}\n\n\nThen the eigenfunctions of $H$ and $H_{0}$ can be related through the shift operator:\n\n\\begin{equation*}\n\\varphi_{n}(x)=\\psi_{n}(x-x_{0})=D_{x}(x_{0}) \\psi_{n}(x) \\tag{12}\n\\end{equation*}\n\n\nConversely,\n\n\\begin{equation*}\n\\psi_{n}(x)=\\varphi_{n}(x+x_{0})=D_{x}(-x_{0}) \\varphi_{n}(x) \\tag{$\\prime$}\n\\end{equation*}\n\n\nSolution two, using the raising and lowering operators of the oscillator\n\n\\begin{equation*}\na=\\sqrt{\\frac{m \\omega}{2 \\hbar}}(x+\\frac{\\mathrm{i}}{m \\omega} p), \\quad a^{+}=\\sqrt{\\frac{m \\omega}{2 \\hbar}}(x-\\frac{\\mathrm{i}}{m \\omega} p) \\tag{11}\n\\end{equation*}\n\n\nExpressing $H_{0}$ and $H$ as\n\n\\begin{gather*}\nH_{0}=(a^{+} a+\\frac{1}{2}) \\hbar \\omega \\tag{14}\\\\\nH=(a^{+} a+\\frac{1}{2}) \\hbar \\omega-q \\mathscr{E} \\sqrt{\\frac{\\hbar}{2 m \\omega}}(a+a^{+}) \\tag{15}\n\\end{gather*}\n\n\nIntroducing\n\n\\begin{equation*}\nx_{0}=q \\varepsilon / m \\omega^{2} \n\\end{equation*}\n\n\nThen\n\n\\begin{align*}\nH & =\\hbar \\omega[a^{+} a+\\frac{1}{2}-x_{0} \\sqrt{\\frac{m \\omega}{2 \\hbar}}(a+a^{+})] \\\\\n& =\\hbar \\omega[(a^{+}-x_{0} \\sqrt{\\frac{m \\omega}{2 \\hbar}})(a-x_{0} \\sqrt{\\frac{m \\omega}{2 \\hbar}})+\\frac{1}{2}-\\frac{m \\omega x_{0}^{2}}{2 \\hbar}] \\\\\n& =\\hbar \\omega[(a^{+}-\\alpha_{0})(a-\\alpha_{0})+\\frac{1}{2}]-\\frac{1}{2} m \\omega^{2} x_{0}^{2} \\tag{16}\n\\end{align*}\n\n\nwhere\n\n\\begin{equation*}\n\\alpha_{0}=x_{0} \\sqrt{m \\omega / 2 \\hbar} \\tag{17}\n\\end{equation*}\n\n\nComparing equations (14), (16), the difference between $H$ and $H_{0}$ is $a \\rightarrow a-\\alpha_{0}, a^{+} \\rightarrow a^{+}-\\alpha_{0}$, and an added constant term $(-\\frac{1}{2} m \\omega^{2} x_{0}^{2})$.\n\nStarting from the fundamental commutation relation\n\n\\begin{equation*}\n[a, a^{+}]=1 \\tag{18}\n\\end{equation*}\n\n\nIt was proved that the energy level formula (9) and the recursion relations between eigenstates\n\n\\begin{equation*}\na \\psi_{n}=\\sqrt{n} \\psi_{n-1}, \\quad a^{-} \\psi_{n}=\\sqrt{n+1} \\psi_{n+1} \\tag{19}\n\\end{equation*}\n\n\nAnd the ground state wave function satisfies\n\n\\begin{equation*}\na \\psi_{0}=0 \\tag{20}\n\\end{equation*}\n\n\nAs\n\n$[a-\\alpha_{0}, a^{+}-\\alpha_{0}]=[a, a^{+}]=1$\n\n\nSo the same reasoning logically leads to similar conclusions for the eigenvalues and eigenfunctions of $H$, simply substituting $(a-\\alpha_{0})$ for $a$ in the entire derivation, $(a^{+}-\\alpha_{0})$ for $a^{+}$. The eigenvalue of $H$ is clearly equation (10). The recursion relations and ground state equation for the eigenfunctions are\n\n\\begin{gather*}\n(a-\\alpha_{0}) \\varphi_{n}(x)=\\sqrt{n} \\varphi_{n-1}(x) \\tag{21}\\\\\n(a^{+}-\\alpha_{0}) \\varphi_{n}(x)=\\sqrt{n+1} \\varphi_{n+1}(x) \\\\\n(a-\\alpha_{0}) \\varphi_{0}(x)=0 \\tag{22}\n\\end{gather*}\n\n$a \\rightarrow(a-\\alpha_{0})$ (and $a^{+} \\rightarrow a^{+}-\\alpha_{0})$ are equivalent to $x \\rightarrow(x-x_{0})$, thus replacing $x$ with $(x-x_{0})$ in $\\psi_{n}(x)$ gives\n\n\\begin{equation*}\n\\varphi_{n}(x)=\\psi_{n}(x-x_{0})=D_{x}(x_{0}) \\psi_{n}(x) \\tag{12}\n\\end{equation*}\n\n$\\varphi_{n}$ and $\\varphi_{0}$ can be related through the operator $(a^{+}-\\alpha_{0})$:\n\n\\begin{equation*}\n\\varphi_{n}(x)=\\frac{1}{\\sqrt{n!}}(a^{+}-\\alpha_{0})^{n} \\varphi_{0}(x) \\tag{23}\n\\end{equation*}", + "symbol": { + "$q$": "charge of the oscillator", + "$H_{0}$": "energy operator without external field", + "$p$": "momentum operator", + "$m$": "mass", + "$\\omega$": "angular frequency", + "$x$": "position", + "$\\mathscr{E}$": "uniform electric field", + "$f$": "force on the oscillator", + "$H$": "total energy operator", + "$E_{n}$": "new energy levels", + "$\\hbar$": "reduced Planck's constant", + "$x_{0}$": "displacement caused by the field", + "$x^{\\prime}$": "shifted coordinate", + "$p^{\\prime}$": "shifted momentum operator", + "$\\varphi_{n}$": "eigenfunction of the total energy operator", + "$\\psi_{n}$": "eigenfunction of the initial operator", + "$a$": "lowering operator", + "$a^{+}$": "raising operator", + "$\\alpha_{0}$": "constant related to displacement" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 2, + "topic": "Theoretical Foundations", + "question": "A particle of mass $m$ is in the ground state of a one-dimensional harmonic oscillator potential\n\n\\begin{equation*}\nV_{1}(x)=\\frac{1}{2} k x^{2}, \\quad k>0 \n\\end{equation*}\n\n\nWhen the spring constant $k$ suddenly changes to $2k$, the potential then becomes\n\n\\begin{equation*}\nV_{2}(x)=k x^{2} \n\\end{equation*}\n\n\nImmediately measure the energy of the particle, and find the expression for the probability of the particle being in the ground state of the new potential $V_{2}$.", + "final_answer": [ + "\\frac{2^{5 / 4}}{1+\\sqrt{2}}" + ], + "answer_type": "Expression", + "answer": "(a) The wave function of the particle $\\psi(x, t)$ should satisfy the time-dependent Schrödinger equation\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} \\psi=-\\frac{\\hbar^{2}}{2 m} \\frac{\\partial^{2}}{\\partial x^{2}} \\psi+V \\psi \\tag{3}\n\\end{equation*}\n\n\nWhen $V$ undergoes a sudden change (from $V_{1} \\rightarrow V_{2}$) but with a finite change quantity, $\\psi$ remains a continuous function of $t$, implying that $\\psi$ does not change when $V$ changes abruptly.\n\nDenote $\\psi_{0}(x)$ and $\\phi_{0}(x)$ as the ground state wave functions of the potential $V_{1}$ and $V_{2}$, respectively. After the potential suddenly changes from $V_{1}$ to $V_{2}$, the wave function of the particle remains $\\psi_{0}$. The probability of measuring the particle in the state $\\phi_{0}$ is $|\\langle\\psi_{0} \\mid \\phi_{0}\\rangle|^{2}$.\n\nRewrite $V_{1}$ and $V_{2}$ in standard form:\n\n\\begin{gather*}\nV_{1}(x)=\\frac{1}{2} k x^{2}=\\frac{1}{2} m \\omega_{1}^{2} x^{2} \\tag{$\\prime$}\\\\\nV_{2}(x)=k x^{2}=\\frac{1}{2} m \\omega_{2}^{2} x^{2} \\tag{$\\prime$}\n\\end{gather*}\n\n\nIt is clear that\n\n\\begin{equation*}\n\\omega_{2}=\\sqrt{2} \\omega_{1} \\tag{4}\n\\end{equation*}\n\n$\\psi_{0}$ and $\\phi_{0}$ can be expressed as in formulas (3) and (5) from problem 3.2, namely\n\n\\begin{align}\n\\psi_{0}(x) &= \\left( \\frac{\\alpha}{\\sqrt{\\pi}} \\right)^{\\frac{1}{2}} \\mathrm{e}^{-\\alpha^2 x^2 / 2}, &\n\\alpha^{2} &= m \\omega_{1} / \\hbar \\notag \\\\\n\\phi_{0}(x) &= \\left( \\frac{\\beta}{\\sqrt{\\pi}} \\right)^{\\frac{1}{2}} \\mathrm{e}^{-\\beta^2 x^2 / 2}, &\n\\beta^{2} &= m \\omega_{2} / \\hbar \\tag{6}\n\\end{align}\n\nwhere\n\n\\begin{equation*}\n\\beta^{2} / \\alpha^{2}=\\omega_{2} / \\omega_{1}=\\sqrt{2} \\tag{7}\n\\end{equation*}\n\n\nThus\n\n\\begin{align*}\n\\langle\\psi_{0} \\mid \\phi_{0}\\rangle & =\\sqrt{\\frac{\\alpha \\beta}{\\pi}} \\int_{-\\infty}^{+\\infty} \\mathrm{e}^{-\\frac{1}{2}(\\alpha^{2}+\\beta^{2}) x^{2}} \\mathrm{~d} x=(\\frac{2 \\alpha \\beta}{\\alpha^{2}+\\beta^{2}})^{\\frac{1}{2}} \\\\\n|\\langle\\psi_{0} \\mid \\phi_{0}\\rangle|^{2} & =\\frac{2 \\alpha \\beta}{\\alpha^{2}+\\beta^{2}}=\\frac{2 \\beta / \\alpha}{1+\\beta^{2} / \\alpha^{2}}=\\frac{2^{5 / 4}}{1+\\sqrt{2}}=0.9852 \\tag{8}\n\\end{align*}\n\n\nThis is the required probability.\n(b) Consider the time when the potential changes for the first time $(V_{1} \\rightarrow V_{2})$ as $t=0$, then the wave function is\n\n\\begin{equation*}\n\\psi(x, 0)=\\psi_{0}(x) \\tag{9}\n\\end{equation*}\n\n\nLet $\\phi_{n}(x)$ denote the energy eigenstates of the potential $V_{2}$, corresponding to energy levels\n\n$E_{n}=(n+\\frac{1}{2}) \\hbar \\omega_{2}$\n\n\nExpand $\\psi_{0}$ as a linear combination of $\\phi_{n}$,\n\n\\begin{equation*}\n\\psi_{0}(x)=\\sum_{n} C_{n} \\phi_{n}(x), \\quad(n \\text { can only take even values }) \\tag{10}\n\\end{equation*}\n\n\nFor $0\\tau$, with the total energy operator given by\n\n\\begin{equation*}\nH_{0}=\\frac{1}{2 m} p^{2}+\\frac{1}{2} m \\omega^{2} x^{2} \n\\end{equation*}\n\n\nThe energy eigenstates are denoted by $\\psi_{n}$, and the energy levels $E_{n}^{(0)}=(n+\\frac{1}{2}) \\hbar \\omega$ . When $0 \\leqslant t \\leqslant \\tau$, a uniform electric field $\\mathscr{E}$ is applied, and the total energy operator becomes\n\n\\begin{equation*}\nH^{\\prime}=\\frac{1}{2 m} p^{2}+\\frac{1}{2} m \\omega^{2} x^{2}-q \\mathscr{\\varepsilon} x \n\\end{equation*}\n\nAssuming the oscillator is in the ground state $\\psi_{0}$ at $t \\leqslant 0$, find the expression for the probability $P_n$ that the system is in the energy eigenstate $\\psi_{n}$ at $t>\\tau$.", + "final_answer": [ + "P_n = \\frac{1}{n!}(2 \\alpha_{0} \\sin \\frac{\\omega \\tau}{2})^{2 n} \\mathrm{e}^{-(2 \\alpha_{0} \\sin \\frac{\\omega \\tau}{2})^{2}}" + ], + "answer_type": "Expression", + "answer": "At $t>\\tau$, the external electric field has vanished, and the wavefunction satisfies the Schrödinger equation\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} \\psi(x, t)=H_{0} \\psi(x, t) \\tag{3}\n\\end{equation*}\n\n\nThe general solution is\n\n\\begin{equation*}\n\\psi(x, t)=\\sum_{n} f_{n} \\psi_{n}(x) \\mathrm{e}^{-\\mathrm{i} E_{n}^{(0)}(t-\\tau) / \\hbar} \\tag{4}\n\\end{equation*}\n\n\nwhere the components of the $\\psi_{n}$ states are\n\n\\begin{equation*}\n|\\langle\\psi_{n} \\mid \\psi\\rangle|^{2}=|f_{n}|^{2} \\tag{5}\n\\end{equation*}\n\n\nEach coefficient $f_{n}$ depends on the wavefunction at $t=\\tau$\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\sum_{n} f_{n} \\psi_{n}(x) \\tag{6}\n\\end{equation*}\n\n\nSo the key is to find $\\psi(x, \\tau)$.\nIn the interval $0 \\leqslant t \\leqslant \\tau$, the Schrödinger equation is\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} \\psi(x, t)=H \\psi(x, t) \\tag{7}\n\\end{equation*}\n\n\nThe solution is\n\n\\begin{equation*}\n\\psi(x, t)=\\sum_{n} C_{n} \\varphi_{n}(x) \\mathrm{e}^{-\\mathrm{i} \\mathrm{E}_{n} t / \\hbar} \\tag{8}\n\\end{equation*}\n\n\nwhere $C_{n}$ depends on the initial wavefunction\n\n\\begin{equation*}\n\\psi(x, 0)=\\psi_{0}(x)=\\sum_{n} C_{n} \\varphi_{n}(x) \\tag{9}\n\\end{equation*}\n\n\nIt has been proven that\n\n\\begin{equation*}\n\\psi_{0}(x)=\\varphi_{0}(x+x_{0})=D_{x}(-x_{0}) \\varphi_{0}(x) \\tag{10}\n\\end{equation*}\n\n\nwhere $x_{0}=q \\mathscr{E} / m \\omega^{2}$. If we express the displacement operator $D_{x}(-x_{0})$ in terms of the ladder operators,\n\n\\begin{equation*}\nD_{x}(-x_{0})=\\mathrm{e}^{\\mathrm{i} x_{0} p / \\hbar}=\\mathrm{e}^{-a_{0}(a^{+}-a)} \\tag{11}\n\\end{equation*}\n\n\nUtilizing Glauber's formula\n\n$\\mathrm{e}^{A+B}=\\mathrm{e}^{A} \\mathrm{e}^{B} \\mathrm{e}^{-\\frac{1}{2}[A, B]}$\n\n\nWe obtain\n\n\\begin{align*}\nD_{x}(-x_{0}) & =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha_{0} a^{+}} \\mathrm{e}^{\\alpha_{0} a} \\\\\n& =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha_{0}(a^{+}-\\alpha_{0})} \\mathrm{e}^{\\alpha_{0}(a-\\alpha_{0})} \\tag{$\\prime$}\n\\end{align*}\n\n\nwhere $\\alpha_{0}=x_{0} \\sqrt{m \\omega / 2 \\hbar}$. Substituting equation ( $11^{\\prime}$ ) into equation (10), we obtain\n\n\\begin{equation*}\n\\psi_{0}(x)=\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha_{0}(a^{+}-\\alpha_{0})} \\mathrm{e}^{\\alpha_{0}(a-\\alpha_{0})} \\varphi_{0}(x) \\tag{12}\n\\end{equation*}\n\n\nIt has been shown in item 3.4\n\n\\begin{gather*}\n(a-\\alpha_{0}) \\varphi_{0}=0 \\tag{13}\\\\\n(a^{+}-\\alpha_{0})^{n} \\varphi_{0}=\\sqrt{n!} \\varphi_{n} \\tag{14}\n\\end{gather*}\n\n\nThus,\n\n\\begin{align*}\n& \\mathrm{e}^{\\alpha_{0}(a-\\alpha_{0})} \\varphi_{0}=\\sum_{n=0}^{\\infty} \\frac{\\alpha_{0}^{n}}{n!}(a-\\alpha_{0})^{n} \\varphi_{0}=\\varphi_{0} \\tag{$\\prime$}\\\\\n& \\begin{aligned}\n\\psi_{0}(x) & =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha_{0}(a^{+}-\\alpha_{0})} \\varphi_{0} \\\\\n& =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\sum_{n} \\frac{(-\\alpha_{0})^{n}}{n!}(a^{+}-\\alpha_{0})^{n} \\varphi_{0} \\\\\n& =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\sum_{n} \\frac{(-\\alpha_{0})^{n}}{\\sqrt{n!}} \\varphi_{n}(x)\n\\end{aligned}\n\\end{align*}\n\n\nnamely, $\\psi_{0}(x)$ is a coherent state wavefunction composed of the basis vectors $\\varphi_{n}$. Comparing equations (9) and (15), we obtain\n\n\\begin{equation*}\nC_{n}=\\frac{(-\\alpha_{0})^{n}}{\\sqrt{n!}} \\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\tag{16}\n\\end{equation*}\n\n\nSubstituting equation (16) into equation (8), and using the energy formula derived in item 3.4\n\n$E_{n}=(n+\\frac{1}{2}) \\hbar \\omega-\\frac{1}{2} m \\omega^{2} x_{0}^{2}$\n\nwe obtain\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\mathrm{e}^{\\mathrm{i} \\delta} \\mathrm{e}^{-\\frac{1}{2} a_{0}^{2}} \\sum_{n} \\frac{[-\\alpha(\\tau)]^{n}}{\\sqrt{n!}} \\varphi_{n}(x) \\tag{17}\n\\end{equation*}\n\n\nwhere\n\n\\begin{gather*}\n\\alpha(\\tau)=\\alpha_{0} \\mathrm{e}^{-\\mathrm{i} \\omega \\tau} \\tag{18}\\\\\n\\delta=\\frac{m \\omega^{2} x_{0}^{2} \\tau}{2 \\hbar}-\\frac{\\omega \\tau}{2} \\tag{19}\n\\end{gather*}\n\n$\\psi(x, \\tau)$ is also a coherent state wavefunction composed of the $\\varphi_{n}$ states.\nUsing equation (14), equation (17) can be expressed as\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\mathrm{e}^{\\mathrm{i} \\delta} \\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha(\\tau)(a^{+}-\\alpha_{0})} \\varphi_{0}(x) \\tag{20}\n\\end{equation*}\n\n\nFrom equation (12 ${ }^{\\prime}$ ), it follows that\n\n$\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\varphi_{0}=\\mathrm{e}^{\\alpha_{0}(a^{+}-\\alpha_{0})} \\psi_{0}$\n\n\nSubstituting into equation (20), we obtain\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\mathrm{e}^{\\mathrm{i} \\delta} \\mathrm{e}^{[\\alpha_{0}-\\alpha(\\tau)](a^{+}-a_{0})} \\psi_{0}=\\mathrm{e}^{\\wp \\delta} \\mathrm{e}^{-\\alpha_{0}^{2}+\\alpha_{0} \\alpha(\\tau)} \\mathrm{e}^{[\\alpha_{0}-\\alpha(\\tau)] a^{+}} \\psi_{0} \\tag{21}\n\\end{equation*}\n\n\nTo express $\\psi(x, \\tau)$ as a linear superposition of the $\\psi_{n}$, we use the formula\n\n\\begin{equation*}\n(a^{+})^{n} \\psi_{0}=\\sqrt{n!} \\psi_{n} \\tag{22}\n\\end{equation*}\n\n\nThus, we have\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\mathrm{e}^{\\mathrm{i} \\delta} \\mathrm{e}^{\\alpha_{0} \\alpha(\\tau)-\\alpha_{0}^{2}} \\sum_{n} \\frac{[\\alpha_{0}-\\alpha(\\tau)]^{n}}{\\sqrt{n!}} \\psi_{n}(x) \\tag{23}\n\\end{equation*}\n\n\nwhere the component fraction of the $\\psi_{n}$ state is\n\n\\begin{align*}\n|\\langle\\psi_{n} \\mid \\psi(x, \\tau)\\rangle|^{2} & =|\\mathrm{e}^{\\alpha_{0} \\alpha(\\tau)-\\alpha_{0}^{2}}|^{2} \\frac{|\\alpha_{0}-\\alpha(\\tau)|^{2 n}}{n!} \\\\\n& =\\frac{1}{n!}(2 \\alpha_{0} \\sin \\frac{\\omega \\tau}{2})^{2 n} \\mathrm{e}^{-(2 \\alpha_{0} \\sin \\frac{\\omega \\tau}{2})^{2}} \\tag{24}\n\\end{align*}\n\n\nIt can be easily verified that\n\n\\begin{equation*}\n\\sum_{n}|\\langle\\psi_{n} \\mid \\psi(x, \\tau)\\rangle|^{2}=1 \\tag{25}\n\\end{equation*}\n\n\nThis is a specific manifestation of the conservation of total probability.\nBy changing the external electric field duration $\\tau$, each time\n\n$\\tau=k 2 \\pi / \\omega, \\quad k=1,2,3, \\cdots$\n\n\nIn $\\psi(x, \\tau)$, the components of each excited state $(n \\geqslant 1)$ become 0, and the ground state $(\\psi_{0})$ component is 1. Each time\n\n$\\tau=(2 k+1) \\pi / \\omega, \\quad k=0,1,2, \\cdots$\n\nthe ground state component in $\\psi(x, \\tau)$ reaches a minimum value of $\\mathrm{e}^{-4 \\alpha_{0}^{2}}, \\psi_{n}$ state components are\n\n$$|f_{n}|^{2}=\\frac{(4 \\alpha_{0}^{2})^{n}}{n!} \\mathrm{e}^{-4 \\alpha_{0}^{2}}$$", + "symbol": { + "$q$": "charge", + "$t$": "time", + "$\\tau$": "duration period for electric field", + "$H_{0}$": "initial energy operator", + "$m$": "mass", + "$p$": "momentum", + "$\\omega$": "angular frequency", + "$x$": "position", + "$\\psi_{n}$": "energy eigenstate", + "$E_{n}^{(0)}$": "energy level without electric field", + "$\\hbar$": "reduced Planck's constant", + "$\\mathscr{E}$": "electric field", + "$H^{\\prime}$": "energy operator with electric field", + "$\\mathscr{\\varepsilon}$": "electric field", + "$\\psi_{0}$": "ground state wavefunction", + "$P_n$": "probability of the system being in the energy eigenstate $\\psi_{n}$", + "$f_{n}$": "coefficients for the wavefunction at a specific time", + "$E_{n}$": "energy level with electric field", + "$C_{n}$": "initial coefficients in wavefunction expansion", + "$\\varphi_{n}$": "basis vectors", + "$x_{0}$": "displacement due to electric field", + "$D_{x}$": "displacement operator", + "$a$": "annihilation operator", + "$a^{+}$": "creation operator", + "$\\alpha_{0}$": "dimensionless displacement parameter", + "$\\alpha(\\tau)$": "time-dependent displacement parameter", + "$\\delta$": "phase factor", + "$\\wp$": "phase factor" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 4, + "topic": "Theoretical Foundations", + "question": "Calculate the result of the commutator $[\\boldsymbol{p}, \\frac{1}{r}]$.", + "final_answer": [ + "\\mathrm{i} \\hbar \\frac{\\boldsymbol{r}}{r^{3}}" + ], + "answer_type": "Expression", + "answer": "Using the commutator\n\n\\begin{equation*}\n[\\boldsymbol{p}, F(\\boldsymbol{r})]=-\\mathrm{i} \\hbar \\nabla F \\tag{1}\n\\end{equation*}\n\n\nWe obtain\n\n\\begin{equation*}\n[p, \\frac{1}{r}]=-\\mathrm{i} \\hbar(\\nabla \\frac{1}{r})=\\mathrm{i} \\hbar \\frac{r}{r^{3}} \\tag{2}\n\\end{equation*}\n\n\nUsing the formula (see question 4.2)\n\n$[\\boldsymbol{A} \\cdot \\boldsymbol{B}, F]=[\\boldsymbol{A}, F] \\cdot \\boldsymbol{B}+\\boldsymbol{A} \\cdot[\\boldsymbol{B}, F]$\n\n\nWe obtain\n\n\\[\n\\begin{aligned}\n{[\\boldsymbol{p}^{2}, \\frac{1}{r}] } & =[\\boldsymbol{p}, \\frac{1}{r}] \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot[\\boldsymbol{p}, \\frac{1}{r}]=\\mathrm{i} \\hbar(\\frac{1}{r^{3}} \\boldsymbol{r} \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot \\frac{\\boldsymbol{r}}{r^{3}}) \\\\\n& =2 \\mathrm{i} \\hbar \\frac{1}{r^{3}} \\boldsymbol{r} \\cdot \\boldsymbol{p}+\\hbar^{2}(\\nabla \\cdot \\frac{\\boldsymbol{r}}{r^{3}})\n\\end{aligned}\n\\]\n\nHowever, since\n\n$\\nabla \\cdot \\frac{\\boldsymbol{r}}{r^{3}}=\\frac{1}{r^{3}} \\nabla \\cdot \\boldsymbol{r}+\\boldsymbol{r} \\cdot(\\nabla \\frac{1}{r^{3}})=\\frac{3}{r^{3}}-\\boldsymbol{r} \\cdot \\frac{3 \\boldsymbol{r}}{r^{5}}=0$\n\n\nSo\n\n\\begin{equation*}\n[\\boldsymbol{p}^{2}, \\frac{1}{r}]=2 \\mathrm{i} \\hbar \\frac{1}{r^{3}} \\boldsymbol{r} \\cdot \\boldsymbol{p}=2 \\hbar^{2} \\frac{1}{r^{2}} \\frac{\\partial}{\\partial r} \\tag{3}\n\\end{equation*}\n\n\nSubsequently, using equation (1), we get\n\n\\begin{align*}\n{[\\boldsymbol{p}, r^{2}] } & =-i \\hbar \\nabla(r^{2})=-2 i \\hbar \\boldsymbol{r} \\tag{4}\\\\\n{[\\boldsymbol{p}^{2}, r^{2}] } & =[\\boldsymbol{p}, r^{2}] \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot[\\boldsymbol{p}, r^{2}]=-2 i \\hbar(\\boldsymbol{r} \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot \\boldsymbol{r}) \\\\\n& =-4 i \\hbar \\boldsymbol{r} \\cdot \\boldsymbol{p}-2 \\hbar^{2} \\nabla \\cdot \\boldsymbol{r}=-4 i \\hbar \\boldsymbol{r} \\cdot \\boldsymbol{p}-6 \\hbar^{2} \\\\\n& =-4 \\hbar^{2} r \\frac{\\partial}{\\partial r}-6 \\hbar^{2} \\tag{5}\n\\end{align*}\n\n\nFinally, using the commutator\n\n\\begin{equation*}\n[\\boldsymbol{p}^{2}, \\boldsymbol{r}]=-\\mathrm{i} \\hbar \\frac{\\partial p^{2}}{\\partial \\boldsymbol{p}}=-2 \\mathrm{i} \\hbar \\boldsymbol{p} \\tag{6}\n\\end{equation*}\n\n\nAnd equation (3), we get\n\n\\begin{align*}\n{[\\boldsymbol{p}^{2}, \\frac{\\boldsymbol{r}}{r}] } & =[\\boldsymbol{p}^{2}, \\boldsymbol{r}] \\frac{1}{r}+\\boldsymbol{r}[\\boldsymbol{p}^{2}, \\frac{1}{r}] \\\\\n& =-2 \\mathrm{i} \\hbar \\boldsymbol{p} \\frac{1}{r}+2 \\mathrm{i} \\hbar \\boldsymbol{r} \\frac{1}{r^{3}}(\\boldsymbol{r} \\cdot \\boldsymbol{p}) \\\\\n& =-2 \\mathrm{i} \\hbar[\\frac{1}{r} \\boldsymbol{p}-\\mathrm{i} \\hbar(\\nabla \\frac{1}{r})]+2 \\mathrm{i} \\hbar \\frac{\\boldsymbol{r}}{r^{3}}(\\boldsymbol{r} \\cdot \\boldsymbol{p}) \\\\\n& =2 \\hbar^{2} \\frac{\\boldsymbol{r}}{r^{3}}+2 \\mathrm{i} \\hbar[\\frac{\\boldsymbol{r}}{r^{3}}(\\boldsymbol{r} \\cdot \\boldsymbol{p})-\\frac{1}{r} \\boldsymbol{p}] \\\\\n& =2 \\hbar^{2}(\\frac{\\boldsymbol{r}}{r^{3}}+\\boldsymbol{r} \\frac{1}{r^{2}} \\frac{\\partial}{\\partial r}-\\frac{1}{r} \\nabla) \\tag{7}\n\\end{align*}", + "symbol": { + "$\\boldsymbol{p}$": "momentum operator", + "$\\hbar$": "reduced Planck's constant", + "$\\boldsymbol{r}$": "position vector", + "$r$": "magnitude of the position vector $\\boldsymbol{r}$" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 5, + "topic": "Theoretical Foundations", + "question": "For the hydrogen-like ion (nuclear charge $Z e$) with the $(H, l^{2}, l_{z})$ common eigenstate $\\psi_{n l m}$, it is known that the various $\\langle r^{\\lambda}\\rangle$ satisfy the following recursion relation (Kramers' formula):\n\n\\begin{equation*}\n\\frac{\\lambda+1}{n^{2}}\\langle r^{\\lambda}\\rangle-(2 \\lambda+1) \\frac{a_{0}}{Z}\\langle r^{\\lambda-1}\\rangle+\\frac{\\lambda}{4}[(2 l+1)^{2}-\\lambda^{2}] \\frac{a_{0}^{2}}{Z^{2}}\\langle r^{\\lambda-2}\\rangle=0 \n\\end{equation*}\n\n\nIt is known that $\\langle r^{0}\\rangle=1$. Use this formula to calculate the expression for $\\langle r\\rangle_{n l m}$.", + "final_answer": [ + "\\langle r\\rangle_{n l m}=\\frac{1}{2}[3 n^{2}-l(l+1)] \\frac{a_{0}}{Z}" + ], + "answer_type": "Expression", + "answer": "The spherical coordinate expression of $\\psi_{n l m}$ is\n\n\\begin{equation*}\n\\psi_{n l m}=R_{n l}(r) \\mathrm{Y}_{l m}(\\theta, \\varphi)=\\frac{1}{r} u_{n l}(r) \\mathrm{Y}_{l m}(\\theta, \\varphi) \\tag{2}\n\\end{equation*}\n\nThe expectation value of $r^{\\lambda}$ is\n\n\\begin{equation*}\n\\langle r^{\\lambda}\\rangle_{n l m}=\\int r^{\\lambda}|\\psi_{n l m}|^{2} \\mathrm{~d}^{3} x=\\int_{0}^{\\infty} r^{\\lambda}(u_{n l})^{2} \\mathrm{~d} r \\tag{3}\n\\end{equation*}\n\n$u_{n l}$ satisfies the radial equation\n\n\\begin{equation*}\n-\\frac{\\hbar^{2}}{2 \\mu} u^{\\prime \\prime}+[l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}-\\frac{Z e^{2}}{r}] u=E_{n} u \\tag{4}\n\\end{equation*}\n\n\nBecause\n\n\\begin{equation*}\nE_{n}=-\\frac{Z^{2} e^{2}}{2 n^{2} a_{0}}, \\quad a_{0}=\\frac{\\hbar^{2}}{\\mu e^{2}} \\tag{5}\n\\end{equation*}\n\n\nEquation (4) can be rewritten as\n\n\\begin{equation*}\nu^{\\prime \\prime}+[\\frac{2 Z}{a_{0} r}-\\frac{l(l+1)}{r^{2}}-(\\frac{Z}{n a_{0}})^{2}] u=0 \\tag{$\\prime$}\n\\end{equation*}\n\n\nMultiply each term of equation ($4^{\\prime}$) by $r^{\\lambda} u$ and integrate $\\int_{0}^{\\infty} \\cdots \\mathrm{d} r$, the last three terms clearly yield the expectation values of $r^{\\lambda-1}, ~ r^{\\lambda-2}$ and $r^{\\lambda}$, while\n\nThe first term gives\n\n\\begin{align*}\n\\int_{0}^{\\infty} r^{\\lambda} u u^{\\prime \\prime} \\mathrm{d} r & =r^{\\lambda} u u^{\\prime}|_{0} ^{\\infty}-\\int_{0}^{\\infty}(r^{\\lambda} u^{\\prime}+\\lambda r^{\\lambda-1} u) u^{\\prime} \\mathrm{d} r \\\\\n& =(r^{\\lambda} u u^{\\prime}-\\frac{\\lambda}{2} r^{\\lambda-1} u^{2})|_{0} ^{\\infty}+\\frac{\\lambda(\\lambda-1)}{2}\\langle r^{\\lambda-2}\\rangle-\\int_{0}^{\\infty} r^{\\lambda}(u^{\\prime})^{2} \\mathrm{~d} r \\tag{6}\n\\end{align*}\n\n\nIf the value of $\\lambda$ ensures that\n\n\\begin{equation*}\nr^{\\lambda} u u^{\\prime}|_{0} ^{\\infty}=0,\\quad r^{\\lambda-1} u^{2}|_{0} ^{\\infty}=0 \\tag{7}\n\\end{equation*}\n\n\nWe obtain the following preliminary result:\n\n\\begin{equation*}\n[\\frac{\\lambda(\\lambda-1)}{2}-l(l+1)]\\langle r^{\\lambda-2}\\rangle+\\frac{2 Z}{a_{0}}\\langle r^{\\lambda-1}\\rangle-(\\frac{Z}{n a_{0}})^{2}\\langle r^{\\lambda}\\rangle=\\int_{0}^{\\infty} r^{\\lambda}(u^{\\prime})^{2} \\mathrm{~d} r \\tag{8}\n\\end{equation*}\n\n\nMoreover, multiply each term of equation ($4^{\\prime}$) by $2 r^{\\lambda+1} u^{\\prime}$ and integrate, successively obtaining\n\n\\[\n\\begin{aligned}\n& \\int_{0}^{\\infty} 2 r^{\\lambda+1} u^{\\prime} u^{\\prime \\prime} \\mathrm{d} r=r^{\\lambda+1}(u^{\\prime})^{2}|_{0} ^{\\infty}-\\int_{0}^{\\infty}(\\lambda+1) r^{\\lambda}(u^{\\prime})^{2} \\mathrm{~d} r \\\\\n& \\int_{0}^{\\infty} 2 r^{\\lambda+1} u^{\\prime} u \\mathrm{~d} r=r^{\\lambda+1} u^{2}|_{0} ^{\\infty}-(\\lambda+1)\\langle r^{\\lambda}\\rangle \\\\\n& \\int_{0}^{\\infty} 2 r^{\\lambda+1} u^{\\prime} \\frac{u}{r} \\mathrm{~d} r=r^{\\lambda} u^{2}|_{0} ^{\\infty}-\\lambda\\langle r^{\\lambda-1}\\rangle \\\\\n& \\int_{0}^{\\infty} 2 r^{\\lambda+1} u^{\\prime} \\frac{u}{r^{2}} \\mathrm{~d} r=r^{\\lambda-1} u^{2}|_{0} ^{\\infty}-(\\lambda-1)\\langle r^{\\lambda-2}\\rangle\n\\end{aligned}\n\\]\n\n\nUnder the conditions guaranteed by equation (7), all first terms in the above equations are zero, thus\n\n\\begin{equation*}\n(\\lambda-1) l(l+1)\\langle r^{\\lambda-2}\\rangle-2 \\lambda \\frac{Z}{a_{0}}\\langle r^{\\lambda-1}\\rangle+(\\lambda+1)(\\frac{Z}{n a_{0}})^{2}\\langle r^{\\lambda}\\rangle=(\\lambda+1) \\int_{0}^{\\infty} r^{\\lambda}(u^{\\prime})^{2} \\mathrm{~d} r \\tag{9}\n\\end{equation*}\n\n\nCombine equations (8) and (9), eliminating the integrals on the right to obtain equation (1). The condition for the validity of equation (1) is equation (7). Given ${ }^{(1)}$\n\n\\begin{align*}\n& r \\rightarrow 0, \\quad u \\sim r^{l+1} \\\\\n& r \\rightarrow \\infty, \\quad u \\sim r^{n} \\mathrm{e}^{-Z r / n a_{0}} \\tag{10}\n\\end{align*}\n\n\nIt is evident that the necessary and sufficient condition for equation (7) to hold is\n\n\\begin{equation*}\n\\lambda>-(2 l+1) \\tag{11}\n\\end{equation*}\n\n\nIn equation (1), taking $\\lambda=0$ and noting $\\langle r^{0}\\rangle=1$ immediately yields\n\n\\begin{equation*}\n\\langle\\frac{1}{r}\\rangle_{n l m}=\\frac{Z}{n^{2} a_{0}} \\tag{12}\n\\end{equation*}\n\n\n\\footnotetext{\n(1) Refer to Zeng Jin-Yan. Quantum Mechanics Volume I. Beijing: Science Press, 1997. §6.3.\n}\n\nThis result has been obtained using the virial theorem.\nSequentially taking $\\lambda=1, 2$, leads to\n\n\\begin{gather*}\n\\langle r\\rangle_{n l m}=\\frac{1}{2}[3 n^{2}-l(l+1)] \\frac{a_{0}}{Z} \\tag{13}\\\\\n\\langle r^{2}\\rangle_{n l m}=\\frac{n^{2}}{2}[1+5 n^{2}-3 l(l+1)](\\frac{a_{0}}{Z})^{2} \\tag{14}\n\\end{gather*}\n\n\nFor example\n\\[\n\\begin{array}{rc}\n1 \\mathrm{~s} \\text { state (ground state), } & \\langle r\\rangle_{100}=\\frac{3}{2},\\langle r^{2}\\rangle_{100}=3 \\\\\n2 \\mathrm{~s} \\text { state, } & \\langle r\\rangle_{200}=6,\\langle r^{2}\\rangle_{200}=42 \\tag{15}\\\\\n2 \\mathrm{p} \\text { state, } & \\langle r\\rangle_{21 m}=5,\\langle r^{2}\\rangle_{21 m}=30\n\\end{array}\n\\]\n\nIn equation (15), $\\langle r\\rangle$ is measured in units of $a_{0} / Z$ and $\\langle r^{2}\\rangle$ is measured in units of $a_{0}^{2} / Z^{2}$.\nNote that in the context of this problem, equation (1) cannot be used to calculate $\\langle r^{-2}\\rangle$, but if the result from the previous problem on $\\langle r^{-2}\\rangle$ is used, then by substituting $\\lambda=-1$ into equation (1), $\\langle r^{-3}\\rangle$ can be calculated, with results consistent with the previous problem. Furthermore, by taking $\\lambda=-2(l \\geqslant 1)$, it is possible to calculate\n\n\\begin{equation*}\n\\langle r^{-4}\\rangle=(\\frac{Z}{a_{0}})^{4} \\frac{3 n^{2}-l(l+1)}{2 n^{5}(l-\\frac{1}{2}) l(l+\\frac{1}{2})(l+1)(l+\\frac{3}{2})} \\tag{16}\n\\end{equation*}\n\n\nCalculating other expectations $\\langle r^{\\lambda}\\rangle$ can follow this analogy ${ }^{(1)}$.", + "symbol": { + "$Z$": "nuclear charge divided by elementary charge", + "$e$": "elementary charge", + "$H$": "hydrogen atom", + "$l^{2}$": "squared orbital angular momentum", + "$l_{z}$": "z component of angular momentum", + "$\\psi_{n l m}$": "wavefunction in the common eigenstate basis", + "$n$": "principal quantum number", + "$a_{0}$": "Bohr radius", + "$r$": "radial distance", + "$\\lambda$": "exponent of the radial distance", + "$E_{n}$": "energy of the level n", + "$\\hbar$": "reduced Planck constant", + "$\\mu$": "reduced mass", + "$u_{n l}$": "radial component of the wavefunction", + "$R_{n l}$": "radial wavefunction", + "$\\mathrm{Y}_{l m}$": "spherical harmonics", + "$\\theta$": "polar angle", + "$\\varphi$": "azimuthal angle", + "$l$": "orbital quantum number", + "$m$": "magnetic quantum number", + "$n^{2}$": "principal quantum number squared" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 6, + "topic": "Theoretical Foundations", + "question": "Three-dimensional isotropic harmonic oscillator, the total energy operator is\n\n\\begin{equation*}\nH=\\frac{\\boldsymbol{p}^{2}}{2 \\mu}+\\frac{1}{2} \\mu \\omega^{2} r^{2}=-\\frac{\\hbar^{2}}{2 \\mu} \\nabla^{2}+\\frac{1}{2} \\mu \\omega^{2} r^{2} \n\\end{equation*}\n\n\nFor the common eigenstates of $(H, l^{2}, l_{z})$\n\n\\begin{equation*}\n\\psi_{n_{r} l m}=R_{n_{r} l}(r) \\mathrm{Y}_{l m}(\\theta, \\varphi)=\\frac{1}{r} u_{n_{r} l}(r) \\mathrm{Y}_{l m}(\\theta, \\varphi) \n\\end{equation*}\n\nCalculate the expression for $\\langle r^{-2}\\rangle_{n_{r} l m}$.", + "final_answer": [ + "\\langle\\frac{1}{r^{2}}\\rangle_{n, l m}=\\frac{1}{l+\\frac{1}{2}} \\frac{\\mu \\omega}{\\hbar}" + ], + "answer_type": "Expression", + "answer": "The energy levels of the three-dimensional isotropic harmonic oscillator are\n\n\\begin{equation*}\nE_{n, l m}=E_{N}=\\left(N+\\frac{3}{2}\\right) \\hbar \\omega, \\quad N=l+2 n_{r} . \\tag{3}\n\\end{equation*}\n\n\nFor $\\psi_{n_{r}, l m}, H$ is equivalent to\n\n\\footnotetext{\n(1) Refer to H. A. Kramers. Quantum Mechanics. Amsterdam: North-Holland, 1958. § 59.\n}\n\n\\begin{equation*}\nH \\rightarrow-\\frac{\\hbar^{2}}{2 \\mu} \\frac{1}{r} \\frac{\\partial^{2}}{\\partial r^{2}} r+l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}+\\frac{1}{2} \\mu \\omega^{2} r^{2} \\tag{4}\n\\end{equation*}\n\n\nAccording to the Hellmann theorem, there should be\n\n\\begin{equation*}\n\\frac{\\partial E_{n_{l} l m}}{\\partial l}=\\langle\\frac{\\partial H}{\\partial l}\\rangle_{n_{r} l m}=(l+\\frac{1}{2}) \\frac{\\hbar^{2}}{\\mu}\\langle\\frac{1}{r^{2}}\\rangle_{n_{r}, l m} \\tag{5}\n\\end{equation*}\n\n\nFrom equation (3) it is evident\n\n\\begin{equation*}\n\\frac{\\partial E_{n, l m}}{\\partial l}=\\frac{\\partial E_{N}}{\\partial N}=\\hbar \\omega \\tag{6}\n\\end{equation*}\n\n\nSubstituting into equation (5), we get\n\n\\begin{equation*}\n\\langle\\frac{1}{r^{2}}\\rangle_{n, l m}=\\frac{1}{l+\\frac{1}{2}} \\frac{\\mu \\omega}{\\hbar}=\\frac{\\alpha^{2}}{l+\\frac{1}{2}}, \\quad \\alpha=\\sqrt{\\frac{\\mu \\omega}{\\hbar}} \\tag{7}\n\\end{equation*}\n\n\nThe average value of the centrifugal potential energy is\n\n\\begin{equation*}\n\\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{n_{r}, m}=\\frac{l(l+1)}{2 l+1} \\hbar \\omega \\tag{8}\n\\end{equation*}\n\n\nNote that the average value of the centrifugal potential energy is directly determined by the angular quantum number $l$ and is independent of the principal quantum number $N$. Among the states with the same energy level $E_{N}$, the centrifugal potential energy is highest in the state $l=N$.\n\n\\begin{equation*}\n\\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{l=N}=\\frac{N(N+1)}{2 N+1} \\hbar \\omega \\tag{9}\n\\end{equation*}\n\n\nThe corresponding radial kinetic energy is only\n\n\\begin{align*}\n\\langle\\frac{p_{r}^{2}}{2 \\mu}\\rangle_{l=N} & =\\langle\\frac{\\boldsymbol{p}^{2}}{2 \\mu}-\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{l=N}=\\frac{E_{N}}{2}-\\frac{N(N+1)}{2 N+1} \\hbar \\omega \\\\\n& =(1+\\frac{1}{4 N+2}) \\frac{\\hbar \\omega}{2} \\tag{10}\n\\end{align*}\n\n\nWhen $N \\gg 1$, it follows\n\n\\begin{equation*}\n\\langle\\frac{p_{r}^{2}}{2 \\mu}\\rangle_{l=N} \\approx \\frac{1}{2} \\hbar \\omega \\tag{$\\prime$}\n\\end{equation*}\n\n\nThis situation corresponds to the circular orbit in Bohr's quantum theory. The general formula for the average value of radial kinetic energy is\n\n\\begin{equation*}\n\\langle\\frac{p_{r}^{2}}{2 \\mu}\\rangle_{n_{r} l m}=\\frac{E_{N}}{2}-\\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{n_{r} l m}=[N+\\frac{3}{2}-\\frac{l(l+1)}{l+1 / 2}] \\frac{\\hbar \\omega}{2} \\tag{11}\n\\end{equation*}\n\n\nIf $n_{r} \\ggg 1$ (quasi-classical case), the approximate treatment can be made as follows:\n\n$$\\frac{l(l+1)}{l+\\frac{1}{2}} \\approx l+\\frac{1}{2}$$\n\n\nEquations (8) and (11) become\n\n\\begin{align*}\n& \\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{n_{r} l m} \\approx(l+\\frac{1}{2}) \\frac{\\hbar \\omega}{2}=(\\frac{l}{2}+\\frac{1}{4}) \\hbar \\omega \\tag{$\\prime$}\\\\\n& \\langle\\frac{p_{r}^{2}}{2 \\mu}\\rangle_{n_{r} l m} \\approx(N+1-l) \\frac{\\hbar \\omega}{2}=(n_{r}+\\frac{1}{2}) \\hbar \\omega \\tag{11'}\n\\end{align*}", + "symbol": { + "$H$": "Hamiltonian (total energy operator)", + "$\\boldsymbol{p}$": "momentum vector", + "$\\mu$": "reduced mass", + "$\\omega$": "angular frequency", + "$r$": "radial coordinate", + "$\\hbar$": "reduced Planck's constant", + "$l$": "orbital angular momentum quantum number", + "$m$": "magnetic quantum number", + "$n_{r}$": "radial quantum number", + "$\\theta$": "polar angle", + "$\\varphi$": "azimuthal angle", + "$\\psi_{n_{r} l m}$": "wave function for specified quantum numbers", + "$R_{n_{r} l}(r)$": "radial wave function component", + "$\\mathrm{Y}_{l m}(\\theta, \\varphi)$": "spherical harmonics", + "$u_{n_{r} l}(r)$": "radial wave function in alternative form", + "$N$": "total quantum number, related to principal energy level", + "$E_{n, l m}$": "energy of the state defined by quantum numbers", + "$E_{N}$": "energy associated with total quantum number", + "$\\langle\\frac{1}{r^{2}}\\rangle_{n_{r} l m}$": "average inverse-square radial distance" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 7, + "topic": "Theoretical Foundations", + "question": "A particle with mass $\\mu$ moves in a \"spherical square well\" potential, $$V(r)= \\begin{cases}0, & r0, & r \\geqslant a.\\end{cases}$$ Consider only the bound state $(0a(\\text { outside the well }) \\tag{4b} \\end{align*} \nThe equation inside the well (4a) is precisely the spherical Bessel equation, and the physically allowed solution is the spherical Bessel function \n\\begin{equation*} R(r)=j_{l}(k_{0} r), \\quad k_{0}=\\sqrt{2 \\mu V_{0}} / \\hbar \\tag{5a} \\end{equation*} \nThe solution outside the well (4b) is \n\\begin{equation*} R(r)=C / r^{l+1} \\tag{5b} \\end{equation*} \n(The other solution $r^{l}$ does not satisfy the bound state boundary condition $R \\rightarrow 0$ as $r \\rightarrow \\infty$, so it is discarded.) For the wave function outside the well (5b), it is evident that \n\\begin{equation*} \\frac{\\mathrm{d}}{\\mathrm{~d} r}[r^{l+1} R(r)]=0, \\quad r \\geqslant a \\tag{6} \\end{equation*} \nAt $r=a$, $R$ and $R^{\\prime}$ should both be continuous. Therefore, for the wave function inside the well (5a), as $r \\rightarrow a$, it should also satisfy condition (6), that is, \n\\begin{equation*} \\frac{\\mathrm{d}}{\\mathrm{~d} r}[r^{l+1} j_{l}(k_{0} r)]_{r=a}=0 \\tag{$\\prime$} \\end{equation*} \nUsing the formula \n$\\frac{\\mathrm{d}}{\\mathrm{~d} x}[x^{l+1} j_{l}(x)]=x^{l+1} j_{l-1}(x)$ Equation $(6^{\\prime})$ \ncan be transformed into \n\\begin{equation*} j_{l-1}(k_{0} a)=0 \\tag{7} \\end{equation*} \nThis is the condition for a new bound state (angular quantum number $l$, energy level $E_{n l} \\approx V_{0}$) to appear. For the first bound state $l=0$, considering that $j_{-1}(k_{0} a)=\\frac{\\cos k_{0} a}{k_{0} a}$ The condition for the appearance of a new s-state $(l=0)$ energy level $(E \\approx V_{0})$ is \\begin{equation*} \\cos k_{0} a=0, \\quad k_{0} a=\\frac{\\pi}{2}, \\frac{3 \\pi}{2}, \\frac{5 \\pi}{2}, \\cdots \\tag{8} \\end{equation*} When the first bound state appears, $k_{0} a=\\pi / 2$, which means \\begin{equation*} V_{0} a^{2}=\\frac{\\pi^{2} \\hbar^{2}}{8 \\mu} \\tag{9} \\end{equation*} As $V_{0}$ gradually increases, whenever condition (7) is satisfied, a new energy level $E_{n l} \\approx V_{0}$ appears. The order of appearance of each energy level can be determined based on the zeros of the spherical Bessel function $j_{l}(x)$:\n\n\\[\n\\begin{aligned}\n&1~\\mathrm{s},\\quad 1~\\mathrm{p},\\quad 1~\\mathrm{d},\\quad 2~\\mathrm{s},\\quad 1~\\mathrm{f},\\quad 2~\\mathrm{p},\\quad 1~\\mathrm{g},\\quad 2~\\mathrm{d},\\quad 3~\\mathrm{s},\\\\\n&1~\\mathrm{h},\\quad 2~\\mathrm{f},\\quad 1~\\mathrm{i},\\quad 3~\\mathrm{p},\\quad 2~\\mathrm{g},\\quad 1~\\mathrm{k},\\quad 3~\\mathrm{d},\\quad 4~\\mathrm{s},\\ \\cdots\n\\end{aligned}\n\\]\n\nThe $l$ values corresponding to each spectral notation are:\n\n\\[\n\\begin{array}{ccccccccc}\nl & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\text{Letter} & \\mathrm{s} & \\mathrm{p} & \\mathrm{d} & \\mathrm{f} & \\mathrm{g} & \\mathrm{h} & \\mathrm{i} & \\mathrm{j}\n\\end{array}\n\\]\n\nWhen $V_0$ is large, the total number of bound states can be approximated by “one bound state per $h^3$ volume in phase space.” The maximum momentum for a particle in a bound state inside the well is:\n\\begin{equation*} p_{0}=\\hbar k_{0}=\\sqrt{2 \\mu V_{0}} \\tag{10} \\end{equation*} \nTherefore, the total phase space volume occupied by the bound states is $\\frac{4 \\pi}{3} a^{3} \\cdot \\frac{4 \\pi}{3} p_{0}^{3}=\\frac{16}{9} \\pi^{2}(a \\hbar k_{0})^{3}=\\Omega$ The total number of bound states is \\begin{align*} N & \\approx \\Omega /(2 \\pi \\hbar)^{3}=\\frac{2 \\pi^{2}}{9}(\\frac{a k_{0}}{\\pi})^{3} \\\\ & =\\frac{2 \\pi^{2}}{9}(\\frac{a}{\\pi \\hbar})^{3}(2 \\mu V_{0})^{3 / 2} \\tag{11} \\end{align*} For example, when $a k_{0}=7 \\pi / 2$, the highest energy level is 4 s, and counting all the states from 1 s to 4 s (consider the degeneracy of energy levels $E_{n l}$ as $2 l+1$) gives 99, while equation (11) yields $N \\approx 94$, which is indeed very close ${ }^{(1)}$.", + "symbol": { + "$\\mu$": "mass of the particle", + "$V_{0}$": "potential outside the well", + "$a$": "radius of the spherical square well", + "$E$": "energy of the bound state", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 8, + "topic": "Theoretical Foundations", + "question": "For the common eigenstate $|l m\\rangle$ of $l^{2}$ and $l_{z}$, calculate the expectation value $\\overline{l_{n}}$ of $l_{n}=\\boldsymbol{n} \\cdot \\boldsymbol{l}$. Here, $\\boldsymbol{n}$ is a unit vector in an arbitrary direction, and its angle with the $z$-axis is $\\gamma$.", + "final_answer": [ + "m \\hbar \\cos \\gamma" + ], + "answer_type": "Expression", + "answer": "Using the basic commutation relation $\\boldsymbol{l} \\times \\boldsymbol{l}=\\mathrm{i} \\hbar \\boldsymbol{l}$, we have\n\\[\n\\begin{aligned}\n& \\mathrm{i} \\hbar l_{x} l_{y}=(l_{y} l_{z}-l_{z} l_{y}) l_{y}=l_{y} l_{z} l_{y}-l_{z} l_{y}^{2} y \\\\\n& \\mathrm{i} \\hbar l_{y} l_{x}=l_{y}(l_{y} l_{z}-l_{z} l_{y})=l_{y}^{2} l_{z}-l_{y} l_{z} l_{y}\n\\end{aligned}\n\\]\n\nCalculating the expectation value in the state $|l m\\rangle$, since\n\n$\\overline{l_{z} l_{y}^{2}}=\\overline{l_{y}^{2} l_{z}}=m \\hbar \\overline{l_{y}^{2}}$\n\n\nTherefore\n\n$\\overline{l_{y} l_{x}}=-\\overline{l_{x} l_{y}}$\n\n\nThus\n\n\\begin{align*}\n& \\overline{l_{x} l_{y}}-\\overline{l_{y} l_{x}}=2 \\overline{l_{x} l_{y}}=\\mathrm{i} \\hbar \\overline{l_{z}}=\\mathrm{i} \\hbar^{2} m \\\\\n& \\overline{l_{x} l_{y}}=\\mathrm{i}^{2} m / 2, \\quad \\overline{l_{y} l_{x}}=-\\mathrm{i}^{2} m / 2 \\tag{1}\n\\end{align*}\n\nThe projection operator of $\\boldsymbol{l}$ in the direction of $\\boldsymbol{n}$ is\n\n$l_{n}=\\boldsymbol{n} \\cdot \\boldsymbol{l}=l_{x} \\cos \\alpha+l_{y} \\cos \\beta+l_{z} \\cos \\gamma$\n\n\nFor the state $|l m\\rangle$, since $\\overline{l_{x}}=0, \\overline{l_{y}}=0$, we have\n\n\\begin{gather*}\n\\overline{l_{n}}=m \\hbar \\cos \\gamma \\tag{2}\\\\\nl_{n}^{2}=l_{x}^{2} \\cos ^{2} \\alpha+l_{y}^{2} \\cos ^{2} \\beta+l_{z}^{2} \\cos ^{2} \\gamma \\\\\n\\\\\n+(l_{x} l_{y}+l_{y} l_{x}) \\cos \\alpha \\cos \\beta+\\cdots \\text { (similar terms with cyclic permutations) }\n\\end{gather*}\n\n\nCalculating the expectation value in the state $|l m\\rangle$, since\n\n\\begin{gather*}\n\\overline{l_{x} l_{y}}+\\overline{l_{y} l_{x}}=0, \\quad \\overline{l_{y} l_{z}}+\\overline{l_{z} l_{y}}=2 m \\hbar \\overline{l_{y}}=0, \\cdots \\\\\n\\overline{l_{x}^{2}}=\\overline{l_{y}^{2}}=\\frac{1}{2}(\\overline{l^{2}}-\\overline{l_{z}^{2}})=\\frac{\\hbar^{2}}{2}{l(l+1)-m^{2}} \\tag{3}\n\\end{gather*}\n\n\nThus\n\n\\begin{align*}\n\\overline{l_{n}^{2}} & =m^{2} \\hbar^{2} \\cos ^{2} \\gamma+\\frac{\\hbar^{2}}{2}{l(l+1)-m^{2}}(\\cos ^{2} \\alpha+\\cos ^{2} \\beta) \\\\\n& =m^{2} \\hbar^{2} \\cos ^{2} \\gamma+\\frac{\\hbar^{2}}{2}{l(l+1)-m^{2}}(1-\\cos ^{2} \\gamma) \\\\\n& =\\frac{\\hbar^{2}}{2}{l(l+1)(1-\\cos ^{2} \\gamma)+m^{2}(3 \\cos ^{2} \\gamma-1)} \\tag{4}\n\\end{align*}\n\n\nIf $\\boldsymbol{n}$ is orthogonal to the $z$-axis, that is, $\\cos \\gamma=0$, then $\\overline{l_{n}^{2}}=\\overline{l_{x}^{2}}=\\overline{l_{y}^{2}}, \\overline{l_{n}}=0$.", + "symbol": { + "$|l m\\rangle$": "common eigenstate of $l^{2}$ and $l_{z}$", + "$l_{z}$": "component of angular momentum along the z-axis", + "$l_{x}$": "component of angular momentum along the x-axis", + "$l_{y}$": "component of angular momentum along the y-axis", + "$l_{n}$": "projection of angular momentum in direction of $\\boldsymbol{n}$", + "$\\boldsymbol{n}$": "unit vector in an arbitrary direction", + "$\\gamma$": "angle between the unit vector $\\boldsymbol{n}$ and z-axis", + "$\\overline{l_{n}}$": "expectation value of projection of angular momentum in direction $\\boldsymbol{n}$", + "$m$": "magnetic quantum number", + "$\\hbar$": "reduced Planck's constant", + "$l$": "angular momentum quantum number" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 9, + "topic": "Theoretical Foundations", + "question": "For an electron's spin state $\\chi_{\\frac{1}{2}}(\\sigma_{z}=1)$ (i.e., the state where the Pauli matrix $\\sigma_z$ has an eigenvalue of $+1$), if we measure its spin projection in an arbitrary direction $\\boldsymbol{n}$, $\\sigma_n = \\boldsymbol{\\sigma} \\cdot \\boldsymbol{n}$, where $\\boldsymbol{n}$ is a unit vector and $n_z$ is its component along the $z$-axis. Find the expression for the probability of measuring $\\sigma_n = +1$ (expressed in terms of $n_z$).", + "final_answer": [ + "\\frac{1}{2}(1+n_z)" + ], + "answer_type": "Expression", + "answer": "One Using the eigenfunctions of $\\sigma_{n}$ obtained from the previous problem, it is easy to find\n(a) In the spin state $\\chi_{\\frac{1}{2}}=[\\begin{array}{l}1 \\\\ 0\\end{array}]$,\nthe probability of $\\sigma_{n}=1$ is\n\n\\begin{equation*}\n|\\langle\\phi_{1} \\lvert\\, \\chi_{\\frac{1}{2}}\\rangle|^{2}=\\cos ^{2} \\frac{\\theta}{2}=\\frac{1}{2}(1+n_{z}) \\tag{1}\n\\end{equation*}\n\nthe probability of $\\sigma_{n}=-1$ is\n\n\\begin{equation*}\n|\\langle\\phi_{-1} \\lvert\\, \\chi_{\\frac{1}{2}}\\rangle|^{2}=\\sin ^{2} \\frac{\\theta}{2}=\\frac{1}{2}(1-n_{z}) \\tag{2}\n\\end{equation*}\n\n(b) In the spin state $\\phi_{1}(\\sigma_{n}=1)$,\nthe probability of $\\sigma_{z}=1$ is\n\n\\begin{equation*}\n|\\langle\\chi_{\\frac{1}{2}} \\rvert\\, \\phi_{1}\\rangle|^{2}=\\frac{1}{2}(1+n_{z}) \\tag{3}\n\\end{equation*}\n\nthe probability of $\\sigma_{z}=-1$ is\n\n\\begin{equation*}\n1-\\frac{1}{2}(1+n_{z})=\\frac{1}{2}(1-n_{z}) \\tag{4}\n\\end{equation*}\n\n\n\\begin{equation*}\n\\langle\\sigma_{z}\\rangle=\\frac{1}{2}(1+n_{z})-\\frac{1}{2}(1-n_{z})=n_{z} \\tag{5}\n\\end{equation*}\n\nConsidering\n\n$\\sigma_{n}=\\sigma_{x} n_{x}+\\sigma_{y} n_{y}+\\sigma_{z} n_{z}$\n\nThe components of $\\boldsymbol{\\sigma}$ and $\\boldsymbol{n}$ have symmetric roles in the construction of $\\sigma_{n}$, so using formulas (3), (4), (5), and cyclically permuting $x, ~ y$, $z$, the following can be deduced:\n\n\\begin{gather*}\n\\sigma_{x}= \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{x}) \\tag{6}\\\\\n\\langle\\sigma_{x}\\rangle=n_{x} \\tag{7}\\\\\n\\sigma_{y}= \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{y}) \\tag{8}\\\\\n\\langle\\sigma_{y}\\rangle=n_{y} \\tag{9}\n\\end{gather*}\n\nBy combining equations (5), (7), (9) in vector form as follows:\nIn the spin state $\\phi_{1}(\\sigma_{n}=1)$,\n\n\\begin{equation*}\n\\langle\\boldsymbol{\\sigma}\\rangle=\\boldsymbol{n} \\tag{10}\n\\end{equation*}\n\nSimilarly, it is easy to calculate\nIn the spin state $\\phi_{-1}(\\sigma_{n}=-1)$,\n\n\\begin{equation*}\n\\langle\\boldsymbol{\\sigma}\\rangle=-\\boldsymbol{n} \\tag{11}\n\\end{equation*}\n\nSolution two (a) In the spin state $\\chi_{\\frac{1}{2}}$ where $\\sigma_{z}=1$, the possible measured values of $\\sigma_{n}$ are the eigenvalues $\\pm 1$; let the corresponding probabilities be $w_{+}$ and $w_{-}$, then\n\n\\begin{equation*}\n\\langle\\sigma_{n}\\rangle=w_{+} \\times 1+w_{-} \\times(-1)=w_{+}-w_{-} \\tag{12}\n\\end{equation*}\n\nSince\n\n\\begin{equation*}\n\\sigma_{n}=\\sigma_{x} n_{x}+\\sigma_{y} n_{y}+\\sigma_{z} n_{z} \\tag{13}\n\\end{equation*}\n\nConsidering that in the eigenstate of $\\sigma_{z}$ the average value of $\\sigma_{x}$ and $\\sigma_{y}$ is zero, and the average value of $\\sigma_{z}$ is the eigenvalue, hence in the state $\\chi_{\\frac{1}{2}}$,\n\n\\begin{equation*}\n\\langle\\sigma_{n}\\rangle=\\langle\\sigma_{z}\\rangle n_{z}=n_{z}=\\cos \\theta \\tag{14}\n\\end{equation*}\n\nFrom equations (12), (14), and using $w_{+}+w_{-}=1$, it can be found that\n\n\\begin{equation*}\nw_{+}=\\frac{1}{2}(1+n_{z}), \\quad w_{-}=\\frac{1}{2}(1-n_{z}) \\tag{15}\n\\end{equation*}\n\nThese are the equations (1), (2) in solution one.\n(b) In equation (14), $\\theta$ is the angle parameter in the $z$-axis and $\\boldsymbol{n}$. The choices of the $z$-axis and $\\boldsymbol{n}$ are arbitrary, and the original $z$-axis can be taken as the new $\\boldsymbol{n}$, while the original $\\boldsymbol{n}$ is taken as the new $z$-axis. Thus, it can be known that in the spin state where $\\sigma_{n}=1$\n\nThe average value of $\\sigma_{z}$ remains $\\cos \\theta$, which is $n_{z}$. By letting $x, ~ y, ~ z$ permute, we obtain\n\n\\begin{equation*}\n\\text { In the spin state } \\phi_{1}(\\sigma_{n}=1) \\text {, }\\langle\\boldsymbol{\\sigma}\\rangle=\\boldsymbol{n} \\tag{10}\n\\end{equation*}\n\n\nIn the state $\\phi_{1}$, the values of each component of $\\boldsymbol{\\sigma}$ are of course all $\\pm 1$, and their probabilities can be written similarly to those in (a), thus\n\n\\begin{align*}\n\\sigma_{x} & = \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{x}) \\tag{6}\\\\\n\\sigma_{y} & = \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{y}) \\tag{8}\\\\\n\\sigma_{z} & = \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{z}) \\tag{3,4}\n\\end{align*}", + "symbol": { + "$\\chi_{\\frac{1}{2}}$": "electron's spin state (spin-up along z-axis)", + "$\\sigma_{z}$": "Pauli matrix corresponding to spin measurement along the z-axis", + "$\\boldsymbol{n}$": "arbitrary unit vector direction for spin projection", + "$n_{z}$": "component of vector \\boldsymbol{n} along the z-axis", + "$\\sigma_{n}$": "spin projection in the direction of vector \\boldsymbol{n}", + "$\\boldsymbol{\\sigma}$": "Pauli spin operator as a vector", + "$\\sigma_{x}$": "Pauli matrix corresponding to spin measurement along the x-axis", + "$\\sigma_{y}$": "Pauli matrix corresponding to spin measurement along the y-axis", + "$\\phi_{1}$": "spin state with eigenvalue +1 for \\sigma_{n}" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 10, + "topic": "Theoretical Foundations", + "question": "Express the operator $(I+\\sigma_{x})^{1 / 2}$ (where $\\sigma_x$ is the Pauli matrix) as a linear combination of the $2 \\times 2$ identity matrix (denoted as $1$ in the expression) and $\\sigma_x$. The operation takes the principal square root.", + "final_answer": [ + "\\frac{1}{\\sqrt{2}}(1+\\sigma_x)" + ], + "answer_type": "Expression", + "answer": "(a) $(I+\\sigma_{x})^{I / 2}$. The eigenvalues of $\\sigma_{x}$ are $\\pm 1$, and for each eigenvalue, $(I+\\sigma_{x})^{1 / 2}$ gives a clear value (principal root is taken), so it can be concluded that $(I+\\sigma_{x})^{1 / 2}$ exists, and it is a function of $\\sigma_{x}$. According to the argument in problem 6.14, we can set\n\n\\begin{equation*}\n(I+\\sigma_{x})^{1 / 2}=C_{0}+C_{1} \\sigma_{x} \\tag{1}\n\\end{equation*}\n\n\nSquaring this equation, we get\n\n$I+\\sigma_{x}=(C_{0}+C_{1} \\sigma_{x})^{2}=C_{0}^{2}+C_{1}^{2}+2 C_{0} C_{1} \\sigma_{x}$\n\n\nTherefore\n\\[\n\\begin{gathered}\nC_{0}^{2}+C_{1}^{2}=1 \\\\\n2 C_{0} C_{1}=1\n\\end{gathered}\n\\]\n\nAdding and subtracting the two equations, we get\n\\[\n\\begin{aligned}\n& (C_{0}+C_{1})^{2}=2 \\\\\n& (C_{0}-C_{1})^{2}=0\n\\end{aligned}\n\\]\n\nIf we set $(C_{0}+C_{1})$ to take positive value, we can solve\n\n$C_{0}=C_{1}=1 / \\sqrt{2}$\n\n\nSubstituting into equation (1), we obtain\n\n\\begin{equation*}\n(1+\\sigma_{z})^{1 / 2}=\\frac{1}{\\sqrt{2}}(I+\\sigma_{x}) \\tag{2}\n\\end{equation*}\n\n\nIt is easy to verify that for any eigenvalue $( \\pm 1)$ of $\\sigma_{x}$, this equation holds true.", + "symbol": { + "$\\sigma_x$": "Pauli matrix (x-component)" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 11, + "topic": "Theoretical Foundations", + "question": "For a spin $1/2$ particle, $\\langle\\boldsymbol{\\sigma}\\rangle$ is often called the polarization vector, denoted as $\\boldsymbol{P}$, which is the spatial orientation of the spin angular momentum. Given the initial spin wave function at $t=0$ (in the $\\sigma_{z}$ representation) as:\n\n\\chi(0)=[\\begin{array}{c}\n\\cos \\delta \\mathrm{e}^{-\\mathrm{i} \\alpha} \\\\\n\\sin \\delta \\mathrm{e}^{\\mathrm{i} \\alpha}\n\\end{array}]\n\n\nwhere $\\delta, ~ \\alpha$ are positive real numbers (or 0), $\\delta \\leqslant \\pi / 2, \\alpha \\leqslant \\pi$. Determine the azimuthal angle $\\theta_{0}$ (polar angle) of the initial polarization vector $\\boldsymbol{P}(t=0)$.", + "final_answer": [ + "2 \\delta" + ], + "answer_type": "Expression", + "answer": "Any definite spin state is an eigenstate (with eigenvalue 1) of the projection $\\sigma_{n}$ of $\\boldsymbol{\\sigma}$ in some direction $(\\theta, \\varphi)$, and\n\n\\begin{equation*}\n\\boldsymbol{P}=\\langle\\boldsymbol{\\sigma}\\rangle=\\boldsymbol{n} \\tag{3}\n\\end{equation*}\n\n\nwhere $\\boldsymbol{n}$ is the unit vector in the direction $(\\theta, \\varphi)$, the eigenfunction of $\\sigma_{n}=1$ is\n\\begin{equation}\n \\phi_{1}(\\theta, \\varphi) =[\\begin{array}{l}\n\\cos \\frac{\\theta}{2} \\mathrm{e}^{-\\mathrm{i} \\varphi / 2} \\tag{4}\\\\\n\\sin \\frac{\\theta}{2} \\mathrm{e}^{\\mathrm{i} \\varphi / 2}\n\\end{array}]\n\\end{equation}\n\n\n\nBy comparing Equation (1) and (4), we find the azimuthal angle of the initial polarization vector $\\boldsymbol{P}(t=0)$\n\n\\begin{equation*}\n\\theta_{0}=2 \\delta, \\quad \\varphi_{0}=2 \\alpha \\tag{5}\n\\end{equation*}", + "symbol": { + "$\\langle\\boldsymbol{\\sigma}\\rangle$": "polarization vector", + "$\\boldsymbol{P}$": "polarization vector", + "$\\boldsymbol{n}$": "unit vector direction", + "$\\sigma_{n}$": "projection of spin operator in direction n", + "$\\theta_{0}$": "azimuthal angle of initial polarization vector", + "$\\theta$": "azimuthal angle", + "$\\varphi$": "polar angle", + "$\\phi_{1}$": "eigenfunction of spin operator", + "$\\delta$": "angle parameter", + "$\\alpha$": "angle parameter" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 12, + "topic": "Theoretical Foundations", + "question": "For a system composed of two spin $1 / 2$ particles, let $s_{1}$ and $s_{2}$ denote their spin angular momentum operators. Calculate and simplify the triple product $s_{1} \\cdot (s_{1} \\times s_{2})$ (take $\\hbar=1$ ).", + "final_answer": [ + "\\mathrm{i} s_{1} \\cdot s_{2}" + ], + "answer_type": "Expression", + "answer": "The basic relationships are as follows (the single particle formula is only written for particle 1)\n(a) $s_{1}^{2}=\\frac{3}{4}, \\boldsymbol{\\sigma}_{1}^{2}=3 ;(s_{1 x})^{2}=\\frac{1}{4},(\\sigma_{1 x})^{2}=1$, and so on.\n(b) $s_{1} \\times s_{1}=\\mathrm{i} s_{1}, \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{1}=2 \\mathrm{i} \\boldsymbol{\\sigma}_{1}$\n(c) $\\sigma_{1 x} \\sigma_{1 y}=-\\sigma_{1}, \\sigma_{1 x}=\\mathrm{i} \\sigma_{1 z}$, and so on.\n(d) $s_{1}$ commutes with $s_{2}$, $s_{1} \\cdot s_{2}=s_{2} \\cdot s_{1}, s_{1} \\times s_{2}=-s_{2} \\times s_{1}$\nFor the triple product, there are the following types:\n\n\\begin{equation*}\n1^{\\circ} \\tag{5}\n\\end{equation*}\n\n\n\\begin{align*}\ns_{1} \\cdot(s_{1} \\times s_{2}) & =(s_{1} \\times s_{1}) \\cdot s_{2}=\\mathrm{i} s_{1} \\cdot s_{2} \\\\\ns_{2} \\cdot(s_{1} \\times s_{2}) & =-s_{2} \\cdot(s_{2} \\times s_{1}) \\\\\n& =-(s_{2} \\times s_{2}) \\cdot s_{1}=-\\mathrm{i} s_{1} \\cdot s_{2} \\tag{6}\\\\\n\\boldsymbol{\\sigma}_{1} \\cdot(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) & =2 \\mathrm{i} \\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2} \\tag{7}\\\\\n\\boldsymbol{\\sigma}_{2} \\cdot(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) & =-2 \\mathrm{i} \\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2} \\tag{8}\n\\end{align*}\n\n$\\mathbf{2}^{\\circ} \\boldsymbol{\\sigma}_{1}(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2})$ type. Since $\\boldsymbol{\\sigma}_{1}$ and $\\boldsymbol{\\sigma}_{2}$ commute, using the formula proven in question 6.21\n\n\\begin{aligned}\n\\boldsymbol{\\sigma}(\\boldsymbol{\\sigma} \\cdot \\boldsymbol{A})-\\boldsymbol{A} & =\\boldsymbol{A}-(\\boldsymbol{\\sigma} \\cdot \\boldsymbol{A}) \\boldsymbol{\\sigma} \\\\\n& =\\mathrm{i} \\boldsymbol{A} \\times \\boldsymbol{\\sigma}, \\quad(\\boldsymbol{\\sigma}, \\boldsymbol{A} \\text { commute })\n\\end{aligned}\n\n\nLet $\\boldsymbol{\\sigma}$ and $\\boldsymbol{A}$ be equal to $\\boldsymbol{\\sigma}_{1}$ and $\\boldsymbol{\\sigma}_{2}$ respectively, and obtain\n\n\\begin{gather*}\n\\boldsymbol{\\sigma}_{1}(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2})=\\boldsymbol{\\sigma}_{2}-\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2} \\tag{9}\\\\\n\\boldsymbol{\\sigma}_{2}(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2})=\\boldsymbol{\\sigma}_{1}+\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2} \\tag{10}\\\\\n(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}) \\boldsymbol{\\sigma}_{1}=\\boldsymbol{\\sigma}_{2}+\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2} \\tag{11}\\\\\n(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}) \\boldsymbol{\\sigma}_{2}=\\boldsymbol{\\sigma}_{1}-\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2} \\tag{12}\\\\\ns_{1}(s_{1} \\cdot s_{2})=\\frac{1}{4} s_{2}-\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2}, \\text { etc. } \\tag{13}\\\\\n{[s_{1} \\cdot s_{2}, s_{1}]=\\mathrm{i} s_{1} \\times s_{2}} \\tag{14}\\\\\n{[s_{1} \\cdot s_{2}, s_{2}]=-\\mathrm{i} s_{1} \\times s_{2}} \\tag{15}\n\\end{gather*}\n\n$3^{\\circ} s_{1} \\times(s_{1} \\times s_{2})$ type. Using the vector operator formula (see question 4.1)\n\n\\boldsymbol{A} \\times(\\boldsymbol{B} \\times \\boldsymbol{C})=\\overparen{\\boldsymbol{A} \\cdot(\\boldsymbol{B C})}-(\\boldsymbol{A} \\cdot \\boldsymbol{B}) \\boldsymbol{C}\n\n\n(\\boldsymbol{A} \\times \\boldsymbol{B}) \\times \\boldsymbol{C}=\\boldsymbol{A \\cdot ( B C )}-\\boldsymbol{A}(\\boldsymbol{B} \\cdot \\boldsymbol{C})\n\n\nHence, we obtain\n\n\\begin{align*}\ns_{1} \\times(s_{1} \\times s_{2}) & =(s_{1} \\cdot s_{2}) s_{1}-\\frac{3}{4} s_{2}=\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2}-\\frac{1}{2} s_{2} \\tag{16}\\\\\n(s_{1} \\times s_{2}) \\times s_{1} & =\\frac{3}{4} s_{2}-s_{1}(s_{1} \\cdot s_{2})=\\frac{1}{2} s_{2}+\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2} \\tag{17}\\\\\ns_{2} \\times(s_{1} \\times s_{2}) & =\\frac{1}{2} s_{1}+\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2} \\tag{18}\\\\\n(s_{1} \\times s_{2}) \\times s_{2} & =\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2}-\\frac{1}{2} s_{1} \\tag{19}\\\\\n\\boldsymbol{\\sigma}_{1} \\times(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) & =\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}-2 \\boldsymbol{\\sigma}_{2} \\tag{20}\\\\\n(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) \\times \\boldsymbol{\\sigma}_{1} & =\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}+2 \\boldsymbol{\\sigma}_{2} \\tag{21}\\\\\n\\boldsymbol{\\sigma}_{2} \\times(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) & =\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}+2 \\boldsymbol{\\sigma}_{1} \\tag{22}\\\\\n(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) \\times \\boldsymbol{\\sigma}_{2} & =\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}-2 \\boldsymbol{\\sigma}_{1} \\tag{23}\n\\end{align*}\n\n$4^{\\circ}$ Total spin $\\boldsymbol{S}=s_{1}+s_{2}$,\n\n\\begin{align*}\n& \\boldsymbol{S}^{2}=2 s_{1} \\cdot s_{2}+\\frac{3}{2} \\tag{24}\\\\\n& {[\\boldsymbol{S}^{2}, s_{1}]=2[s_{1} \\cdot s_{2}, s_{1}]=2 \\mathrm{i} s_{1} \\times s_{2}} \\tag{25}\\\\\n& {[\\boldsymbol{S}^{2}, s_{2}]=-2 \\mathrm{i} s_{1} \\times s_{2}} \\tag{26}\\\\\n& \\boldsymbol{S} \\cdot(s_{1} \\times s_{2})=(s_{1} \\times s_{2}) \\cdot \\boldsymbol{S}=0 \\tag{27}\\\\\n& \\boldsymbol{S}(s_{1} \\cdot s_{2})=(s_{1} \\cdot s_{2}) \\boldsymbol{S}=\\frac{1}{4} \\boldsymbol{S} \\tag{28}\\\\\n& \\boldsymbol{S S}^{2}=\\boldsymbol{S}^{2} \\boldsymbol{S}=2 \\boldsymbol{S} \\tag{29}\n\\end{align*}", + "symbol": { + "$s_{1}$": "spin angular momentum operator for particle 1", + "$s_{2}$": "spin angular momentum operator for particle 2", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 13, + "topic": "Theoretical Foundations", + "question": "Two localized non-identical particles with spin $1/2$ (ignoring orbital motion) have an interaction energy given by (setting $\\hbar=1)$\n\n\\begin{equation*}\nH=A \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2} \n\\end{equation*}\n\nAt $t=0$, particle 1 has spin 'up' $(s_{1 z}=1 / 2)$, and particle 2 has spin 'down' $(s_{2 z}=-\\frac{1}{2})$. Find the probability that particle 1 has spin 'up' $(s_{1 z}=1 / 2)$ at any time $t>0$.", + "final_answer": [ + "\\cos^{2}(\\frac{A t}{2})" + ], + "answer_type": "Expression", + "answer": "Start by finding the spin wave function of the system. Since\n\n\\begin{equation*}\nH=A s_{1} \\cdot s_{2}=\\frac{A}{2}(\\boldsymbol{S}^{2}-\\frac{3}{2}) \\tag{$\\prime$}\n\\end{equation*}\n\n\nIt is evident that the total spin $\\boldsymbol{S}$ is a conserved quantity, so the stationary wave function can be chosen as a common eigenfunction of $\\boldsymbol{S}^{2}, ~ S_{z}$. According to different values of the total spin quantum number $S$, the eigenfunctions and energy levels are\n\n\\begin{array}{ll}\nS=1, & \\chi_{1 M_{s}}, \\quad E_{1}=A / 4 \\tag{2}\\\\\nS=0, & \\chi_{00}, \\quad E_{0}=-3 A / 4\n\\end{array}}\n\nAt $t=0$, the spin state of the system is\n\n\\begin{equation*}\n\\chi(0)=\\alpha(1) \\beta(2)=\\frac{1}{\\sqrt{2}}(\\chi_{10}+\\chi_{00}) \\tag{3}\n\\end{equation*}\n\n\nTherefore, the wave function at $t>0$ is\n\n\\begin{equation*}\n\\chi(t)=\\frac{1}{\\sqrt{2}} \\chi_{10} \\mathrm{e}^{-\\mathrm{i} E_{1} t}+\\frac{1}{\\sqrt{2}} \\chi_{00} \\mathrm{e}^{-\\mathrm{i} E_{0} t} \\tag{4}\n\\end{equation*}\n\n\nThat is\n\n\\begin{align*}\n\\chi(t) & =\\frac{1}{2}[\\alpha(1) \\beta(2)+\\beta(1) \\alpha(2)] \\mathrm{e}^{-\\mathrm{i} A / 4}+\\frac{1}{2}[\\alpha(1) \\beta(2)-\\beta(1) \\alpha(2)] \\mathrm{e}^{3 \\mathrm{i} t / 4} \\\\\n& =[\\alpha(1) \\beta(2) \\cos \\frac{A t}{2}-\\mathrm{i} \\beta(1) \\alpha(2) \\sin \\frac{A t}{2}] \\mathrm{e}^{\\mathrm{i} A / 4} \\tag{$4^\\prime$}\n\\end{align*}\n\nFrom formula ($4^{\\prime}$), it can be seen that at time $t$, the probability of particle 1 having spin 'up' [while particle 2 has spin 'down', corresponding to the $\\alpha(1) \\beta(2)$ term] is $\\cos ^{2}(\\frac{A t}{2})$.", + "symbol": { + "$A$": "interaction constant", + "$\\boldsymbol{s}_{1}$": "spin vector of particle 1", + "$\\boldsymbol{s}_{2}$": "spin vector of particle 2", + "$t$": "time", + "$s_{1 z}$": "z-component of spin vector of particle 1", + "$s_{2 z}$": "z-component of spin vector of particle 2", + "$\\hbar$": "reduced Planck's constant", + "$\\boldsymbol{S}$": "total spin vector", + "$S$": "total spin quantum number", + "$S_{z}$": "z-component of total spin vector", + "$\\chi_{1 M_{s}}$": "eigenfunction for total spin S=1 with magnetic quantum number $M_s$", + "$E_{1}$": "energy level for total spin S=1", + "$\\chi_{00}$": "eigenfunction for total spin S=0", + "$E_{0}$": "energy level for total spin S=0", + "$\\chi(t)$": "wave function at time $t$" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 14, + "topic": "Theoretical Foundations", + "question": "Consider a system consisting of three distinguishable particles each with spin $1/2$, with the Hamiltonian given by\n\n\\begin{equation*}\nH=A(s_{1} \\cdot s_{2}+s_{2} \\cdot s_{3}+s_{3} \\cdot s_{1}) \\quad \\text { ( } A \\text { is real) } \n\\end{equation*}\n\nLet $S$ denote the total spin quantum number of the system. Determine the expression for the energy level $E_{3/2}$ when $S=3/2$ (taking $\\hbar=1$).", + "final_answer": [ + "E_{3/2} = \\frac{3}{4}A" + ], + "answer_type": "Expression", + "answer": "The Hamiltonian $H$ can be written as\n\n\\begin{equation*}\nH=\\frac{A}{2}(\\boldsymbol{S}_{123}^{2}-3 \\times \\frac{3}{4}) \\tag{$\\prime$}\n\\end{equation*}\n\n\nTherefore, the energy levels (taking $\\hbar=1$) are\n\n\\begin{equation*}\nE_{S}=\\frac{A}{2}[S(S+1)-\\frac{9}{4}] \\tag{2}\n\\end{equation*}\n\n\nA complete set of conserved quantities can be taken as $[S_{123}^{2}, ~(S_{123})_{z}, S_{12}^{2}]$, with eigenvalues\n\n\\begin{array}{l}\n\\boldsymbol{S}_{12}^{2}=S^{\\prime}(S^{\\prime}+1), \\quad S^{\\prime}=1,0 \\tag{3}\\\\\n\\boldsymbol{S}_{123}^{2}=S(S+1), \\quad S=\\frac{3}{2}, \\frac{1}{2}, \\frac{1}{2} \\\\\n(S_{123})_{z}=M=S, S-1, \\cdots,(-S)\n\\end{array}}\n\n\nThe possible combinations for the quantum numbers $S^{\\prime}, ~ S$ are $S=S^{\\prime} \\pm \\frac{1}{2}>0$, i.e.,\n\n\\begin{array}{l}\nS=3 / 2, \\quad S^{\\prime}=1 \\tag{4}\\\\\nS=1 / 2, \\quad S^{\\prime}=1,0\n\\end{array}}\n\n\nFor each pair $(S, S^{\\prime})$, the degeneracy of the energy level is $(2 S+1)$, so\n\n\\begin{align*}\n& E_{3 / 2}=\\frac{3}{4} A, \\quad S=\\frac{3}{2}, \\quad S^{\\prime}=1, \\quad \\text { Degeneracy }=4 \\\\\n& E_{1 / 2}=-\\frac{3}{4} A, \\quad S=\\frac{1}{2}, \\quad S^{\\prime}=1,0, \\quad \\text { Degeneracy }=4 \\tag{5}\n\\end{align*}\n\n\nWe now aim to determine the common eigenstates of $[\\boldsymbol{S}_{123}^{2},(S_{123})_{z}, \\boldsymbol{S}_{12}^{2}]$. The spin \"up\" and \"down\" states of the $k$-th particle $(k=1,2,3)$ are denoted as $\\alpha(k), ~ \\beta(k)$, corresponding to $s_{k z}=\\frac{1}{2}, ~-\\frac{1}{2}$. As is known, the common eigenstates of $S_{12}^{2}$ and $(S_{12})_{z}$ are given by:\n\n\\begin{aligned}\n& \\chi_{11}(1,2)=\\alpha(1)_{\\alpha}(2), \\quad S^{\\prime}=1, \\quad(S_{12})_{z}=1 \\\\\n& \\chi_{10}(1,2)=\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2)+\\beta(1)_{\\alpha}(2)], \\quad S^{\\prime}=1, \\quad(S_{12})_{z}=0 \\\\\n& \\chi_{1,-1}(1,2)=\\beta(1) \\beta(2), \\quad S^{\\prime}=1, \\quad(S_{12})_{z}=-1 \\\\\n& \\chi_{00}(1,2)=\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2)-\\beta(1) \\alpha(2)], \\quad S^{\\prime}=0, \\quad(S_{12})_{z}=0\n\\end{aligned}\n\nThe common eigenstates of $\\boldsymbol{S}_{12}^{2}, ~ \\boldsymbol{S}_{123}^{2}, ~(S_{123})_{z}$ are denoted as $\\chi_{S_{S M}}(1,2,3)$. When all three quantum numbers take their maximum values, the eigenstate is clearly\n\n\\begin{equation*}\n\\chi_{1 \\frac{3}{2} \\frac{3}{2}}=\\chi_{11}(1,2)_{\\alpha}(3)=\\alpha(1)_{\\alpha}(2) \\alpha(3) \\tag{6a}\n\\end{equation*}\n\nWhen $S=3 / 2$, $M$ has 4 possible values; corresponding eigenstates for $M=\\frac{1}{2},-\\frac{1}{2},-\\frac{3}{2}$ can be obtained by repeatedly applying the ladder operator $[(S_{123})_{x}-\\mathrm{i}(S_{123})_{y}]$ to $\\chi_{1 \\frac{3}{2}} \\frac{3}{2}$. Since the ladder operator and $\\chi_{1 \\frac{3}{2} \\frac{3}{2}}$ are symmetric under permutation of the particles, each $\\chi_{1 \\frac{3}{2} M}$ obtained from this is a symmetric function. Based on this observation and considering the values of $M$, the only possible structure of these functions can be immediately written:\n\n\\begin{align*}\n\\chi_{1 \\frac{3}{2} \\frac{1}{2}} & =\\frac{1}{\\sqrt{3}}[\\alpha(1) \\alpha(2) \\beta(3)+\\beta(1) \\alpha(2) \\alpha(3)+\\alpha(1) \\beta(2) \\alpha(3)] \\tag{6b}\\\\\n\\chi_{1 \\frac{3}{2},-\\frac{1}{2}} & =\\frac{1}{\\sqrt{3}}[\\alpha(1) \\beta(2) \\beta(3)+\\beta(1) \\alpha(2) \\beta(3)+\\beta(1) \\beta(2) \\alpha(3)] \\tag{6c}\\\\\n\\chi_{1 \\frac{3}{2},-\\frac{3}{2}} & =\\beta(1) \\beta(2) \\beta(3) \\tag{6d}\n\\end{align*}\n\nWhen $S^{\\prime}=0$, the part of the system wave function concerning particles $1, ~ 2$ can only be $\\chi_{00}$, and considering the values of $M$, it can be immediately concluded that\n\n\\begin{align*}\n\\chi_{0 \\frac{1}{2} \\frac{1}{2}} & =\\chi_{00}(1,2) \\alpha(3) \\\\\n& =\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2) \\alpha(3)-\\beta(1) \\alpha(2) \\alpha(3)] \\tag{7a}\\\\\n\\chi_{0 \\frac{1}{2},-\\frac{1}{2}} & =\\chi_{00}(1,2) \\beta(3) \\\\\n& =\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2) \\beta(3)-\\beta(1) \\alpha(2) \\beta(3)] \\tag{7b}\n\\end{align*}\n\n\nNow only $\\chi_{1 \\frac{1}{2} \\frac{1}{2}}$ and $\\chi_{1 \\frac{1}{2},-\\frac{1}{2}}$ remain to be determined. In the construction of $\\chi_{1 \\frac{1}{2} \\frac{1}{2}}$, each term should include two $\\alpha$ states and one $\\beta$ state. It must be a linear combination of $\\chi_{11}(1,2) \\beta(3)$ and $\\chi_{10}(1,2) \\alpha(3)$, and should be orthogonal to $\\chi_{1 \\frac{3}{2} \\frac{1}{2}}$ since they have different $S$ values. Expression (6b) for $\\chi_{1 \\frac{3}{2} \\frac{1}{2}}$ can be written as\n\n\\begin{equation*}\n\\chi_{1 \\frac{3}{2} \\frac{1}{2}}=\\frac{1}{\\sqrt{3}}[\\chi_{11}(1,2) \\beta(3)+\\sqrt{2} \\chi_{10}(1,2)_{\\alpha}(3)] \\tag{$\\prime$}\n\\end{equation*}\n\n\nThus, the construction of $\\chi_{1 \\frac{1}{2} \\frac{1}{2}}$ can only be\n\n\\begin{align*}\n\\chi_{1 \\frac{1}{2} \\frac{1}{2}} & =\\frac{1}{\\sqrt{3}}[\\sqrt{2} \\chi_{11}(1,2) \\beta(3)-\\chi_{10}(1,2) \\alpha(3)] \\\\\n& =\\frac{1}{\\sqrt{6}}[2 \\alpha(1) \\alpha(2) \\beta(3)-\\alpha(1) \\beta(2) \\alpha(3)-\\beta(1) \\alpha(2) \\beta(3)] \\tag{8a}\n\\end{align*}\n\n\nSimilarly, expression (6c) can be written as\n\n\\begin{equation*}\n\\chi_{1 \\frac{3}{2},-\\frac{1}{2}}=\\frac{1}{\\sqrt{3}}[\\sqrt{2} \\chi_{10}(1,2) \\beta(3)+\\chi_{1-1}(1,2)_{\\alpha}(3)] \\tag{$\\prime$}\n\\end{equation*}\n\n$\\chi_{1 \\frac{1}{2},-\\frac{1}{2}}$ should be formed with the same terms but orthogonal to the above expression, so\n\n\n$$\\chi_{1 \\frac{1}{2},-\\frac{1}{2}}=\\frac{1}{\\sqrt{3}}[\\chi_{\\mathrm{i} 0}(1,2) \\beta(3)-\\sqrt{2} \\chi_{1,-1}(1,2) \\alpha(3)]$$\n\n\n\\begin{equation*}\n=\\frac{1}{\\sqrt{6}}[\\alpha(1) \\beta(2) \\beta(3)+\\beta(1) \\alpha(2) \\beta(3)-2 \\beta(1) \\beta(2) \\alpha(3)] \\tag{8b}\n\\end{equation*}\n\n\nExpressions (6), (7), and (8) above are the complete set of common eigenfunctions for $\\boldsymbol{S}_{12}^{2}, ~ \\boldsymbol{S}_{123}^{2}, ~(S_{123})_{z}$. It is easy to verify that they are indeed orthogonal to each other. Since each particle has two spin states $\\alpha$ and $\\beta$, the system of three particles has a total of 8 independent states, thus expressions (6) to (8) form the sought orthogonal and complete set of energy eigenstates.\n\nThe reader can easily verify that according to the theory of angular momentum coupling, similar results would be obtained using the C.G. coefficient table.", + "symbol": { + "$H$": "Hamiltonian", + "$A$": "real constant in the Hamiltonian", + "$s_{1}$": "spin component of the first particle", + "$s_{2}$": "spin component of the second particle", + "$s_{3}$": "spin component of the third particle", + "$S$": "total spin quantum number of the system", + "$E_{3/2}$": "energy level for total spin quantum number 3/2", + "$\\hbar$": "reduced Planck's constant", + "$\\boldsymbol{S}_{123}$": "total spin operator for particles 1, 2, and 3", + "$S_{123}^{2}$": "squared total spin operator for particles 1, 2, and 3", + "$(S_{123})_{z}$": "z-component of the total spin operator for particles 1, 2, and 3", + "$S_{12}^{2}$": "squared spin operator for particles 1 and 2", + "$S^{\\prime}$": "intermediate spin quantum number", + "$M$": "magnetic quantum number", + "$\\alpha(k)$": "spin up state of the k-th particle", + "$\\beta(k)$": "spin down state of the k-th particle", + "$s_{k z}$": "z-component of the spin for the k-th particle", + "$\\chi_{11}$": "state with specific spin configuration for two particles", + "$\\chi_{10}$": "state with specific spin configuration for two particles", + "$\\chi_{1,-1}$": "state with specific spin configuration for two particles", + "$\\chi_{00}$": "state with specific spin configuration for two particles", + "$\\chi_{1 \\frac{3}{2} \\frac{3}{2}}$": "common eigenstate when all quantum numbers are at maximum values", + "$\\chi_{1 \\frac{3}{2} \\frac{1}{2}}$": "common eigenstate for total spin 3/2 and magnetic quantum number 1/2", + "$\\chi_{1 \\frac{3}{2},-\\frac{1}{2}}$": "common eigenstate for total spin 3/2 and magnetic quantum number -1/2", + "$\\chi_{1 \\frac{3}{2},-\\frac{3}{2}}$": "common eigenstate for total spin 3/2 and magnetic quantum number -3/2", + "$\\chi_{0 \\frac{1}{2} \\frac{1}{2}}$": "common eigenstate for intermediate spin 0 and magnetic quantum number 1/2", + "$\\chi_{0 \\frac{1}{2},-\\frac{1}{2}}$": "common eigenstate for intermediate spin 0 and magnetic quantum number -1/2", + "$\\chi_{1 \\frac{1}{2} \\frac{1}{2}}$": "common eigenstate for total spin 1/2 and magnetic quantum number 1/2", + "$\\chi_{1 \\frac{1}{2},-\\frac{1}{2}}$": "common eigenstate for total spin 1/2 and magnetic quantum number -1/2" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 15, + "topic": "Theoretical Foundations", + "question": "An electron moves freely in a one-dimensional region $-L/2 \\leqslant x \\leqslant L/2$, with the wave function satisfying periodic boundary conditions $\\psi(x)=\\psi(x+L)$. Its unperturbed energy eigenvalue is $E_n^{(0)}$. A perturbation $H^{\\prime}=\\varepsilon \\cos q x$ is applied to this system where $L q=4 \\pi N$ ($N$ is a large positive integer). Consider the degenerate state where the electron's momentum is $|p|=q \\hbar / 2$ (corresponding to energy $E_N^{(0)}$) without perturbation. Find the expression for the second-order energy correction $E_N^{(2)}$ of these degenerate states caused by the perturbation $H^{\\prime}$.", + "final_answer": [ + "-\\frac{\\varepsilon^{2}}{32 E_{N}^{(0)}}" + ], + "answer_type": "Expression", + "answer": "(a) For a free particle, the common eigenfunction of the energy $(H_{0})$ and momentum $(p)$ satisfying the periodic condition is\n\n\\begin{equation*}\n\\psi_{n}^{(0)}=\\frac{1}{\\sqrt{L}} \\mathrm{e}^{\\mathrm{i} 2 \\pi x / L}, \\quad n=0, \\pm 1, \\pm 2, \\cdots \\tag{1}\n\\end{equation*}\n\n\nThe eigenvalue is\n\n\\begin{align*}\np_{n} & =\\hbar k_{n}=2 \\pi n \\hbar / L \\\\\nE_{n}^{(0)} & =\\frac{p_{n}^{2}}{2 m}=\\frac{(2 \\pi n \\hbar)^{2}}{2 m L^{2}} \\tag{2}\n\\end{align*}\n\n(b) The perturbation operator is\n\n\\begin{equation*}\nH^{\\prime}=\\varepsilon \\cos q x=\\frac{\\varepsilon}{2}(\\mathrm{e}^{\\mathrm{i} \\pi \\lambda N_{x} / L}+\\mathrm{e}^{-\\mathrm{i} 4 \\pi N_{x} / L}) \\tag{3}\n\\end{equation*}\n\n\nFor $|p|=q \\hbar / 2=2 \\pi N \\hbar / L$ , the corresponding zeroth-order wave function is\n\n\\begin{equation*}\n\\psi^{(0)}=C_{N} \\psi_{N}^{(0)}+C_{-, ~} \\psi_{-, ~}^{(0)} \\tag{4}\n\\end{equation*}\n\n\nIt can be easily calculated that\n\n\\begin{equation*}\nH_{\\mathrm{vv}}^{\\prime}=H_{-\\mathrm{v},-\\mathrm{N}}^{\\prime}=0, \\quad H_{v,-\\mathrm{v}}^{\\prime}=H_{-\\mathrm{N}, \\mathrm{v}}^{\\prime}=\\frac{\\varepsilon}{2} \\tag{5}\n\\end{equation*}\n\n\nTherefore, in the subspace ${\\psi_{N}^{(0)}, \\psi_{-N}^{(0)}}$, the matrix representation of $H^{\\prime}$ is\n\nH^{\\prime}=\\frac{\\varepsilon}{2}[\\begin{array}{ll}\n0 & 1 \\tag{$\\prime$}\\\\\n1 & 0\n\\end{array}]\n\n\nIn this subspace, $\\psi^{(0)}$ should satisfy the eigenvalue equation\n\n$H^{\\prime} \\psi^{(0)}=E_{N}^{(1)} \\psi^{(0)}\n\n\nThat is\n\\begin{equation}\n\\frac{\\varepsilon}{2}\\left[\\begin{array}{ll}\n0 & 1 \\tag{6}\\\\\n1 & 0\n\\end{array}\\right]\\left[\\begin{array}{l}\nC_{\\mathrm{N}} \\\\\nC_{-\\mathrm{N}}\n\\end{array}\\right]=E_{N}^{(1)}\\left[\\begin{array}{l}\nC_{\\mathrm{N}} \\\\\nC_{-N}\n\\end{array}\\right]\n\\end{equation}\n\n\n\nIt is easy to solve\n\n\\begin{align*}\n& E_{\\mathrm{N}+}^{(1)}=\\varepsilon / 2 \\tag{7}\\\\\n& \\psi^{(0)}=\\frac{1}{\\sqrt{2}}[\\psi_{N}^{(0)}+\\psi_{-\\mathrm{N}}^{(0)}]=\\psi_{N+}^{(0)} \\\\\n& E_{\\mathrm{N}-}^{(1)}=-\\varepsilon / 2 \\\\\n& \\psi^{(0)}=\\frac{1}{\\sqrt{2}}[\\psi_{\\mathrm{N}}^{(0)}-\\psi_{-\\mathrm{N}}^{(0)}]=\\psi_{N-}^{(0)}\n\\end{align*}\n\n\nThe first-order correction to the wave function is\n\n\\begin{equation*}\n\\psi^{(1)}=\\sum_{n}^{\\prime} \\frac{1}{E_{N}^{(0)}-E_{n}^{(0)}}\\langle\\psi_{n}^{(0)}| H^{\\prime}|\\psi^{(0)}\\rangle \\psi_{n}^{(0)} \\quad(n \\neq \\pm N) \\tag{8}\n\\end{equation*}\n\n\nThe matrix elements contributing to $\\psi^{(1)}$ are (corresponding to $n= \\pm 3 N$)\n\n$$ H_{3, \\mathrm{~V} . \\mathrm{N}}^{\\prime}=H_{-3, \\mathrm{~V},-\\mathrm{N}}^{\\prime}=\\varepsilon / 2$$\n\n\nThe corresponding energy difference is\n\n$$ E_{N}^{(0)}-E_{3 N}^{(0)}=-8 E_{N}^{(0)}=-\\frac{8(2 \\pi N \\hbar)^{2}}{2 m L^{2}}$$ \n\n\nThus, the first-order correction to the wave function is\n\n\\begin{align}\nE^{(1)} & =E_{N+}^{(1)}=\\frac{\\varepsilon}{2} \\\\\n\\psi^{(1)} & =-\\frac{\\varepsilon}{2} \\frac{2 m L^{2}}{8(2 \\pi N \\hbar)^{2}} \\frac{1}{\\sqrt{2}}[\\psi_{3 N}^{(0)}+\\psi_{-3 N}^{(0)}]\n\\end{align}\n\n\n\\begin{align*}\nE^{(1)} & =E_{N-}^{(1)}=-\\frac{\\varepsilon}{2} \\\\\n\\psi^{(1)} & =-\\frac{\\varepsilon}{2} \\frac{2 m L^{2}}{8(2 \\pi N \\hbar)^{2}} \\frac{1}{\\sqrt{2}}[\\psi_{3 N}^{(0)}-\\psi_{-3 N}^{(0)}] \\tag{9}\n\\end{align*}\n\n(c) The second-order correction to the energy is\n\n\\begin{equation*}\nE_{N}^{(2)}=\\langle\\psi^{(0)}| H^{\\prime}|\\psi^{(1)}\\rangle \\tag{10}\n\\end{equation*}\n\n\nSubstituting equations (7) and (9) into the above formula, we obtain\n\n\\begin{equation*}\nE_{N}^{(2)}=-\\frac{\\varepsilon^{2}}{32} \\frac{2 m L^{2}}{(2 \\pi N \\hbar)^{2}}=-\\frac{\\varepsilon^{2}}{32 E_{N}^{(0)}} \\tag{11}\n\\end{equation*}\n\n\nCombining equations (7) and (11), the conclusion for the energy level up to $\\varepsilon^{2}$ order is:\n\n\\begin{equation*}\nE_{N}=E_{N}^{(0)}+E_{N}^{(1)}+E_{N}^{(2)}=E_{\\mathrm{N}}^{(0)} \\pm \\frac{\\varepsilon}{2}-\\frac{\\varepsilon^{2}}{32 E_{N}^{(0)}} \\tag{12}\n\\end{equation*}\n\n\nCorresponding to\n\n$$\\psi^{(0)}=\\frac{1}{\\sqrt{2}}[\\psi_{N}^{(0)} \\pm \\psi_{-, .}^{(0)}] $$", + "symbol": { + "$x$": "position", + "$L$": "length of the region", + "$\\psi(x)$": "wave function", + "$\\psi(x+L)$": "wave function after a period", + "$E_n^{(0)}$": "unperturbed energy eigenvalue", + "$H^{\\prime}$": "perturbation to the system", + "$\\varepsilon$": "perturbation strength", + "$q$": "wave number associated with the perturbation", + "$N$": "large positive integer", + "$p$": "momentum of the electron", + "$\\hbar$": "reduced Planck's constant", + "$E_N^{(0)}$": "energy corresponding to the momentum state before perturbation", + "$E_N^{(2)}$": "second-order energy correction", + "$H_{0}$": "Hamiltonian for unperturbed system", + "$p_{n}$": "momentum eigenvalue", + "$E_{n}^{(0)}$": "energy eigenvalue for state n", + "$H_{\\mathrm{vv}}^{\\prime}$": "matrix element of the perturbation", + "$H_{-\\mathrm{v},-\\mathrm{N}}^{\\prime}$": "off-diagonal matrix element of perturbation", + "$H_{v,-\\mathrm{v}}^{\\prime}$": "off-diagonal matrix element of perturbation", + "$H_{-\\mathrm{N},\\mathrm{v}}^{\\prime}$": "off-diagonal matrix element of perturbation", + "$E_{N+}^{(1)}$": "first-order energy correction (positive)", + "$\\psi^{(0)}$": "zeroth-order wave function", + "$C_{N}$": "coefficient for positive momentum state", + "$C_{-N}$": "coefficient for negative momentum state", + "$\\psi_{N+}^{(0)}$": "positive superposition state", + "$E_{N-}^{(1)}$": "first-order energy correction (negative)", + "$\\psi_{N-}^{(0)}$": "negative superposition state", + "$E_{3 N}^{(0)}$": "energy of 3N state", + "$E^{(1)}$": "first-order correction", + "$\\psi^{(1)}$": "first-order correction to wave function" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 16, + "topic": "Theoretical Foundations", + "question": "For the $n$-th bound state $\\psi_{n}, ~ E_{n}$ of a square well (depth $V_{0}$, width $a$), under the condition $V_{0} \\gg E_{n}$, calculate the probability of the particle appearing outside the well.", + "final_answer": [ + "\\frac{2 \\hbar E_n}{a V_0 \\sqrt{2 m V_0}}" + ], + "answer_type": "Expression", + "answer": "Take the even parity state as an example. The energy eigenvalue equation can be written as\n\n\\begin{equation}\n \\begin{array}{lll}\n\\psi^{\\prime \\prime}+k^{2} \\psi=0, & |x| \\leqslant a / 2 & \\text { (inside the well) } \\tag{1}\\\\\n\\psi^{\\prime \\prime}-\\beta^{2} \\psi=0, & |x| \\geqslant a / 2 & \\text { (outside the well) }\n\\end{array}\n\\end{equation}\n\n\n\nwhere\n\n\\begin{equation*}\nk=\\sqrt{2 m E} / \\hbar, \\quad \\beta=\\sqrt{2 m(V_{0}-E)} / \\hbar \\tag{2}\n\\end{equation*}\n\n\nNote that under the condition $V_{0} \\gg E$, $\\beta \\gg k$.\nThe even parity solution of equation (1) is\n\n\\begin{array}{ll}\n\\psi=\\cos k x, & |x| \\leqslant a / 2 \\\\\n\\psi=C \\mathrm{e}^{-\\beta|x|}, & |x| \\geqslant a / 2 \\tag{3}\n\\end{array}\n\nAt $x=a / 2$, $\\psi$ should be continuous, thus yielding\n\n\\begin{equation*}\nC=\\mathrm{e}^{\\beta a / 2} \\cos \\frac{k a}{2} \\tag{4}\n\\end{equation*}\n\nAt $x=a / 2$, $\\psi^{\\prime}$ should also be continuous, thus yielding\n\n$$ C=\\frac{k}{\\beta} \\mathrm{e}^{\\beta a / 2} \\sin \\frac{k a}{2} $$\n\n\nDividing by equation (4), we obtain the energy level formula\n\n\\begin{equation*}\n\\tan \\frac{k a}{2}=\\frac{\\beta}{k} \\tag{5}\n\\end{equation*}\n\n\nUnder the condition $\\beta \\gg k$, the solution of equation (5) is\n\n\\begin{equation*}\nk a \\approx n \\pi, \\quad n=1,3,5, \\cdots \\tag{6}\n\\end{equation*}\n\n\nSubstituting into equation (2), the energy levels are\n\n\\begin{equation*}\nE_{n}=\\frac{\\hbar^{2} k^{2}}{2 m} \\approx \\frac{1}{2 m}(\\frac{n \\pi \\hbar}{a})^{2} \\tag{7}\n\\end{equation*}\n\n\nThis is precisely the energy level formula for an infinitely deep potential well.\nNow calculate the probability of the particle appearing inside and outside the well. From equations (3) and (4), it is easy to find\n\n\\begin{align*}\n& \\int_{\\text {outside }}|\\psi|^{2} \\mathrm{~d} x=2 C^{2} \\int_{a / 2}^{\\infty} \\mathrm{e}^{-2 \\beta x} \\mathrm{~d} x=\\frac{C^{2}}{\\beta} \\mathrm{e}^{-\\beta a}=\\frac{1}{\\beta} \\cos ^{2} \\frac{k a}{2} \\tag{8}\\\\\n& \\int_{\\text {inside }}|\\psi|^{2} \\mathrm{~d} x=2 \\int_{0}^{a / 2} \\cos ^{2} k x \\mathrm{~d} x=\\frac{a}{2}(1+\\frac{\\sin k a}{k a}) \\tag{9}\n\\end{align*}\n\n\nConsidering $k a \\approx n \\pi(n=1,3,5, \\cdots), \\sin k a$ and $\\cos (k a / 2)$ are both close to zero, it is understood that the probability of the particle appearing outside the well is much smaller than that inside the well. Additionally,\n\n\\begin{gather*}\n\\int_{-\\infty}^{+\\infty}|\\psi|^{2} \\mathrm{~d} x \\approx \\int_{\\text {inside }}|\\psi|^{2} \\mathrm{~d} x \\approx \\frac{a}{2} \\\\\n\\text { Outside probability }=\\frac{\\int_{\\text {outside }}|\\psi|^{2} \\mathrm{~d} x}{\\int_{-\\infty}^{+\\infty}|\\psi|^{2} \\mathrm{~d} x} \\approx \\frac{2}{\\beta a} \\cos ^{2} \\cdot \\frac{k a}{2} \\tag{10}\n\\end{gather*}\n\n\nUsing the energy level formula (5), it is easy to obtain\n\n\\begin{gather*}\n1+\\tan ^{2} \\frac{k a}{2}=\\frac{k^{2}+\\beta^{2}}{k^{2}}=\\frac{V_{0}}{E} \\\\\n\\cos ^{2} \\frac{k a}{2}=\\frac{E}{V_{0}} \\tag{11}\n\\end{gather*}\n\n\nSubstituting into equation (10), we get\n\n\\begin{equation*}\n\\text { Outside probability }=\\frac{2 E}{a \\beta V_{0}} \\approx \\frac{2 \\hbar}{a \\sqrt{2 m V_{0}}} \\frac{E}{V_{0}} \\tag{12}\n\\end{equation*}\n\n\nConsidering $V_{0} \\gg E$ and the energy level formula (7), it is easily seen\n\n\\begin{equation*}\n\\sqrt{2 m V_{0}} \\gg n \\pi \\hbar / a \\tag{13}\n\\end{equation*}\n\n\nThus,\n\n\\begin{equation*}\n\\text { Outside probability } \\ll \\frac{2 E}{n \\pi V_{0}} \\tag{14}\n\\end{equation*}", + "symbol": { + "$n$": "quantum number representing the bound state", + "$\\psi_{n}$": "wave function for the n-th bound state", + "$E_{n}$": "energy of the n-th bound state", + "$V_{0}$": "depth of the potential well", + "$a$": "width of the potential well", + "$k$": "wave number inside the well", + "$m$": "mass of the particle", + "$E$": "energy", + "$\\beta$": "exponential decay constant outside the well", + "$C$": "normalization constant for the wave function" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 17, + "topic": "Theoretical Foundations", + "question": "A particle moves freely, and the initial wave function at $t=0$ is given as\n$$ \\psi(x, 0)=(2 \\pi a^{2})^{-1 / 4} \\exp [\\mathrm{i} k_{0}(x-x_{0})-(\\frac{x-x_{0}}{2 a})^{2}], \\quad a>0 $$\n\nFind the wave function $\\varphi(p)$ in the $p$ representation at $t=0$;", + "final_answer": [ + "\\varphi(p) = (\\frac{2 a^{2}}{\\pi})^{\\frac{1}{4}} \\exp [-\\mathrm{i} \\frac{p x_{0}}{\\hbar}-a^{2}(\\frac{p}{\\hbar}-k_{0})^{2}]" + ], + "answer_type": "Expression", + "answer": "To simplify, we set $\\hbar=1$ during calculations, then add it to the final result. First, consider the shape of the wave packet at $t=0$\n\n\\begin{equation*}\n|\\psi(x, 0)|^{2}=(2 \\pi a^{2})^{-1 / 2} \\mathrm{e}^{-(x-x_{0})^{2} / 2 a^{2}} \\tag{1}\n\\end{equation*}\n\n\nwhich is a Gaussian distribution. According to the mean value formula\n\\begin{equation}\n \\overline{f(x)}=\\int_{-\\infty}^{+\\infty}|\\psi|^{2} f(x) \\mathrm{d} x\n\\end{equation}\n\nit is easy to calculate that at $t=0$\n\n\\begin{align*}\n& \\bar{x}=x_{0}, \\quad \\bar{x}^{2}=a^{2}+x_{0}^{2} \\\\\n& \\Delta x=(\\bar{x}^{2}-\\bar{x}^{2})^{1 / 2}=a \\tag{2}\n\\end{align*}\n\nwhich means at $t=0$, the center of the wave packet is at $x_{0}$, and the width of the wave packet is $a$. Let\n\n\\begin{equation*}\n\\psi(x, 0)=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty} \\varphi(k) \\mathrm{e}^{\\mathrm{i} k x} \\mathrm{~d} k \\tag{3}\n\\end{equation*}\n\nwhere $\\varphi(k)$ is the initial wave function in the momentum representation. According to the Fourier transform formula\n\\begin{align*}\n \\varphi(k)&=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty} \\psi(x, 0) \\mathrm{e}^{-\\mathrm{i} k x} \\mathrm{~d} x\\\\\n & =(\\frac{1}{2 \\pi})^{3 / 4} \\frac{1}{\\sqrt{a}} \\mathrm{e}^{-\\mathrm{k} x_{0}} \\int_{-\\infty}^{+\\infty} \\mathrm{d} x \\exp [-(\\frac{x-x_{0}}{2 a})^{2}-\\mathrm{i}(k-k_{0})(x-x_{0})] \\\\\n& =(\\frac{2 a^{2}}{\\pi})^{\\frac{1}{4}} \\exp [-\\mathrm{i} k x_{0}-a^{2}(k-k_{0})^{2}] \\tag{4}\n\\end{align*}\n\nThe integration formula is used in the calculation (refer to the note at the end of the question). From equation (4)\n\n\\begin{equation*}\n|\\varphi(k)|^{2}=\\sqrt{\\frac{2}{\\pi}} a \\mathrm{e}^{-2 a^{2}(k-k_{0})^{2}} \\tag{5}\n\\end{equation*}\n\n\nwhich indicates that the probability distribution of momentum is also a Gaussian distribution. According to the mean value formula\n\\begin{equation}\n \\overline{f(k)}=\\int_{-\\infty}^{+\\infty}|\\varphi(k)|^{2} f(k) \\mathrm{d} k \n\\end{equation}\n\nit is readily calculated that\n\n\\begin{equation*}\n\\bar{k}=k_{0}, \\quad \\overline{k^{2}}=k_{0}^{2}+\\frac{1}{4 a^{2}} \\tag{6}\n\\end{equation*}\n\n\nSince $p=\\hbar k$, thus\n\n\\begin{align*}\n& \\bar{p}=\\hbar k_{0}, \\quad \\bar{p}^{2}=\\hbar^{2} k_{0}^{2}+\\frac{\\hbar^{2}}{4 a^{2}} \\tag{7}\\\\\n& \\Delta p=(\\overline{p^{2}}-\\bar{p}^{2})^{1 / 2}=\\hbar / 2 a\n\\end{align*}\n\n\nThese are the characteristics of the momentum distribution at $t=0$. Since it is a free particle, momentum conservation implies that the probability distribution of momentum is also conserved. Therefore, equation (7) applies to any time. From equations (2) and (7), it follows that\n\n\\begin{equation*}\n\\Delta x \\cdot \\Delta p=\\hbar / 2 \\quad(t=0 \\text { instant }) \\tag{8}\n\\end{equation*}", + "symbol": { + "$t$": "time", + "$\\psi$": "wave function in position representation", + "$x$": "position", + "$a$": "width of the wave packet", + "$k_{0}$": "initial wave number", + "$x_{0}$": "initial position", + "$\\varphi$": "wave function in momentum representation", + "$p$": "momentum", + "$\\hbar$": "reduced Planck's constant", + "$k$": "wave number" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 18, + "topic": "Theoretical Foundations", + "question": "The particle moves freely, with the initial wave function at $t=0$ given as\n\n$$ \\psi(x, 0)=(2 \\pi a^{2})^{-1 / 4} \\exp [\\mathrm{i} k_{0}(x-x_{0})-(\\frac{x-x_{0}}{2 a})^{2}], \\quad a>0 $$\n\nFind the wave function $\\psi(x, t)$ for $t>0$", + "final_answer": [ + "\\psi(x, t) = \\frac{\\exp [\\mathrm{i} k_{0}(x-x_{0})-\\mathrm{i} t k_{0}^{2} / 2 m]}{(2 \\pi)^{1 / 4}(a+\\mathrm{i} t / 2 m a)^{1 / 2}} \\exp [-\\frac{1}{4}(x-x_{0}-\\frac{k_{0} t}{m})^{2} \\frac{1-\\mathrm{i} t / 2 m a^{2}}{a^{2}+(t / 2 m a)^{2}}]" + ], + "answer_type": "Expression", + "answer": "In equation \\begin{equation*}\n\\psi(x, 0)=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty} \\varphi(k) \\mathrm{e}^{\\mathrm{i} k x} \\mathrm{~d} k, \\tag{3}\n\\end{equation*} let\n\n$$ \\mathrm{e}^{\\mathrm{i} k x} \\rightarrow \\mathrm{e}^{\\mathrm{i}(k x-\\omega t)}, \\quad \\omega=\\frac{\\hbar k^{2}}{2 m}=\\frac{k^{2}}{2 m} \\quad(\\hbar=1) $$\n\n\nthus obtaining the wave function $\\psi(x, t)$ for $t>0$, i.e.,\n\n\\begin{equation*}\n\\psi(x, t)=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty} \\varphi(k) \\exp (\\mathrm{i} k x-\\frac{\\mathrm{i} k^{2} t}{2 m}) \\mathrm{d} k \\tag{9}\n\\end{equation*}\n\n\nSubstitute equation \\begin{align*}\n \\varphi(k)&= (\\frac{2 a^{2}}{\\pi})^{\\frac{1}{4}} \\exp [-\\mathrm{i} k x_{0}-a^{2}(k-k_{0})^{2}] \n\\end{align*} into the above expression and we obtain\n\\begin{align*}\n& \\psi(x, t) \\\\\n= & \\frac{\\exp [\\mathrm{i} k_{0}(x-x_{0})-\\mathrm{i} t k_{0}^{2} / 2 m]}{(2 \\pi)^{1 / 4}(a+\\mathrm{i} t / 2 m a)^{1 / 2}} \\exp [-\\frac{1}{4}(x-x_{0}-\\frac{k_{0} t}{m})^{2} \\frac{1-\\mathrm{i} t / 2 m a^{2}}{a^{2}+(t / 2 m a)^{2}}] \\tag{10}\\\\\n\\end{align*}", + "symbol": { + "$t$": "time", + "$\\psi$": "wave function", + "$x$": "position", + "$a$": "positive constant related to wave packet width", + "$k_{0}$": "initial wave number", + "$x_{0}$": "initial position", + "$\\varphi$": "function in wave number space", + "$\\omega$": "angular frequency", + "$m$": "mass" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 19, + "topic": "Theoretical Foundations", + "question": "Using the raising and lowering operators $a^{+}, ~ a$, find the energy eigenfunctions of the harmonic oscillator (in the $x$ representation), and briefly discuss their mathematical properties.", + "final_answer": [ + "\\psi_n(x) = N_n H_n(\\alpha x) e^{-\\frac{1}{2}(\\alpha x)^2}", + "\\psi_n(x) = (\\frac{\\alpha}{\\sqrt{\\pi} 2^n n!})^{1/2} H_n(\\alpha x) e^{-\\frac{1}{2}(\\alpha x)^2}" + ], + "answer_type": "Expression", + "answer": "Start with the ground state wave function $\\psi_{0}(x)$. We have,\n\n\\begin{equation*}\na|0\\rangle=0 \\tag{1}\n\\end{equation*}\n\n\nIn the $x$ representation this reads as\n\n\\begin{equation*}\n(\\mathrm{i} \\hat{p}+m \\omega x) \\psi_{0}(x)=(\\hbar \\frac{\\mathrm{d}}{\\mathrm{~d} x}+m \\omega x) \\psi_{0}(x)=0 \\tag{2}\n\\end{equation*}\n\n\nLet $\\alpha=\\sqrt{m \\omega / \\hbar}$, the above equation can be written as\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} x} \\psi_{0}+\\alpha^{2} x \\psi_{0}=0 \\tag{2'}\n\\end{equation*}\n\n\nIt is easy to solve\n\n\\begin{equation*}\n\\psi_{0}(x)=N_{0} \\mathrm{e}^{-\\alpha^{2} x^{2} / 2} \\tag{3}\n\\end{equation*}\n\n\nwhere $N_{0}$ is the normalization constant. From the normalization condition\n\n\\begin{equation*}\n\\int_{-\\infty}^{+\\infty}|\\psi_{n}(x)|^{2} \\mathrm{~d} x=1 \\tag{4}\n\\end{equation*}\n\n\nFind (taking as real)\n\n\\begin{equation*}\nN_{0}=\\sqrt{\\alpha} / \\pi^{1 / 4} \\tag{5}\n\\end{equation*}\n\n\nSecondly, we have\n\n$$ a^{+}|0\\rangle=|1\\rangle $$\n\nThat is\n\n\\begin{equation*}\n\\psi_{1}(x)=a^{+} \\psi_{0}(x)=\\frac{1}{\\sqrt{2}}(\\alpha x-\\frac{1}{\\alpha} \\frac{\\mathrm{~d}}{\\mathrm{~d} x}) \\psi_{0}(x) \\tag{6}\n\\end{equation*}\n\n\nSubstituting equation (3) and (5) into equation (6), it's easy to find\n\n\\begin{equation*}\n\\psi_{1}=\\sqrt{2} \\alpha x \\psi_{0}=\\frac{\\sqrt{2 \\alpha}}{\\pi^{1 / 4}} \\alpha x \\mathrm{e}^{-\\alpha^{2} x^{2} / 2} \\tag{7}\n\\end{equation*}\n\n\nGenerally, we have\n\n\\begin{equation*}\n\\psi_{n}=\\frac{1}{\\sqrt{n}} a^{+} \\psi_{n-1}=\\frac{1}{\\sqrt{2 n}}(\\alpha x-\\frac{1}{\\alpha} \\frac{\\mathrm{~d}}{\\mathrm{~d} x}) \\psi_{n-1} \\tag{8}\n\\end{equation*}\n\n\nIntroducing the dimensionless variable\n\n\\begin{equation*}\n\\xi=\\alpha x, \\tag{9}\n\\end{equation*}\n\nwe thus obtain\n\n\\begin{equation*}\n\\psi_{n}=\\frac{1}{\\sqrt{2 n}}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi}) \\psi_{n-1} \\tag{10}.\n\\end{equation*}\n\n\nRecursively, we get\n\n\\begin{equation*}\n\\psi_{n}=(\\frac{1}{2^{n} n!})^{\\frac{1}{2}}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{n} \\psi_{0}=(\\frac{\\alpha}{\\sqrt{\\pi} 2^{n} n!})^{\\frac{1}{2}}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{n} \\mathrm{e}^{-\\xi^{2} / 2}=N_{n} H_{n}(\\xi) \\mathrm{e}^{-\\xi^{2} / 2} \\tag{11}\n\\end{equation*}\n\n\nWhere\n\n\\begin{equation*}\nN_{n}=(\\frac{\\alpha}{\\sqrt{\\pi} 2^{n} n!})^{\\frac{1}{2}} \\tag{12}\n\\end{equation*}\n\n\nis the normalization constant. Notice\n\n\\begin{equation*}\nN_{n}=N_{n-1} / \\sqrt{2 n} \\tag{13}\n\\end{equation*}\n\n\nIn equation (11),\n\n\\begin{equation*}\nH_{n}(\\xi)=\\mathrm{e}^{\\xi^{2} / 2}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{n} \\mathrm{e}^{-\\xi^{2} / 2} \\tag{1}\n\\end{equation*}\n\nIt is obvious that $H_{n}(\\xi)$ is an $n$-th degree polynomial in $\\xi$, called the Hermite polynomial.\nBelow is a brief discussion of the mathematical properties of $H_{n}(\\xi)$.\n\n{Parity}\n\nIt is obvious from equation (14)\n\n\\begin{equation*}\nH_{n}(-\\xi)=(-1)^{n} H_{n}(\\xi) \\tag{15}\n\\end{equation*}\n\n\nWhen $n$ is even, $H_{n}(\\xi)$ has even parity; when $n$ is odd, $H_{n}(\\xi)$ has odd parity.\n\n{Recursion relations}\n\nWe also have,\n\n$$ a \\psi_{n}=\\frac{1}{\\sqrt{2}}(\\xi+\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi}) \\psi_{n}=\\sqrt{n} \\psi_{n-1}$$\n\nSubstituting equation (11) into this equation, and using equation (13) to eliminate the normalization constant, we get\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi} H_{n}(\\xi)=2 n H_{n-1}(\\xi) \\tag{16},\n\\end{equation*}\n\nWe also have:\n$$ \\xi \\psi_{n}=\\sqrt{\\frac{n+1}{2}} \\psi_{n+1}+\\sqrt{\\frac{n}{2}} \\psi_{n-1} $$\n\nSubstituting equations (11), (13) into this equation, we get\n\n\\begin{equation*}\n2 \\xi H_{n}(\\xi)=H_{n+1}(\\xi)+2 n H_{n-1}(\\xi) \\tag{17}\n\\end{equation*}\n\n\n{Differential equation}\n\nCombining equations (16) and (17), eliminate $H_{n-1}$, then differentiate once more, and using equation (16) to eliminate $\\mathrm{d} H_{n+1} / \\mathrm{d} \\xi$, we find that $H_{n}$ satisfies the following differential equation:\n\n\\begin{equation*}\n\\frac{\\mathrm{d}^{2}}{\\mathrm{~d} \\xi^{2}} H_{n}(\\xi)-2 \\xi \\frac{\\mathrm{~d}}{\\mathrm{~d} \\xi} H_{n}(\\xi)+2 n H_{n}(\\xi)=0 \\tag{18}\n\\end{equation*}\n\n\nThis equation is known as the Hermite equation. Equation (14) is the unique polynomial solution to this equation.\n$4{ }^{\\circ}$ Differential expression\nEquation (14) is equivalent to the following:\n\n\\begin{equation*}\nH_{n}(\\xi)=(-1)^{n} e^{\\xi^{2}}(\\frac{d}{d \\xi})^{n} e^{-\\xi^{2}} \\tag{$\\prime$}\n\\end{equation*}\n\n\nThis can be proved by mathematical induction. First, for $n=0,1$, both equations (14) and ($14'$) give\n\n$$H_{0}(\\xi)=1, \\quad H_{1}(\\xi)=2 \\xi $$\n\n\nAssume the proposition holds for $n=k$, i.e., assume\n\n\\mathrm{e}^{\\xi^{2}}(-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k} \\mathrm{e}^{-\\xi^{2}}=\\mathrm{e}^{\\xi^{2} / 2}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k} \\mathrm{e}^{-\\xi^{2} / 2}=H_{k}(\\xi)\n\n\nThen\n\n\\begin{gathered}\n(-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k+1} \\mathrm{e}^{-\\xi^{2}}=-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi}(\\mathrm{e}^{-\\xi^{2}} H_{k})=(2 \\xi H_{k}-H_{k}^{\\prime}) \\mathrm{e}^{-\\xi^{2}} \\\\\n(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k+1} \\mathrm{e}^{-\\xi^{2} / 2}=(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})(\\mathrm{e}^{-\\xi^{2} / 2} H_{k})=(2 \\xi H_{k}-H_{k}^{\\prime}) \\mathrm{e}^{-\\xi^{2} / 2}\n\\end{gathered}\n\n\nThus\n\n\\mathrm{e}^{\\xi^{2}}(-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k+1} \\mathrm{e}^{-\\xi^{2}}=2 \\xi H_{k}-H_{k}^{\\prime}=\\mathrm{e}^{\\xi^{2} / 2}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k+1} \\mathrm{e}^{-\\xi^{2} / 2}\n\n\nThe proposition holds for $n=k+1$. Proof complete.\n$5^{\\circ} H_{n}(0)$ and $\\psi_{n}(0)$\nWhen $n=2 k+1$ (odd), from equation (15), we have\n\nH_{2 k+1}(-\\xi)=-H_{2 k+1}(\\xi)\n\n\nLet $\\xi \\rightarrow 0$, we get\n\n\\begin{equation*}\nH_{2 k+1}(0)=0, \\quad k=1,2,3, \\cdots \\tag{19}\n\\end{equation*}\n\n\nWhen $n=2 k$ (even), in equation (17) let $\\xi \\rightarrow 0$, and change $n$ to ( $n-1$ ), we get\n\nH_{2 k}(0)=-2(2 k-1) H_{2 k-2}(0)\n\n\nUsing this repeatedly, and noting that $H_{0}=1$, we get\n\n\\begin{equation*}\nH_{2 k}(0)=(-1)^{k} 2^{k}(2 k-1)!!=(-1)^{k}(2 k)!/ k! \\tag{20}\n\\end{equation*}\n\n\nUsing equations (11), (12) again, we get\n\n\\begin{gather*}\n\\psi_{2 k+1}(0)=0 \\tag{21}\\\\\n\\psi_{2 k}(0)=(-1)^{k} \\frac{\\sqrt{\\alpha}}{\\pi^{1 / 4}} \\frac{\\sqrt{(2 k)!}}{(2 k)!!} \\tag{22}\n\\end{gather*}\n\n\n\\begin{equation*}\n[\\psi_{2 k}(0)]^{2}=\\frac{\\alpha}{\\sqrt{\\pi}} \\frac{(2 k)!}{[(2 k)!!]^{2}}=\\frac{\\alpha}{\\sqrt{\\pi}} \\frac{(2 k-1)!!}{(2 k)!!} \\tag{23}\n\\end{equation*}", + "symbol": { + "$a$": "lowering operator", + "$a^{+}$": "raising operator", + "$x$": "position variable", + "$m$": "mass of the particle", + "$\\omega$": "angular frequency of the oscillator", + "$\\hbar$": "reduced Planck's constant", + "$\\alpha$": "dimensionless constant related to mass, frequency, and Planck's constant", + "$N_{0}$": "normalization constant of the ground state wave function", + "$N_{n}$": "normalization constant of the nth energy eigenfunction", + "$\\psi_{0}(x)$": "ground state wave function in the position representation", + "$\\psi_{n}(x)$": "nth energy eigenfunction in the position representation", + "$\\xi$": "dimensionless position variable", + "$H_{n}(\\xi)$": "Hermite polynomial of degree n" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 20, + "topic": "Theoretical Foundations", + "question": "The emission of recoil-free $\\gamma$ radiation by nuclei bound in a lattice is a necessary condition for the Mössbauer effect. The potential acting on the nuclei in the lattice can be approximated as a harmonic oscillator potential\n\n$$ V(x)=\\frac{1}{2} M \\omega^{2} x^{2} $$\n\nwhere $M$ is the mass of the nucleus, $x$ is the coordinate of the nucleus's center of mass, and $\\omega$ is the vibration frequency. Assume that initially, the nucleus's center of mass motion (harmonic vibration) is in its ground state, and at $t=0$, due to a transition of energy levels within the nucleus, a photon is emitted along the $x$-axis with energy $E_{\\gamma}$, and momentum $E_{\\gamma} / c$. Since the $\\gamma$ radiation occurs suddenly, the only effect on the nucleus's center of mass motion is that its momentum eigenvalue changes from $p$ to $(p-E_{\\gamma} / c)$. Determine the probability that the nucleus's center of mass motion remains in the ground state after the photon is emitted. For instance, for a ${ }^{57} \\mathrm{Fe}$ nucleus, if $E_{\\gamma}=18 \\mathrm{keV}, \\omega=10^{12} \\mathrm{~Hz}$, calculate the probability of this 'recoil-free emission' (i.e., no energy is transferred to the atom).", + "final_answer": [ + "P=\\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})" + ], + "answer_type": "Expression", + "answer": "Due to the harmonic oscillator potential, the center of mass motion of the nucleus is harmonic, initially (for $t<0$) in the ground state $\\psi_{0}(x)$. Expanding $\\psi_{0}(x)$ using momentum eigenfunctions gives:\n\n\\begin{equation*}\n\\psi_{0}(x)=(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\varphi(p) \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\mathrm{d} p \\tag{1}\n\\end{equation*}\n\n\nDue to the emission of $\\gamma$ photon, momentum changes (total momentum is conserved!) from $p$ to $p-p_{0},(p_{0}=E_{\\gamma} / c)$, which means the wave function $\\varphi(p)$ transforms as follows:\n\n\\begin{equation*}\n\\varphi(p) \\rightarrow \\varphi(p+p_{0}) \\tag{2}\n\\end{equation*}\n\n\nTherefore, after $\\gamma$ emission, the wave function of the nucleus's center of mass motion becomes\n\n\\begin{align*}\n\\psi(x) & =(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\varphi(p+p_{0}) \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\mathrm{d} p \\\\\n& =(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\varphi(p^{\\prime}) \\mathrm{e}^{\\mathrm{i}(p^{\\prime}-p_{0}) x / \\hbar} \\mathrm{d} p^{\\prime} \\\\\n& =\\psi_{0}(x) \\mathrm{e}^{-\\mathrm{i} p_{0} x / \\hbar} \\tag{3}\n\\end{align*}\n\n\nHere, the exponential factor indicates that the nucleus's center of mass momentum is reduced by $p_{0}$. The probability that the nucleus's center of mass vibration remains in the ground state after photon emission is\n\n\\begin{equation*}\nP=|\\langle\\psi_{0} \\mid \\psi\\rangle|^{2}=|\\int_{-\\infty}^{+\\infty} \\psi_{0}^{2}(x) \\mathrm{e}^{-i p_{0} x / \\hbar} \\mathrm{d} x|^{2} \\tag{4}\n\\end{equation*}\n\n\nUsing the explicit form of the ground state wave function $\\psi_{0}$\n\n\\begin{equation*}\n\\psi_{0}(x)=\\frac{\\sqrt{\\alpha}}{\\pi^{1 / 4}} \\mathrm{e}^{-\\alpha^{2} x^{2} / 2}, \\quad \\alpha=\\sqrt{\\frac{M_{\\omega}}{\\hbar}} \\tag{5}\n\\end{equation*}\n\n\nIt is straightforward to calculate\n\n\\begin{align*}\n\\langle\\psi_{0} \\mid \\psi\\rangle & =\\frac{\\alpha}{\\sqrt{\\pi}} \\int_{-\\infty}^{+\\infty} \\mathrm{e}^{-\\alpha^{2} x^{2}-i p_{0} x / \\hbar} \\mathrm{d} x \\\\\n& =\\frac{\\alpha}{\\sqrt{\\pi}} \\int_{-\\infty}^{+\\infty} \\cos (\\frac{p_{0} x}{\\hbar}) \\mathrm{e}^{-\\alpha^{2} x^{2}} \\mathrm{~d} x \\\\\n& =\\exp (-\\frac{p_{0}^{2}}{4 \\hbar^{2} \\alpha^{2}})=\\exp (-\\frac{E_{\\gamma}^{2}}{4 \\hbar \\omega M c^{2}}) \\tag{6}\n\\end{align*}\n\n\nSubstituting into equation (4), we obtain the probability, which is\n\n\\begin{equation*}\nP=|\\langle\\psi_{0} \\mid \\psi\\rangle|^{2}=\\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}}) \\tag{7}\n\\end{equation*}\n\n\nCalculated numerical values are as follows:\n\n\\begin{gathered}\n{ }^{57} \\mathrm{Fe} \\text { nucleus, } M c^{2} \\approx 57 m_{\\mathrm{p}} c^{2} \\approx 57 \\times 938 \\mathrm{MeV}=5.35 \\times 10^{10} \\mathrm{eV} \\\\\nE_{\\gamma}=18 \\mathrm{keV}=1.8 \\times 10^{4} \\mathrm{eV} \\\\\n\\hbar \\omega=\\hbar c \\cdot \\frac{\\omega}{c} \\approx 1.97 \\times 10^{-5} \\mathrm{eV} \\cdot \\mathrm{~cm} \\times \\frac{2 \\pi \\times 10^{12}}{3 \\times 10^{10}} \\mathrm{~cm}^{-1} \\\\\n=4.13 \\times 10^{-3} \\mathrm{eV} \\\\\nP=\\exp [-\\frac{(1.8 \\times 10^{4})^{2}}{2 \\times 4.13 \\times 10^{-3} \\times 5.35 \\times 10^{10}}] \\\\\n=\\mathrm{e}^{-0.733}=0.48\n\\end{gathered}", + "symbol": { + "$\\gamma$": "gamma radiation", + "$M$": "mass of the nucleus", + "$x$": "coordinate of the nucleus's center of mass", + "$\\omega$": "vibration frequency", + "$t$": "time", + "$E_{\\gamma}$": "energy of the emitted photon", + "$c$": "speed of light", + "$p$": "momentum", + "$p_{0}$": "momentum of the photon (E_{\\gamma}/c)", + "$\\hbar$": "reduced Planck's constant", + "$\\psi_{0}(x)$": "ground state wave function", + "$\\alpha$": "related to the oscillator strength (\\sqrt{M_{\\omega}/\\hbar})" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 21, + "topic": "Theoretical Foundations", + "question": "Calculate the probability that the nucleus is in various energy eigenstates after $\\gamma$ radiation, as in the previous problem.", + "final_answer": [ + "P_n = \\frac{1}{n!}(\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})^{n} \\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})" + ], + "answer_type": "Expression", + "answer": "From the previous problem, the center-of-mass wave function of the nucleus after $\\gamma$ radiation is\n\n\\begin{equation*}\n\\psi(x)=\\mathrm{e}^{-\\mathrm{i} p_{0} x / \\hbar} \\psi_{0}(x), \\quad p_{0}=E_{\\gamma} / c \\tag{1}\n\\end{equation*}\n\n\nThe probability of being in the ${ }^{\\prime} n$-th vibrational excited state $\\psi_{n}(x)$ is\n\n\\begin{equation*}\nP_{n}=|\\langle\\psi_{n} \\mid \\psi\\rangle|^{2}=|\\langle\\psi_{n} \\mid \\mathrm{e}^{-\\mathrm{i} p_{0} x / \\hbar} \\psi_{0}\\rangle|^{2} \\tag{2}\n\\end{equation*}\n\n\nLet $\\lambda=-p_{0} / \\hbar$, calculate $\\langle\\psi_{n} \\mid \\mathrm{e}^{\\mathrm{i} \\lambda x} \\psi_{0}\\rangle$ where $\\mathrm{e}^{\\mathrm{i} \\lambda x}$ can be viewed as an operator, expressed in terms of creation and annihilation operators, with the following relations:\n\n\\begin{align*}\n& \\mathrm{i} \\lambda x=\\alpha a^{+}-\\alpha^{*} a, \\quad \\alpha=\\mathrm{i} \\lambda \\sqrt{\\frac{\\hbar}{2 M \\omega}} \\tag{3}\\\\\n& \\mathrm{e}^{\\mathrm{i} \\lambda x}=\\mathrm{e}^{\\alpha a^{+}-\\alpha^{*} a}=\\mathrm{e}^{\\alpha a^{+}} \\mathrm{e}^{-\\alpha^{*} a} \\mathrm{e}^{-\\frac{1}{2}|\\alpha|^{2}} \\tag{4}\\\\\n& \\mathrm{e}^{\\mathrm{i} \\lambda x}|\\psi_{0}\\rangle=\\mathrm{e}^{-\\frac{1}{2}|\\alpha|^{2}} \\mathrm{e}^{\\alpha a^{+}} \\mathrm{e}^{-\\alpha^{*} a}|\\psi_{0}\\rangle=|\\psi_{\\alpha}\\rangle \\\\\n&= \\mathrm{e}^{-\\frac{1}{2}|\\alpha|^{2}} \\sum_{n} \\frac{\\alpha^{n}}{\\sqrt{n!}}|\\psi_{n}\\rangle \\tag{5}\n\\end{align*}\nwhere $\\psi_{\\alpha}$ is the coherent state of the harmonic oscillator. Utilizing equation (5), we get\n\n\\begin{gather*}\n\\langle\\psi_{n} \\mid \\mathrm{e}^{\\mathrm{i} \\lambda x} \\psi_{0}\\rangle=\\frac{\\alpha^{n}}{\\sqrt{n!}} \\mathrm{e}^{-\\frac{1}{2}|\\alpha|^{2}} \\tag{6}\\\\\nP_{n}=\\frac{|\\alpha|^{2 n}}{n!} \\mathrm{e}^{-|\\alpha|^{2}}=\\frac{1}{n!}(\\frac{\\lambda^{2} \\hbar}{2 M \\omega})^{n} \\mathrm{e}^{-\\frac{\\lambda^{2} \\hbar}{2 M \\omega}} \\\\\n= \\frac{1}{n!}(\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})^{n} \\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}}) \\tag{7}\n\\end{gather*}\n\n\nIt is easy to verify $\\sum_{n} P_{n}=1$. For the example of the ${ }^{57} \\mathrm{Fe}$ nucleus, the probabilities for the first few states are\n\n$$ P_{0}=0.48, \\quad P_{1}=0.35, \\quad P_{2}=0.13 $$", + "symbol": { + "$\\gamma$": "gamma radiation energy", + "$\\psi$": "wave function", + "$p_{0}$": "momentum of the nucleus", + "$x$": "position", + "$\\hbar$": "reduced Planck constant", + "$\\psi_{0}$": "initial wave function", + "$P_{n}$": "probability of being in the nth vibrational excited state", + "$\\psi_{n}$": "nth vibrational eigenstate", + "$\\lambda$": "parameter related to momentum and reduced Planck constant", + "$M$": "mass of the nucleus", + "$\\omega$": "angular frequency", + "$E_{\\gamma}$": "energy of the gamma photon", + "$a^{+}$": "creation operator", + "$a$": "annihilation operator", + "$\\alpha$": "displacement parameter" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 22, + "topic": "Theoretical Foundations", + "question": "Suppose the operator $\\hat{H}$ has continuous eigenvalues $\\omega$, and its eigenfunctions $u_{\\omega}(\\boldsymbol{x})$ form an orthonormal complete system, i.e.\n\n\\begin{gather*}\n\\hat{H} u_{\\omega}(\\boldsymbol{x})=\\omega u_{\\omega}(\\boldsymbol{x}) \\tag{1}\\\\\n\\int u_{\\omega^{*}}^{*}(\\boldsymbol{x}) u_{\\omega}(\\boldsymbol{x}) \\mathrm{d}^{3} x=\\delta(\\omega-\\omega^{\\prime}) \\tag{2}\\\\\n\\int u_{\\omega}^{*}(\\boldsymbol{x}^{\\prime}) u_{\\omega}(\\boldsymbol{x}) \\mathrm{d} \\omega=\\delta(\\boldsymbol{x}-\\boldsymbol{x}^{\\prime}) \\tag{3}\n\\end{gather*}\n\n\nSolve the equation\n\n\\begin{equation*}\n(\\hat{H}-\\omega_{0}) u(\\boldsymbol{x})=F(\\boldsymbol{x}) \\tag{4}\n\\end{equation*}\n\n\nwhere $F(\\boldsymbol{x})$ is a known function, and $\\omega_{0}$ is a specific eigenvalue of $H$.\n\nIf the answer exists in an integral, then find the integrand", + "final_answer": [ + "\\frac{F_{\\omega}}{\\omega-\\omega_{0}} u_{\\omega}(\\boldsymbol{x})" + ], + "answer_type": "Expression", + "answer": "Since $u_{\\omega}(\\boldsymbol{x})$ is a complete system, the solution of equation (4) can always be expressed as\n\n\\begin{equation*}\\nu(x)=\\int C_{\\omega} u_{\\omega}(x) \\mathrm{d} \\omega \\tag{5}\n\\end{equation*}\n\n\nSubstitute into equation (4), we obtain\n\n\\begin{equation*}\n\\int(\\omega-\\omega_{0}) C_{\\omega} u_{\\omega}(\\boldsymbol{x}) \\mathrm{d} \\omega=F(\\boldsymbol{x}) \\tag{6}\n\\end{equation*}\n\n\nExpand $F(\\boldsymbol{x})$ as\n\n\\begin{align*}\nF(\\boldsymbol{x}) & =\\int F(\\boldsymbol{x}^{\\prime}) \\delta(\\boldsymbol{x}-\\boldsymbol{x}^{\\prime}) \\mathrm{d}^{3} x^{\\prime} \\\\\n& =\\iint F(\\boldsymbol{x}^{\\prime}) u_{\\omega}^{*}(\\boldsymbol{x}^{\\prime}) u_{\\omega}(\\boldsymbol{x}) \\mathrm{d}^{3} x^{\\prime} \\mathrm{d} \\omega \\\\\n& =\\int F_{\\omega} u_{\\omega}(\\boldsymbol{x}) \\mathrm{d} \\omega \\tag{7}\n\\end{align*}\n\n\nwhere\n\n\\begin{equation*}\nF_{\\omega}=\\int F(\\boldsymbol{x}^{\\prime}) u_{\\omega}^{*}(\\boldsymbol{x}^{\\prime}) \\mathrm{d}^{3} x^{\\prime}=\\int F(\\boldsymbol{x}) u_{\\omega}^{*}(\\boldsymbol{x}) \\mathrm{d}^{3} x \\tag{8}\n\\end{equation*}\n\n\nSubstitute equation (7) into equation (6), we obtain\n\n\\begin{equation*}\nC_{\\omega}=F_{\\omega} /(\\omega-\\omega_{0}) \\tag{9}\n\\end{equation*}\n\n\nSubstitute back into equation (5), the solution to equation (4) is\n\n\\begin{equation*}\\nu(\\boldsymbol{x})=\\int \\frac{F_{\\omega}}{\\omega-\\omega_{0}} u_{\\omega}(\\boldsymbol{x}) \\mathrm{d} \\omega \\tag{10}\n\\end{equation*}", + "symbol": { + "$\\hat{H}$": "operator", + "$\\omega$": "continuous eigenvalue", + "$u_{\\omega}(\\boldsymbol{x})$": "eigenfunction corresponding to eigenvalue $\\omega$", + "$F(\\boldsymbol{x})$": "known function", + "$\\omega_{0}$": "specific eigenvalue of $H$", + "$F_{\\omega}$": "coefficient related to function $F(\\boldsymbol{x})$" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 23, + "topic": "Theoretical Foundations", + "question": "For a hydrogen-like ion (nuclear charge $Z e$ ), an electron is in the bound state $\\psi_{n l m}$, calculate $\\langle r^{\\lambda}\\rangle, \\lambda=-3．", + "final_answer": [ + "\\langle\\frac{1}{r^{3}}\\rangle_{n l m}=\\frac{1}{n^{3} l(l+\\frac{1}{2})(l+1)}(\\frac{Z}{a_{0}})^{3}" + ], + "answer_type": "Expression", + "answer": "Known energy levels of a hydrogen-like ion are\n\n\\begin{equation*}\nE_{n l m}=E_{n}=-\\frac{Z^{2} e^{2}}{2 n^{2} a_{0}}, \\quad n=n_{r}+l+1 \\tag{1}\n\\end{equation*}\n\n\nwhere $a_{0}=\\hbar^{2} / \\mu e^{2}$ is the Bohr radius. According to the virial theorem,\n\n\\begin{equation*}\n\\langle\\frac{p^{2}}{2 \\mu}\\rangle_{n l m}=\\langle\\frac{r}{2} \\frac{\\mathrm{~d} V}{\\mathrm{~d} r}\\rangle_{n l m}=\\frac{Z e^{2}}{2}\\langle\\frac{1}{r}\\rangle_{n l m}=-\\frac{1}{2}\\langle V\\rangle_{n l m} \\tag{2}\n\\end{equation*}\n\n\nso\n\nE_{n}=\\frac{1}{2}\\langle V\\rangle_{n l m}=-\\frac{Z e^{2}}{2}\\langle\\frac{1}{r}\\rangle_{n l m}\n\n\n\\begin{equation*}\n\\langle\\frac{1}{r}\\rangle_{n l m}=-\\frac{2 E_{n}}{Z e^{2}}=\\frac{Z}{n^{2} a_{0}} . \\quad n=1,2,3, \\cdots \\tag{3}\n\\end{equation*}\n\n\nFurther, the spherical coordinate representation of $\\psi_{n l m}$ is\n\n\\begin{equation*}\n\\psi_{n l m}(r, \\theta, \\varphi)=R_{n l}(r) Y_{l m}(\\theta, \\varphi) \\tag{4}\n\\end{equation*}\n\n\nIt is a common eigenfunction of $(H, l^{2}, l_{z})$, satisfying the energy eigenvalue equation\n\n\\begin{equation*}\n-\\frac{\\hbar^{2}}{2 \\mu} \\frac{1}{r} \\frac{\\partial^{2}}{\\partial r^{2}} r \\psi_{n l m}+[\\frac{l(l+1) \\hbar^{2}}{2 \\mu r^{2}}-\\frac{Z e^{2}}{r}] \\psi_{n l m}=E_{n} \\psi_{n l m} \\tag{5}\n\\end{equation*}\n\n\nThe total energy operator is equivalent to\n\n\\begin{equation*}\nH \\rightarrow-\\frac{\\hbar^{2}}{2 \\mu} \\frac{1}{r} \\frac{\\partial^{2}}{\\partial r^{2}} r+l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}-\\frac{Z e^{2}}{r} \\tag{6}\n\\end{equation*}\n\n\nConsidering $l$ as a parametric variable, differentiate equation (6) with respect to $l$, using the Hellmann theorem (refer to Chapter 8), we get\n\n\\begin{equation*}\n\\frac{\\partial E_{n}}{\\partial l}=\\langle\\frac{\\partial H}{\\partial l}\\rangle_{n l m}=(l+\\frac{1}{2}) \\frac{\\hbar^{2}}{\\mu}\\langle\\frac{1}{r^{2}}\\rangle_{n l m} \\tag{7}\n\\end{equation*}\n\n\nSince $n=n_{r}+l+1$, we find\n\n\\begin{equation*}\n\\frac{\\partial E_{n}}{\\partial l}=\\frac{\\partial E_{n}}{\\partial n}=\\frac{Z^{2} e^{2}}{n^{3} a_{0}} \\tag{8}\n\\end{equation*}\n\n\nSubstituting into equation (7) and using $a_{0}=\\hbar^{2} / \\mu e^{2}$, we obtain\n\n\\begin{equation*}\n\\langle\\frac{1}{r^{2}}\\rangle_{n l m}=\\frac{1}{(l+\\frac{1}{2}) n^{3}} \\frac{Z^{2}}{a_{0}^{2}} \\tag{9}\n\\end{equation*}\n\n\nFinally, compute $(r^{-3})$．\nFor the s state $(l=0), r \\rightarrow 0$, $\\psi \\rightarrow C$ (constant), therefore\n\n\\begin{equation*}\n\\langle r^{-3}\\rangle_{n 00} \\rightarrow \\infty \\tag{10}\n\\end{equation*}\n\n\nWhen $l \\neq 0$, using formula (7b) in problem (5.7), we get\n\n\\begin{equation*}\n\\langle\\frac{1}{r^{3}}\\rangle_{n l m}=\\frac{Z}{l(l+1) a_{0}}\\langle\\frac{1}{r^{2}}\\rangle_{n l m} \\tag{11}\n\\end{equation*}\n\n\nThus\n\n\\begin{equation*}\n\\langle\\frac{1}{r^{3}}\\rangle_{n l m}=\\frac{1}{n^{3} l(l+\\frac{1}{2})(l+1)}(\\frac{Z}{a_{0}})^{3} \\tag{12}\n\\end{equation*}\n\n\nAs $l \\rightarrow 0$, the right side of the equation $\\rightarrow \\infty$, so this formula is actually applicable for all $l$ values.\nDiscussion: Since both the total energy operator and radial equation are independent of the magnetic quantum number $m$, $\\langle r^{\\lambda}\\rangle$ is independent of $m$. However, $\\langle r^{-1}\\rangle$ is also independent of the angular quantum number $l$, depending only on the principal quantum number $n.$ $\\langle r^{-2}\\rangle$ and $\\langle r^{-3}\\rangle$ depend on both $n$ and $l$, meaning for states with the same energy level but different \"orbital shapes\" (different $l$), $\\langle r^{-2}\\rangle$ or $\\langle r^{-3}\\rangle$ have different values.\n\nUsing formula (9), it can be easily obtained that the average value of the centrifugal potential energy in the state $\\psi_{n l m}$ is\n\n\\begin{equation*}\n\\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{n l m}=\\frac{l(l+1) Z^{2} e^{2}}{(2 l+1) n^{3} a_{0}}=-\\frac{l(l+1)}{(l+\\frac{1}{2}) n} E_{n} \\tag{13}\n\\end{equation*}\n\n\nSince $(-E_{n})$ is the average kinetic energy, the proportion of centrifugal potential energy within kinetic energy is $l(l+1) /(l+\\frac{1}{2}) n$. When $n$ is fixed, as $l$ increases, this proportion grows. When $l$ takes the maximum value $(l=n-1)$, this proportion is $(n-1) /$ $(n-\\frac{1}{2})$, and the radial kinetic energy occupies only $1 /(2 n-1)$ of the kinetic energy in this case. Therefore, if $n \\gg 1,(n, n-1, m)$ implies small radial kinetic energy, corresponding to circular orbits in Bohr's quantum theory.", + "symbol": { + "$Z$": "nuclear charge", + "$e$": "elementary charge", + "$n$": "principal quantum number", + "$l$": "angular quantum number", + "$m$": "magnetic quantum number", + "$E_{n}$": "energy of the state with principal quantum number n", + "$a_{0}$": "Bohr radius", + "$\\mu$": "reduced mass", + "$\\hbar$": "reduced Planck's constant", + "$r$": "radial distance", + "$\\psi_{n l m}$": "wavefunction of the state with quantum numbers n, l, m", + "$R_{n l}$": "radial part of the wavefunction corresponding to n and l", + "$Y_{l m}$": "spherical harmonics corresponding to l and m", + "$H$": "total energy operator", + "$l_{z}$": "z-component of angular momentum" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 24, + "topic": "Theoretical Foundations", + "question": "The potential acting on the valence electron (outermost electron) of a monovalent atom by the atomic nucleus (atomic nucleus and inner electrons) can be approximately expressed as\n\n\\begin{equation*}\nV(r)=-\\frac{e^{2}}{r}-\\lambda \\frac{e^{2} a_{0}}{r^{2}}, \\quad 0<\\lambda \\ll 1 \\tag{1}\n\\end{equation*}\nwhere $a_{0}$ is the Bohr radius. Find the energy level of the valence electron and compare it with the energy level of the hydrogen atom.", + "final_answer": [ + "E_{n l}=-\\frac{e^{2}}{2(n^{\\prime})^{2} a_{0}}" + ], + "answer_type": "Expression", + "answer": "Take the complete set of conserved quantities as $(H, l^{2}, l_{z})$, whose common eigenfunctions are\n\n\\begin{equation*}\n\\psi(r, \\theta, \\varphi)=R(r) \\mathrm{Y}_{l m}(\\theta, \\varphi)=\\frac{u(r)}{r} \\mathrm{Y}_{l m}(\\theta, \\varphi) \\tag{2}\n\\end{equation*}\n\n$u(r)$ satisfies the radial equation\n\n\\begin{equation*}\n-\\frac{\\hbar^{2}}{2 \\mu} u^{\\prime \\prime}+[l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}-\\frac{e^{2}}{r}-\\lambda \\frac{e^{2} a_{0}}{r^{2}}] u=E u \\tag{3}\n\\end{equation*}\n\n\nLet\n\n\\begin{equation*}\nl(l+1)-2 \\lambda=l^{\\prime}(l^{\\prime}+1) \\tag{4}\n\\end{equation*}\n\n\nEquation (3) can then be transformed into\n\n\\begin{equation*}\n-\\frac{\\hbar^{2}}{2 \\mu} u^{\\prime \\prime}+[l^{\\prime}(l^{\\prime}+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}-\\frac{e^{2}}{r}] u=E u \\tag{3'}\n\\end{equation*}\n\n\nThis is equivalent to the radial equation of the hydrogen atom with $l$ replaced by $l^{\\prime}$. Therefore, the solution process for equation ($3^{\\prime}$) is entirely analogous to the hydrogen atom problem. The latter's energy levels are\n\n\\begin{equation*}\nE_{n}=-\\frac{e^{2}}{2 n^{2} a_{0}}, \\quad n=n_{r}+l+1, \\quad n_{r}=0,1,2, \\cdots \\tag{5}\n\\end{equation*}\n\n\nReplacing $l$ with $l^{\\prime}$ gives the energy levels of the valence electron:\n\n\\begin{equation*}\nE_{n l}=-\\frac{e^{2}}{2(n^{\\prime})^{2} a_{0}}, \\quad n^{\\prime}=n_{r}+l^{\\prime}+1 \\tag{6}\n\\end{equation*}\n\n\nIt is generally assumed that\n\n\\begin{equation*}\nl^{\\prime}=l+\\Delta_{l} \\tag{7}\n\\end{equation*}\n\n\n\\begin{equation*}\nn^{\\prime}=n_{r}+l+\\Delta_{l}+1=n+\\Delta_{l} \\tag{8}\n\\end{equation*}\n\n$\\Delta_{l}$ is referred to as the 'correction number' of the quantum numbers $l$ and $n$. Since $\\lambda \\ll 1$, equation (4) can be approximated as follows:\n\n\\begin{aligned}\nl(l+1)-2 \\lambda & =l^{\\prime}(l^{\\prime}+1)=(l+\\Delta_{l})(l+\\Delta_{l}+1) \\\\\n& =l(l+1)+(2 l+1) \\Delta_{l}+(\\Delta_{l})^{2}\n\\end{aligned}\n\n\nNeglecting $(\\Delta_{l})^{2}$, we get\n\n\\begin{equation*}\n\\Delta_{l} \\approx-\\lambda /(l+\\frac{1}{2}) \\tag{9}\n\\end{equation*}\n\n\nSince $\\lambda \\ll 1$, $|\\Delta_{l}| \\ll 1$. Thus, the energy level $E_{n l}$ obtained in this problem has only a small difference from the hydrogen atomic energy level, but the 'degeneracy in $l$' of the energy levels has been removed. Equation (6) is broadly consistent with experimental data on alkali metal spectra; in particular, the correction number $|\\Delta_{l}|$ decreases as $l$ increases, which agrees well with experiments.\n\nThe exact solution for equation (4) is\n\n\\begin{equation*}\nl^{\\prime}=-\\frac{1}{2}+(l+\\frac{1}{2})[1-\\frac{8 \\lambda}{(2 l+1)^{2}}]^{\\frac{1}{2}} \\tag{10}\n\\end{equation*}\n\n\nExpanding the above equation as a binomial series and retaining the $\\lambda$ term while neglecting terms of order $\\lambda^{2}$ and higher, yields equation (9).", + "symbol": { + "$V(r)$": "potential function", + "$e$": "elementary charge", + "$r$": "radial distance", + "$\\lambda$": "dimensionless small parameter", + "$a_{0}$": "Bohr radius", + "$H$": "Hamiltonian operator", + "$l$": "angular momentum quantum number", + "$l_{z}$": "z-component of angular momentum", + "$\\psi$": "wave function", + "$R(r)$": "radial wave function component", + "$u(r)$": "radial function", + "$\\hbar$": "reduced Planck's constant", + "$\\mu$": "reduced mass", + "$E$": "energy", + "$l^{\\prime}$": "modified angular momentum quantum number", + "$E_{n}$": "energy level of hydrogen atom", + "$n$": "principal quantum number", + "$n_{r}$": "radial quantum number", + "$E_{n l}$": "energy level of the valence electron", + "$n^{\\prime}$": "modified principal quantum number", + "$\\Delta_{l}$": "correction number of quantum numbers" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 25, + "topic": "Theoretical Foundations", + "question": "For a particle of mass $\\mu$ moving in a spherical shell $\\delta$ potential well\n\n\\begin{equation*}\nV(r)=-V_{0} \\delta(r-a), \\quad V_{0}>0, a>0 \\tag{1}\n\\end{equation*}\nfind the minimum value of $V_{0}$ required for bound states to exist.", + "final_answer": [ + "V_{0}=\\frac{\\hbar^{2}}{2 \\mu a}" + ], + "answer_type": "Expression", + "answer": "The ground state is an s-state $(l=0)$, and the wave function can be expressed as\n\n\\begin{equation*}\n\\psi(r)=u(r) / r \\tag{2}\n\\end{equation*}\n\n$u(r)$ satisfies the radial equation\n\n\\begin{equation*}\\nu^{\\prime \\prime}+\\frac{2 \\mu E}{\\hbar^{2}} u+\\frac{2 \\mu V_{0}}{\\hbar^{2}} \\delta(r-a) u=0 \\tag{3}\n\\end{equation*}\n\n\nAs $r \\rightarrow \\infty$, $V(r) \\rightarrow 0$, so for a bound state $E<0$. Define\n\n\\begin{equation*}\n\\beta=\\sqrt{-2 \\mu E} / \\hbar \\tag{4}\n\\end{equation*}\n\n\nEquation (3) can be rewritten as\n\n\\begin{equation*}\n\\nu^{\\prime \\prime} - \\beta^{2} u + \\frac{2 \\mu V_{0}}{\\hbar^{2}} \\delta(r-a) u=0 \\tag{3^{\\prime}}\n\\end{equation*}\n\n\nThe boundary conditions are\n\n$$r \\rightarrow 0, \\infty \\text {, } u \\rightarrow 0$$\n\n\nIntegrating equation ($3^{\\prime}$) around $r \\sim a$, we obtain the jump condition for $u^{\\prime}$ (see Problem 2.1)\n\n\\begin{equation*}\n \\nu^{\\prime}(a+0)-u^{\\prime}(a-0)=-\\frac{2 \\mu V_{0}}{\\hbar^{2}} u(a) \\tag{5}\n\\end{equation*}\n\n\nThis means\n\n\\begin{equation*}\n\\frac{u^{\\prime}}{u}|_{r=a-0} ^{r=a+0}=-\\frac{2 \\mu V_{0}}{\\hbar^{2}} \\tag{$\\prime$}\n\\end{equation*}\n\n\nFor $r \\neq a$, equation ($3^{\\prime}$) becomes\n\n\\begin{equation*}\\nu^{\\prime \\prime}-\\beta^{2} u=0 \\tag{\\prime\\prime}\n\\end{equation*}\n\n\nIn the region $r>a$, the solution satisfying the boundary condition at infinity is\n\n\\begin{equation*}\\nu=C \\mathrm{e}^{-\\beta r}, \\quad r>a \\tag{6}\n\\end{equation*}\n\n\nThus\n\n\\begin{equation*}\n(\\frac{u^{\\prime}}{u})_{r=a+0}=-\\beta \\tag{7}\n\\end{equation*}\n\n\nIf the value of $V_{0}$ is just sufficient to form the first bound state, the energy level must be $E=0^{-}$, at which point $\\beta=0$, and equation ($3^{\\prime\\prime}$) becomes\n\n\\begin{equation*}\\nu^{\\prime \\prime}=0 \\quad(E \\rightarrow 0^{-}) \\tag{8}\n\\end{equation*}\n\n\nEquation (7) becomes\n\n\\begin{equation*}\n(\\frac{u^{\\prime}}{u})_{r=a+0}=0 \\quad(E \\rightarrow 0^{-}) \\tag{7'}\n\\end{equation*}\n\nWhen $E \\rightarrow 0^{-}$, the solution of equation (8) in the region $r0$, i.e., $\\langle\\sigma_{z}\\rangle_{t}$.", + "final_answer": [ + "n_{3}^{2}+(1-n_{3}^{2}) \\cos 2 \\omega t" + ], + "answer_type": "Expression", + "answer": "First, determine the spin wave function, then compute $\\langle\\boldsymbol{\\sigma}\\rangle$. Using $\\boldsymbol{n}$ to denote the unit vector in the $(\\theta, \\varphi)$ direction, $\\boldsymbol{e}_{1}, ~ \\boldsymbol{e}_{2}, ~ \\boldsymbol{e}_{3}$ denote the unit vectors in the $x, ~ y, ~ z$ directions, respectively:\n\n\\begin{align*}\n\\boldsymbol{n} & =n_{1} \\boldsymbol{e}_{1}+n_{2} \\boldsymbol{e}_{2}+n_{3} \\boldsymbol{e}_{3} \\\\\n& =\\sin \\theta \\cos \\varphi \\boldsymbol{e}_{1}+\\sin \\theta \\sin \\varphi \\boldsymbol{e}_{2}+\\cos \\theta \\boldsymbol{e}_{3} \\tag{1}\n\\end{align*}\n\nThe Hamiltonian associated with the spin motion is\n\n\\begin{equation*}\nH=-\\boldsymbol{\\mu} \\cdot \\boldsymbol{B}=-\\mu_{0} B \\boldsymbol{\\sigma} \\cdot \\boldsymbol{n}=-\\mu_{0} B_{\\sigma_{n}} \\tag{2}\n\\end{equation*}\n\nIn the $\\sigma_{z}$ representation, the matrix representation of $\\sigma_{n}$ is\n\n\\sigma_{n}=\\sigma_{x} n_{1}+\\sigma_{y} n_{2}+\\sigma_{z} n_{3}=[\\begin{array}{cc}\nn_{3} & n_{1}-\\mathrm{i} n_{2} \\tag{3}\\\\\nn_{1}+\\mathrm{i} n_{2} & -n_{3}\n\\end{array}]\n\nAssume the spin wave function of the particle is\n\n\\chi(t)=[\\begin{array}{l}\na(t) \\tag{4}\\\\\nb(t)\n\\end{array}]\n\n$\\chi(t)$ satisfies the Schrödinger equation\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} t} \\chi(t)=H \\chi(t)=-\\mu_{0} B \\sigma_{n} \\chi(t) \\tag{5}\n\\end{equation*}\n\nThe eigenvalues of $\\sigma_{n}$ are $\\pm 1$, the eigenvalues of $H$ (i.e., stationary energy levels) are\n\n\\begin{gather*}\nE=\\mp \\mu_{0} B=\\mp \\hbar \\omega \\\\\n\\omega=\\mu_{0} B / \\hbar \\tag{6}\n\\end{gather*}\n\n\nSince $\\omega$ and $\\mu_{0}$ have the same sign, if $\\mu_{0}<0$ then $\\omega<0$. The eigenfunctions of $\\sigma_{n}$ and $H$ are\n\n\\begin{gather*}\n\\phi_{1}=\\frac{1}{\\sqrt{2(1+n_{3})}}[\\begin{array}{c}\n1+n_{3} \\\\\nn_{1}+\\mathrm{i} n_{2}\n\\end{array}] \\quad(\\sigma_{n}=1, E=-\\hbar \\omega) \\\\\n\\phi_{-1}=\\frac{1}{\\sqrt{2(1+n_{3})}}[\\begin{array}{c}\nn_{1}-\\mathrm{i} n_{2} \\\\\n-1-n_{3}\n\\end{array}] \\quad(\\sigma_{n}=-1, E=\\hbar \\omega) \\tag{7}\n\\end{gather*}\n\nThe general solution of equation (5) is\n\n\\begin{equation*}\n\\chi(t)=C_{1} \\phi_{1} \\mathrm{e}^{\\mathrm{i} \\omega t}+C_{-1} \\phi_{-1} \\mathrm{e}^{-\\mathrm{i} \\omega t} \\tag{8}\n\\end{equation*}\n\n$C_{1}, ~ C_{-1}$ are determined by initial conditions:\n\n\\begin{equation*}\nC_{1}=\\phi_{1}^{+} \\chi(0), \\quad C_{-1}=\\phi_{-1}^{+} \\chi(0) \\tag{9}\n\\end{equation*}\n\n\nThe initial wave function for this problem is the eigenfunction of $\\sigma_{z}=1$, that is\n\n\\chi(0)=\\chi_{\\frac{1}{2}}=[\\begin{array}{l}\n1 \\tag{10}\\\\\n0\n\\end{array}]\n\n\nThus\n\n\\begin{equation*}\nC_{1}=\\sqrt{\\frac{1+n_{3}}{2}}, \\quad C_{-1}=\\frac{n_{1}+\\mathrm{i} n_{2}}{\\sqrt{2(1+n_{3})}} \\tag{11}\n\\end{equation*}\n\n\nSubstituting into equation (8), we get\n\n\\begin{align*}\n\\chi(t) & =\\frac{1}{\\sqrt{2(1+n_{3})}}[\\begin{array}{l}\n(1+n_{3}) C_{1} \\mathrm{e}^{\\mathrm{i} \\omega t}+(n_{1}-\\mathrm{i} n_{2}) C_{-1} \\mathrm{e}^{-\\mathrm{i} \\omega t} \\\\\n(n_{1}+\\mathrm{i} n_{2}) C_{1} \\mathrm{e}^{\\mathrm{i} \\omega t}-(1+n_{3}) C_{-1} \\mathrm{e}^{-\\mathrm{i} \\omega t}\n\\end{array}] \\\\\n& =[\\begin{array}{l}\n\\cos \\omega t+\\mathrm{i} n_{3} \\sin \\omega t \\\\\n(\\mathrm{i} n_{1}-n_{2}) \\sin \\omega t\n\\end{array}]=[\\begin{array}{l}\na(t) \\\\\nb(t)\n\\end{array}] \\tag{12}\n\\end{align*}\n\n\nThe expectation values of $\\boldsymbol{\\sigma}$ in the $\\chi(t)$ state are\n\n\\begin{align*}\n\\langle\\sigma_{x}\\rangle & =\\chi^{+} \\sigma_{x} \\chi=[a^{*} b^{*}][\\begin{array}{ll}\n0 & 1 \\\\\n1 & 0\n\\end{array}][\\begin{array}{l}\na \\\\\nb\n\\end{array}]=a^{*} b+b^{*} a \\\\\n& =n_{1} n_{3}(1-\\cos 2 \\omega t)-n_{2} \\sin 2 \\omega t \\tag{13a}\\\\\n\\langle\\sigma_{y}\\rangle & =\\chi^{+} \\sigma_{y} \\chi=\\mathrm{i}(b^{*} a-a^{*} b) \\\\\n& =n_{2} n_{3}(1-\\cos 2 \\omega t)+n_{1} \\sin 2 \\omega t \\tag{1;b}\\\\\n\\langle\\sigma_{z}\\rangle & =\\chi^{-} \\sigma_{*} \\chi=a^{*} a-b^{*} b \\\\\n& =n_{3}^{2}+(1-n_{3}^{2}) \\cos 2 \\omega t \\tag{13c}\n\\end{align*}\n\n\nIf the magnetic field $\\boldsymbol{B}$ points in the positive $x$ direction, then\n\nn_{1}=1, \\quad n_{2}=n_{3}=0\n\n\nThen equation (13) becomes\n\n\\langle\\sigma_{x}\\rangle=0,\\langle\\sigma_{y}\\rangle=\\sin 2 \\omega t, \\quad\\langle\\sigma_{z}\\rangle=\\cos 2 \\omega t\n\n\nThis is precisely the result obtained in the previous problem (note that $\\omega$ in the previous problem corresponds to $-\\omega$ in this problem).", + "symbol": { + "$\\hbar$": "reduced Planck's constant", + "$\\boldsymbol{\\mu}$": "magnetic moment", + "$\\mu_{0}$": "proportionality constant for magnetic moment", + "$\\boldsymbol{\\sigma}$": "Pauli spin matrices", + "$\\boldsymbol{B}$": "magnetic field", + "$\\theta$": "polar angle defining direction", + "$\\varphi$": "azimuthal angle defining direction", + "$n_3$": "projection of direction vector on the z-axis", + "$t$": "time", + "$\\sigma_{z}$": "Pauli spin matrix in z direction", + "$\\langle\\boldsymbol{\\sigma}\\rangle$": "expectation value of Pauli matrices", + "$\\langle\\sigma_{z}\\rangle_{t}$": "expectation value of z-component of spin at time t", + "$\\boldsymbol{n}$": "unit vector in specified direction", + "$\\boldsymbol{e}_{1}$": "unit vector in x direction", + "$\\boldsymbol{e}_{2}$": "unit vector in y direction", + "$\\boldsymbol{e}_{3}$": "unit vector in z direction", + "$n_{1}$": "x-component of unit vector", + "$n_{2}$": "y-component of unit vector", + "$H$": "Hamiltonian", + "$\\sigma_{n}$": "component of Pauli spin matrix in direction n", + "$\\chi(t)$": "spin wave function", + "$E$": "energy", + "$\\omega$": "angular frequency", + "$C_{1}$": "coefficient for initial condition related to positive eigenvalue", + "$C_{-1}$": "coefficient for initial condition related to negative eigenvalue", + "$\\chi(0)$": "initial wave function", + "$\\phi_{1}$": "eigenfunction corresponding to eigenvalue 1", + "$\\phi_{-1}$": "eigenfunction corresponding to eigenvalue -1", + "$a(t)$": "component of wave function", + "$b(t)$": "component of wave function", + "$\\langle\\sigma_{x}\\rangle$": "expectation value of x-component of spin", + "$\\langle\\sigma_{y}\\rangle$": "expectation value of y-component of spin" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 28, + "topic": "Theoretical Foundations", + "question": "In a system with a spin of $\\hbar / 2$, the magnetic moment $\\boldsymbol{\\mu}=\\mu_{0} \\boldsymbol{\\sigma}$ is placed in a uniform magnetic field $\\boldsymbol{B}_{0}$ directed along the positive $z$ direction for $t<0$. At $t \\geqslant 0$, an additional rotating magnetic field $\\boldsymbol{B}_{1}(t)$, perpendicular to the $z$ axis, is applied: \n\n$$ \\boldsymbol{B}_{1}(t)=B_{1} \\cos 2 \\omega_{0} t e_{1}-B_{1} \\sin 2 \\omega_{0} t e_{2},$$\n\nwhere $\\omega_{0}=\\mu_{0} B_{0} / \\hbar$. It is known that for $t \\leqslant 0$, the system is in the eigenstate $\\chi_{\\frac{1}{2}}$ with $s_{z}=\\hbar / 2$. Find the expression for the time $\\Delta t$ it takes for the system's spin to first reverse from $s_z = \\hbar/2$ (along the positive $z$ direction) to $s_z = -\\hbar/2$ (along the negative $z$ direction) starting from $t=0$. Express this in terms of $\\omega_1 = \\mu_0 B_1 / \\hbar$ and relevant constants.", + "final_answer": [ + "\\pi \\hbar / 2 \\mu_{0} B_{1}" + ], + "answer_type": "Expression", + "answer": "The Hamiltonian related to the spin motion of the system is\n\n\\begin{equation*}\nH=-\\mu \\cdot[\\boldsymbol{B}_{0}+\\boldsymbol{B}_{1}(t)], \\quad t \\geqslant 0 \\tag{1}\n\\end{equation*}\n\n\nIn the $s_{z}$ representation, the matrix form of $H$ is\n\n\\begin{align}\nH & =-\\mu_{0} B_{1}(\\sigma_{x} \\cos 2 \\omega_{0} t-\\sigma_{y} \\sin 2 \\omega_{0} t)-\\mu_{0} B_{0} \\sigma_{z} \\\\\n& =-\\mu_{0} \\begin{pmatrix}\nB_{0} & B_{1} \\mathrm{e}^{2 i \\omega_{0} t} \\\\\nB_{1} \\mathrm{e}^{-2 i \\omega_{0} t} & -B_{0}\n\\end{pmatrix}\n\\tag{\\prime}\n\\end{align}\n\n\nAssuming the wave function for $t \\geqslant 0$ is\n\n$$\\chi(t)=\\begin{pmatrix}\na(t) \\tag{2}\\\\\nb(t)\n\\end{pmatrix}.$$\n\nSubstituting into the Schrödinger equation\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} t} \\chi(t)=H \\chi(t) \\tag{3},\n\\end{equation*}\nwe have \n\n\\begin{gather*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t} a(t)=\\mathrm{i} \\omega_{0} a(t)+\\mathrm{i} \\omega_{1} b(t) \\mathrm{e}^{2 \\mathrm{i} \\omega_{0} t} \\\\\n\\frac{\\mathrm{~d}}{\\mathrm{~d} t} b(t)=-\\mathrm{i} \\omega_{0} b(t)+\\mathrm{i} \\omega_{1} a(t) \\mathrm{e}^{-2 \\mathrm{i} \\omega_{0} t}. \\tag{4}\n\\end{gather*}\n\n\nwhere\n\n\\begin{equation*}\n\\omega_{0}=\\frac{\\mu_{0} B_{0}}{\\hbar}, \\quad \\omega_{1}=\\frac{\\mu_{0} B_{1}}{\\hbar} \\tag{5}\n\\end{equation*}\n\n\nLet\n\n\\begin{equation*}\na(t)=c_{1}(t) \\mathrm{e}^{\\mathrm{i} \\omega_{0} t}, \\quad b(t)=c_{2}(t) \\mathrm{e}^{-\\mathrm{i} \\omega_{0} t} \\tag{6}\n\\end{equation*}\n\n\nSubstituting into equation (4), yields equations for $c_{1}$ and $c_{2}$\n\n\\begin{align*}\n& \\frac{\\mathrm{d}}{\\mathrm{~d} t} c_{1}(t)=\\mathrm{i} \\omega_{1} c_{2}(t) \\\\\n& \\frac{\\mathrm{d}}{\\mathrm{~d} t} c_{2}(t)=\\mathrm{i} \\omega_{1} c_{1}(t) \\tag{7}\n\\end{align*}\n\n\nThe initial conditions are\n\n\\begin{equation*}\n\\chi(0)=\\begin{pmatrix}{l}\na(0) \\\\\nb(0)\n\\end{pmatrix}=\\begin{pmatrix}\n1 \\\\\n0\n\\end{pmatrix}=\\chi_{\\frac{1}{2}}, \\quad \\text { i.e., } \\quad \\left\\{\\begin{array}{l}\nc_{1}(0)=1 \\\\\nc_{2}(0)=0\n\\end{array}\\right.\n\\tag{8}\n\\end{equation*}\n\nBy adding and subtracting equations in (7), we obtain\n\n\\begin{gather*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(c_{1}+c_{2})=\\mathrm{i} \\omega_{1}(c_{1}+c_{2}) \\\\\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(c_{1}-c_{2})=-\\mathrm{i} \\omega_{1}(c_{1}-c_{2}) \\tag{9}\n\\end{gather*}\n\n\nThe solution is\n\n\\begin{align*}\nc_{1}(t)+c_{2}(t) & =[c_{1}(0)+c_{2}(0)] \\mathrm{e}^{\\mathrm{i} \\omega_{1} t} = \\mathrm{e}^{\\mathrm{i} \\omega_{1} t}\\\\\nc_{1}(t)-c_{2}(t) &= (c_{1}(0)-c_{2}(0)) \\mathrm{e}^{-\\mathrm{i} \\omega_{1} t}=\\mathrm{e}^{-\\mathrm{i} \\omega_{1} t} .\n\\end{align*}\n\n\nAdding and subtracting these, we find\n\n\\begin{equation*}\nc_{1}(t)=\\cos \\omega_{1} t, \\quad c_{2}(t)=\\mathrm{i} \\sin \\omega_{1} t \\tag{11}\n\\end{equation*}\n\n\nSubstituting into equation (6), we obtain\n\n\\begin{equation*}\na(t)=\\cos \\omega_{1} t \\mathrm{e}^{\\mathrm{i} \\omega_{0} t}, \\quad b(t)=\\mathrm{i} \\sin \\omega_{1} t \\mathrm{e}^{-\\mathrm{i} \\omega_{0} t} \\tag{12}\n\\end{equation*}\n\n\nSubstituting into equation (2), we find\n\\begin{equation}\n\\begin{split}\n\\chi(t)&=\\begin{pmatrix}\n\\cos \\omega_{1} t \\mathrm{e}^{\\mathrm{i} \\omega_{0} t} \\\\\n\\mathrm{i} \\sin \\omega_{1} t \\mathrm{e}^{-\\mathrm{i} \\omega_{0} t}\n\\end{pmatrix} \\\\\n&=\\cos \\omega_{1} t \\mathrm{e}^{\\mathrm{i} \\omega_{0} t} \\chi_{\\frac{1}{2}}+\\mathrm{i} \\sin \\omega_{1} t \\mathrm{e}^{-\\mathrm{i} \\omega_{0} t} \\chi_{-\\frac{1}{2}} \\tag{13}\n\\end{split}\n\\end{equation}\n\nClearly\n\\begin{gathered}\nt=0, \\quad \\chi=\\chi_{\\frac{1}{2}}=[\\begin{array}{l}\n1 \\\\\n0\n\\end{array}] \\\\\ns_{z}=\\frac{\\hbar}{2}, \\quad\\langle s\\rangle=\\frac{\\hbar}{2} e_{3} \\\\\nt=\\frac{\\pi}{2 \\omega_{1}}, \\quad \\chi=\\mathrm{ie}^{-\\mathrm{i} \\omega_{0} t} \\chi_{-\\frac{1}{2}}=\\mathrm{ie}^{-\\mathrm{i} \\omega_{0} t}[\\begin{array}{l}\n0 \\\\\n1\n\\end{array}] \\\\\ns_{\\tilde{z}}=-\\frac{\\hbar}{2}, \\quad\\langle\\boldsymbol{s}\\rangle=-\\frac{\\hbar}{2} e_{3} \\\\\nt=\\frac{\\pi}{\\omega_{1}}, \\quad \\chi=-\\mathrm{e}^{\\mathrm{i} \\omega_{0} t} \\chi_{\\frac{1}{2}}=-\\mathrm{e}^{\\mathrm{i} \\omega_{0} t}[\\begin{array}{l}\n1 \\\\\n0\n\\end{array}] \\\\\ns_{z}=\\frac{\\hbar}{2}, \\quad\\langle\\boldsymbol{s}\\rangle=\\frac{\\hbar}{2} e_{3}\n\\end{gathered}\n\nIn other words, the system's spin direction changes once every $\\Delta t=\\pi / 2 \\omega_{1}=\\pi \\hbar / 2 \\mu_{0} B_{1}$ . The spin state of the system undergoes periodic oscillation between $\\chi_{\\frac{1}{2}}$ and $\\chi_{-\\frac{1}{2}}$, with a period $T=2 \\Delta t=\\pi \\hbar / \\mu_{0} B_{1}$. \n\nThis problem illustrates the basic principle of magnetic resonance.", + "symbol": { + "$\\hbar$": "reduced Planck's constant", + "$\\mu_{0}$": "magnetic permeability constant", + "$\\boldsymbol{\\mu}$": "magnetic moment vector", + "$\\boldsymbol{\\sigma}$": "Pauli matrices vector", + "$\\boldsymbol{B}_{0}$": "uniform magnetic field in positive z direction", + "$t$": "time", + "$\\boldsymbol{B}_{1}(t)$": "rotating magnetic field", + "$B_{1}$": "magnitude of the rotating magnetic field", + "$\\omega_{0}$": "precession angular frequency due to $B_{0}$", + "$\\chi_{\\frac{1}{2}}$": "eigenstate with spin $s_{z} = \\hbar/2$", + "$s_{z}$": "spin component along z axis", + "$\\Delta t$": "time for the spin to reverse", + "$\\omega_{1}$": "precession angular frequency due to $B_{1}$", + "$c_{1}(t)$": "time-dependent coefficient in the wave function", + "$c_{2}(t)$": "time-dependent coefficient in the wave function" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 29, + "topic": "Theoretical Foundations", + "question": "The magnetic moment (operator) of an electron is\n\n\\begin{equation*}\n\\boldsymbol{\\mu}=\\boldsymbol{\\mu}_{l}+\\boldsymbol{\\mu}_{s}=-\\frac{e}{2 m_{\\mathrm{e}} c}(\\boldsymbol{l}+2 \\boldsymbol{s}) \\tag{1}\n\\end{equation*}\n\n\nTry to calculate the expectation value of $\\mu_{z}$ for the $|l j m_{j}\\rangle$ state.", + "final_answer": [ + "\\langle l j m_{j}| \\mu_{z}|l j m_{j}\\rangle=-(1+\\frac{j(j+1)-l(l+1)+3 / 4}{2 j(j+1)}) m_{j}" + ], + "answer_type": "Expression", + "answer": "If we use the Bohr magneton $\\mu_{\\mathrm{B}}=e \\hbar / 2 m_{\\mathrm{e}} c$ as the unit of magnetic moment, then the magnetic moment operator of an electron can be written as (here $\\hbar=1$ is taken)\n\n\\begin{equation*}\n\\boldsymbol{\\mu}=-(\\boldsymbol{l}+2 \\boldsymbol{s})=-(\\boldsymbol{j}+\\boldsymbol{s})=-(\\boldsymbol{j}+\\frac{1}{2} \\boldsymbol{\\sigma}) \\tag{2}\n\\end{equation*}\n\n\nThus, we have\n\n\\begin{equation*}\n\\langle l j m_{j}| \\mu_{z}|l j m_{j}\\rangle=-g m_{j} \\tag{3}\n\\end{equation*}\n\n\nwhere\n\n\\begin{equation*}\ng=1+\\frac{j(j+1)-l(l+1)+3 / 4}{2 j(j+1)} \\quad \\text { (Landè } g \\text { factor) } \\tag{4}\n\\end{equation*}\n\n\nThe average value of $\\mu_{z}$ for $m_{j}=j$ (its maximum value) is usually taken as the definition of the magnetic moment observable, denoted as $\\mu$. For an electron,\n\n\\begin{equation*}\n\\mu=\\langle l j j| \\mu_{z}|l j j\\rangle=-g j \\tag{5}\n\\end{equation*}\n\n\nthat is\n\n\\mu= \\begin{cases}-(j+\\frac{1}{2}), & j=l+\\frac{1}{2} \\tag{$\\prime$}\\\\ -j(2 j+1) /(2 j+2), & j=l-\\frac{1}{2}\\end{cases}", + "symbol": { + "$e$": "electron charge", + "$m_{\\mathrm{e}}$": "electron mass", + "$c$": "speed of light in vacuum", + "$l$": "orbital angular momentum", + "$s$": "spin angular momentum", + "$\\mu_{z}$": "z-component of the magnetic moment", + "$\\mu_{\\mathrm{B}}$": "Bohr magneton", + "$\\hbar$": "reduced Planck's constant", + "$j$": "total angular momentum quantum number", + "$m_{j}$": "magnetic quantum number related to total angular momentum", + "$\\sigma$": "Pauli matrices", + "$g$": "Landè g factor", + "$\\mu$": "magnetic moment" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 30, + "topic": "Theoretical Foundations", + "question": "A system composed of two spin-$1 / 2$ particles is placed in a uniform magnetic field, with the magnetic field direction as the $z$-axis. The Hamiltonian of the system related to spin is given by\n\n\\begin{equation*}\nH=a \\sigma_{1 z}+b \\sigma_{2 z}+c_{0} \\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2} \\tag{1}\n\\end{equation*}\n\n\nwhere $a, ~ b$ terms arise from the interaction between the magnetic field and the particles' intrinsic magnetic moments, and the $c_{0}$ term arises from the interaction between the two particles. $a, ~ b, ~ c_{0}$ are real constants. If the system is in the common eigenstate $\\chi_{11}=\\alpha(1) \\alpha(2)$ of the total spin operators $(\\boldsymbol{S}^{2}, S_z)$, find the energy level of the system in this case.", + "final_answer": [ + "c_{0} + a + b" + ], + "answer_type": "Expression", + "answer": "We will solve using matrix methods in spin state vector space. The basis vectors can be chosen as the common eigenstates of $(\\sigma_{1 z}, \\sigma_{2 z})$\n\n$$\\alpha(1) \\alpha(2), \\quad \\alpha(1) \\beta(2), \\quad \\beta(1) \\alpha(2), \\quad \\beta(1) \\beta(2) $$\n\n\nor as the common eigenstates $\\chi_{S M_{s}}$ of the total spin operators $(\\boldsymbol{S}^{2}, S_{z})$. For this problem, considering the diagonalization of $\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}$, it is more convenient to use $\\chi_{S M_{S}}$ as the basis vectors. For convenience, the basis vector order is as follows:\n\n\\begin{array}{l}\n\\chi_{1}=\\chi_{11}=\\alpha(1) \\alpha(2) \\tag{2}\\\\\n\\chi_{2}=\\chi_{1-1}=\\beta(1) \\beta(2) \\\\\n\\chi_{3}=\\chi_{10}=\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2)+\\beta(1) \\alpha(2)] \\\\\n\\chi_{4}=\\chi_{00}=\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2)-\\beta(1) \\alpha(2)]\n\\end{array}}\n\n\nThe Hamiltonian operator can be rewritten as\n\n\\begin{gather*}\nH=c_{0} \\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}+c_{1}(\\sigma_{1 z}+\\sigma_{2 z})+c_{2}(\\sigma_{1 z}-\\sigma_{2 z}) \\tag{1'}\\\\\nc_{1}=\\frac{1}{2}(a+b), \\quad c_{2}=\\frac{1}{2}(a-b) \\tag{3}\n\\end{gather*}\n\n\nAll four basis vectors are common eigenstates of $(\\boldsymbol{S}^{2}, S_{z})$, and also common eigenstates of $\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}$ and $(\\sigma_{1 z}+\\sigma_{2 z})$. It is easy to see that $\\chi_{1}$ and $\\chi_{2}$ are also eigenstates of $(\\sigma_{1 z}-\\sigma_{2 z})$, thus $\\chi_{1}$ and $\\chi_{2}$ are already eigenstates of $H$.\n\n\\begin{align*}\n& H \\chi_{1}=(c_{0}+2 c_{1}) \\chi_{1}=(c_{0}+a+b) \\chi_{1} \\tag{4}\\\\\n& H \\chi_{2}=(c_{0}-2 c_{1}) \\chi_{2}=(c_{0}-a-b) \\chi_{2} \\tag{5}\n\\end{align*}\n\n\nThus, we obtain two energy levels of the system: $E=c_{0} \\pm 2 c_{1}$.\nIt is easy to calculate the action of $(\\sigma_{1 z}-\\sigma_{2 z})$ on the basis vectors, which is\n\n\\begin{array}{cl}\n(\\sigma_{1 z}-\\sigma_{2 z}) \\chi_{1}=0, & (\\sigma_{1 z}-\\sigma_{2 z}) \\chi_{2}=0 \\\\\n(\\sigma_{1 z}-\\sigma_{2 z}) \\chi_{3}=2 \\chi_{4}, & (\\sigma_{1 z}-\\sigma_{2 z}) \\chi_{4}=2 \\chi_{3} \\tag{7}\n\\end{array}\n\n\nTherefore, in the subspace ${\\chi_{3}, \\chi_{4}}$, the matrix elements of $(\\sigma_{1 z}-\\sigma_{2 z})$ are\n\n\\begin{array}{l}\n(\\sigma_{1 z}-\\sigma_{2 z})_{33}=(\\sigma_{1 z}-\\sigma_{2 z})_{44}=0 \\tag{7'}\\\\\n(\\sigma_{1 z}-\\sigma_{2 z})_{34}=(\\sigma_{1 z}-\\sigma_{2 z})_{43}=2\n\\end{array}\n\nAll matrix elements of $(\\sigma_{1 z}+\\sigma_{2 z})$ are zero. The matrix representation of $H$ is\n\\begin{equation}\n H=\\begin{pmatrix}\nc_{0} & 2 c_{2} \\tag{8}\\\\\n2 c_{2} & -3 c_{0}\n\\end{pmatrix}\n\\end{equation}\n\nAssume the energy eigenstate is\n\n\\begin{equation*}\n\\chi=f_{3} \\chi_{3}+f_{4} \\chi_{4} \\tag{9}\n\\end{equation*}\n\n\nSubstitute into the energy eigenvalue equation\n\n\\begin{equation*}\nH \\chi=E \\chi \\tag{10}\n\\end{equation*}\n\n\nresulting in\n\n[\\begin{array}{cc}\nc_{0}-E & 2 c_{2} \\tag{$\\prime$}\\\\\n2 c_{2} & -3 c_{0}-E\n\\end{array}][\\begin{array}{l}\nf_{1} \\\\\nf_{2}\n\\end{array}]=0\n\n\nThe energy level $E$ is determined by:\n\n\\begin{equation*}\n\\operatorname{det}(H-E)=0 \\tag{11}\n\\end{equation*}\n\n\nnamely\n\n|\\begin{array}{cc}\nc_{0}-E & 2 c_{2} \\tag{11'}\\\\\n2 c_{2} & -3 c_{0}-E\n\\end{array}|=(E-c_{0})(E+3 c_{0})-4 c_{2}^{2}=0\n\n\nSolving gives\n\nE=-c_{0} \\pm 2 \\sqrt{c_{0}^{2}+c_{2}^{2}}\n\n\nConclusion: This problem has four energy levels (excluding accidental degeneracy), which are\n\n\\begin{equation*}\nE=c_{0} \\pm 2 c_{1}, \\quad-c_{0} \\pm 2 \\sqrt{c_{0}^{2}+c_{2}^{2}} \\tag{12}\n\\end{equation*}\n\n\nThe energy eigenstates of the first two energy levels are $\\chi_{1}$ and $\\chi_{2}$, respectively, and the energy eigenstates of the last two energy levels are linear combinations of $\\chi_{3}$ and $\\chi_{4}$.", + "symbol": { + "$H$": "Hamiltonian of the system", + "$a$": "real constant related to interaction with magnetic field", + "$b$": "real constant related to interaction with magnetic field", + "$c_{0}$": "real constant related to interaction between particles", + "$\\sigma_{1 z}$": "spin operator for the first particle along the z-axis", + "$\\sigma_{2 z}$": "spin operator for the second particle along the z-axis", + "$\\boldsymbol{\\sigma}_{1}$": "spin operator vector for the first particle", + "$\\boldsymbol{\\sigma}_{2}$": "spin operator vector for the second particle", + "$\\chi_{11}$": "eigenstate of total spin operators for maximum projection", + "$\\chi_{1-1}$": "eigenstate of total spin operators for minimum projection", + "$\\chi_{10}$": "eigenstate of total spin operators with intermediate projection", + "$\\chi_{00}$": "eigenstate of total spin operators for zero projection", + "$c_{1}$": "transformed constant as a function of a and b", + "$c_{2}$": "transformed constant as a function of a and b" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 31, + "topic": "Theoretical Foundations", + "question": "Consider a system composed of three non-identical spin $1/2$ particles, with the Hamiltonian given by\n\n\\begin{equation*}\nH=A \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}+B(\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}) \\cdot \\boldsymbol{s}_{3}, \\tag{1}\n\\end{equation*}\nwhere $A, ~ B$ are real constants. Let $\\boldsymbol{S}_{12}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}$ and $\\boldsymbol{S}_{123}=\\boldsymbol{S}_{12}+\\boldsymbol{s}_{3}$. Try to express the Hamiltonian $H$ as a function of $\\boldsymbol{S}_{12}^{2}$ and $\\boldsymbol{S}_{123}^{2}$. (Take $\\hbar=1$ )", + "final_answer": [ + "H = \\frac{1}{2}(A-B) \\boldsymbol{S}_{12}^{2}+\\frac{B}{2} \\boldsymbol{S}_{123}^{2}-\\frac{3}{8}(2 A+B)" + ], + "answer_type": "Expression", + "answer": "The sum of the spins of particles 1 and 2 is denoted as $\\boldsymbol{S}_{12}$, and the total spin is denoted as $\\boldsymbol{S}_{123}$, that is\n\\begin{equation*}\n\\boldsymbol{S}_{12}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}, \\quad \\boldsymbol{S}_{123}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}+\\boldsymbol{s}_{3}=\\boldsymbol{S}_{12}+\\boldsymbol{s}_{3} \\tag{2}\n\\end{equation*}\n\nEvidently, $\\boldsymbol{S}_{12}$ and $\\boldsymbol{S}_{123}$ both possess the properties of angular momentum, satisfying the commutation relation\n\n\\begin{array}{ll}\n\\boldsymbol{S}_{12} \\times \\boldsymbol{S}_{12}=\\mathrm{i} \\boldsymbol{S}_{12}, & {[\\boldsymbol{S}_{12}^{2}, \\boldsymbol{S}_{12}]=0} \\\\\n\\boldsymbol{S}_{123} \\times \\boldsymbol{S}_{123}=\\mathrm{i} \\boldsymbol{S}_{123}, & {[\\boldsymbol{S}_{123}^{2}, \\boldsymbol{S}_{123}]=0} \\tag{4}\n\\end{array}\n\n$\\boldsymbol{s}_{1}, ~ \\boldsymbol{s}_{2}, ~ \\boldsymbol{s}_{3}$ commute with each other, and\n\n\\begin{equation*}\ns_{1}^{2}=s_{2}^{2}=s_{3}^{2}=\\frac{3}{4} \\tag{5}\n\\end{equation*}\n\n\nTherefore\n\n\\begin{align*}\n& \\boldsymbol{S}_{12}^{2}=\\boldsymbol{s}_{1}^{2}+\\boldsymbol{s}_{2}^{2}+2 \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}=\\frac{3}{2}+2 \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2} \\tag{6}\\\\\n& \\boldsymbol{S}_{123}^{2}=\\frac{9}{4}+2(\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}+\\boldsymbol{s}_{2} \\cdot \\boldsymbol{s}_{3}+\\boldsymbol{s}_{3} \\cdot \\boldsymbol{s}_{1}) \\tag{7}\n\\end{align*}\n\n\nBased on this, $H$ can be written as\n\n\\begin{align*}\nH & =(A-B) \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}+B(\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}+\\boldsymbol{s}_{2} \\cdot \\boldsymbol{s}_{3}+\\boldsymbol{s}_{3} \\cdot \\boldsymbol{s}_{1}) \\\\\n& =\\frac{1}{2}(A-B) \\boldsymbol{S}_{12}^{2}+\\frac{B}{2} \\boldsymbol{S}_{123}^{2}-\\frac{3}{8}(2 A+B) \\tag{$\\prime$}\n\\end{align*}", + "symbol": { + "$H$": "Hamiltonian", + "$A$": "real constant related to interaction", + "$B$": "real constant related to interaction", + "$\\boldsymbol{s}_{1}$": "spin vector of particle 1", + "$\\boldsymbol{s}_{2}$": "spin vector of particle 2", + "$\\boldsymbol{s}_{3}$": "spin vector of particle 3", + "$\\boldsymbol{S}_{12}$": "combined spin vector of particles 1 and 2", + "$\\boldsymbol{S}_{123}$": "combined spin vector of particles 1, 2, and 3" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 32, + "topic": "Theoretical Foundations", + "question": "Same as the previous question, for any value, find\n\n\\begin{equation*}\nd_{j m}^{j}(\\lambda)=\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}|j m\\rangle \\tag{1}\n\\end{equation*}", + "final_answer": [ + "d_{j m}^{j}(\\lambda)=(-1)^{j-m}[\\frac{(2 j)!}{(j+m)!(j-m)!}]^{\\frac{1}{2}}(\\cos \\frac{\\lambda}{2})^{j+m}(\\sin \\frac{\\lambda}{2})^{j-m}" + ], + "answer_type": "Expression", + "answer": "According to the general theory of angular momentum,\n\n\\begin{array}{rl}\nJ_{+}|j m\\rangle & =(J_{x}+i J_{y})|j m\\rangle \\tag{2}\\\\\nJ_{-}|j m\\rangle & =a_{j m}|j m+1\\rangle \\\\\nJ_{x}-i J_{y})|j m\\rangle & =a_{j,-m}|j m-1\\rangle\n\\end{array}}\n\n\nwhere\n\n\\begin{equation*}\na_{j m}=\\sqrt{(j-m)(j+m+1)} \\tag{3}\n\\end{equation*}\n\n\nWhen $m=j$\n\n\\begin{equation*}\nJ_{+}|j j\\rangle=0, \\quad\\langle j j| J_{-}=0 \\tag{4}\n\\end{equation*}\n\n(The second equation is the conjugate of the first equation.) Therefore\n\n\\begin{equation*}\n\\langle j j| J_{-} \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}|j m\\rangle=0. \\tag{5}\n\\end{equation*}\n\nWe also have:\n\\begin{align*}\nJ_{-} \\mathrm{e}^{-\\mathrm{i} \\mathrm{i} J_{y}} & =\\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}(J_{z} \\sin \\lambda+J_{x} \\cos \\lambda-\\mathrm{i} J_{y}) \\\\\n& =\\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}[J_{z} \\sin \\lambda+\\frac{1}{2}(\\cos \\lambda-1) J_{+}+\\frac{1}{2}(\\cos \\lambda+1) J_{-}] \\tag{6}\n\\end{align*}\n\nSubstitute into equation (5), and use equation (2) to obtain\n\n\\begin{gather*}\nm \\sin \\lambda d_{j m}^{j}(\\lambda)+\\frac{1}{2}(\\cos \\lambda-1) a_{j m} d_{j m+1}^{j}(\\lambda) \\\\\n\\quad+\\frac{1}{2}(\\cos \\lambda+1) a_{j,-m} d_{j m-1}^{j}(\\lambda)=0 \\tag{7}\n\\end{gather*}\n\nAdditionally, we can also obtain\n\n\\begin{equation*}\nJ_{z} \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}=\\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}(J_{z} \\cos \\lambda-J_{x} \\sin \\lambda)=\\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}[J_{z} \\cos \\lambda-\\frac{1}{2} \\sin \\lambda(J_{+}+J_{-})] \\tag{8}\n\\end{equation*}\n\n\nMultiply the above equation from the left with $\\langle j|$ and from the right with $|j m\\rangle$, yielding\n\n\\begin{equation*}\n(j-m \\cos \\lambda) d_{j m}^{j}(\\lambda)+\\frac{1}{2} \\sin \\lambda[a_{j m} d_{j m+1}^{j}(\\lambda)+a_{j,-m} d_{j m-1}^{j}(\\lambda)]=0 \\tag{9}\n\\end{equation*}\n\n\nCombine equations (7) and (9), eliminate $d_{j m-1}^{j}(\\lambda)$, and obtain a simpler recursive relation:\n\n\\begin{equation*}\n\\sin \\lambda \\cdot a_{j m} d_{j m+1}^{j}(\\lambda)=-(j-m)(1+\\cos \\lambda) d_{j m}^{j}(\\lambda) \\tag{10}\n\\end{equation*}\n\nThat is\n\\begin{equation*}\nd_{j m}^{j}(\\lambda)=-\\frac{\\sin \\frac{\\lambda}{2}}{\\cos \\frac{\\lambda}{2}}(\\frac{j+m+1}{j-m})^{\\frac{1}{2}} d_{j m+1}^{j}(\\lambda) \\tag{$10^\\prime$}\n\\end{equation*}\n\n\nThus, once $d_{j j}^{j}$ is found, all $d_{j m}^{j}$ can be recursively derived. Let's first find $d_{j j}^{j}$.\n\n\\begin{equation*}\nd_{j j}^{j}(\\lambda)=\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}|j j\\rangle \\tag{11}\n\\end{equation*}\n\n\nNote that $d_{j j}^{j}(0)=1$, differentiating the above equation with respect to $\\lambda$, we get\n\n\\begin{align*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} \\lambda d_{j j}^{j}(\\lambda)} & =\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}(-\\mathrm{i} J_{y})|j j\\rangle \\\\\n& =\\frac{1}{2}\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}(J_{-}-J_{+})|j j\\rangle \\\\\n& =\\frac{1}{2} a_{j-j} d_{j j-1}^{j}(\\lambda) \\tag{12}\n\\end{align*}\n\n\nWhile equation (7) takes $m=j$, noticing $a_{j j}=0$, we get\n\n\\begin{equation*}\n\\frac{1}{2}(\\cos \\lambda+1) a_{j,-j} d_{j j-1}^{j}(\\lambda)=-j \\sin \\lambda \\cdot d_{j j}^{j}(\\lambda) \\tag{13}\n\\end{equation*}\n\n\nSubstituting into equation (12), we get\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} \\lambda} d_{j j}^{j}(\\lambda)=-j \\frac{\\sin \\lambda}{1+\\cos \\lambda} d_{j j}^{j}(\\lambda) \\tag{14}\n\\end{equation*}\n\n\nIntegrate, and using $d_{j j}^{j}(0)=1$, we obtain\n\n\\begin{equation*}\nd_{j j}^{j}(\\lambda)=(\\cos \\frac{\\lambda}{2})^{2 j} \\tag{15}\n\\end{equation*}\n\n\nThis formula can also be derived using the model in another format. Using equation (15) and the recursive relation ( $10^{\\prime}$), we can sequentially obtain\n\n\\begin{aligned}\n& d_{j j-1}^{j}(\\lambda)=-(2 j)^{1 / 2}(\\cos \\frac{\\lambda}{2})^{2 j-1} \\sin \\frac{\\lambda}{2} \\\\\n& d_{j j-2}^{j}(\\lambda)=[\\frac{2 j(2 j-1)}{2!}]^{\\frac{1}{2}}(\\cos \\frac{\\lambda}{2})^{2 j-2}(\\sin \\frac{\\lambda}{2})^{2} \\\\\n& \\vdots\n\\end{aligned}\n\n\n\\begin{align*}\n& d_{j m}^{j}(\\lambda)=(-1)^{j-m}[\\frac{(2 j)!}{(j+m)!(j-m)!}]^{\\frac{1}{2}}(\\cos \\frac{\\lambda}{2})^{j+m}(\\sin \\frac{\\lambda}{2})^{j-m} \\tag{16}\\\\\n& \\vdots \\\\\n& d_{j-j}^{j}(\\lambda)=(-1)^{2 j}(\\sin \\frac{\\lambda}{2})^{2 j}\n\\end{align*}\n\n\nDiscussion $1^{\\circ}$ When $\\lambda \\rightarrow-\\lambda$, clearly\n\nd_{j m}^{j_{m}}(-\\lambda)=(-1)^{\\jmath^{-m}} d_{j m}^{\\prime}(\\lambda)\n\n\nThis result matches the general property of $d_{m^{\\prime} m}^{\\prime}$.\n$\\mathbf{2}^{\\circ}$ When $\\lambda=\\pi$, the only non-zero matrix element is clearly\n\n\\begin{equation*}\nd_{j-\\jmath}^{j}(\\pi)=(-1)^{2 j} \\tag{18}\n\\end{equation*}\n\n\nThis is because\n\n\\begin{align*}\n& \\mathrm{e}^{-i \\pi J_{y}}|j m\\rangle=(-1)^{j-m}|j-m\\rangle \\tag{19}\\\\\n& \\mathrm{e}^{-i \\pi J_{y}}|j-j\\rangle=(-1)^{2 j}|j j\\rangle\n\\end{align*}\n\n[The meaning of operator $\\mathrm{e}^{-\\mathrm{i} \\pi_{y}}$ is to rotate the system by $180^{\\circ}$ about the y-axis, so the state $|j m\\rangle$ becomes the state $|j-m\\rangle$, and equation (19) specifies the relative phase factor for the relationship between $|j m\\rangle$ and $|j-m\\rangle$.]", + "symbol": { + "$d_{j m}^{j}(\\lambda)$": "matrix element in the context of angular momentum theory, parameterized by angular momentum quantum numbers and an angle", + "$j$": "total angular momentum quantum number", + "$m$": "magnetic quantum number", + "$\\lambda$": "rotation angle related to angular momentum", + "$J_{y}$": "y-component of angular momentum operator", + "$J_{+}$": "raising operator for angular momentum", + "$J_{-}$": "lowering operator for angular momentum", + "$a_{j m}$": "coefficient related to angular momentum quantum numbers j and m", + "$J_{z}$": "z-component of angular momentum operator", + "$J_{x}$": "x-component of angular momentum operator" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 33, + "topic": "Theoretical Foundations", + "question": "Let $\\boldsymbol{J}_{1}$ and $\\boldsymbol{J}_{2}$ be angular momenta corresponding to different degrees of freedom, then their sum $\\boldsymbol{J}=\\boldsymbol{J}_{1}+\\boldsymbol{J}_{2}$ is also an angular momentum. Try to compute the expectation values of $J_{1 z}$ for the common eigenstate $|j_{1} j_{2} j m\\rangle$ of $(\\boldsymbol{J}_{1}^{2}, \\boldsymbol{J}_{2}^{2}, \\boldsymbol{J}^{2}, J_{z})$. (Take $\\hbar=1$)", + "final_answer": [ + "m \\frac{j(j+1)+j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}{2 j(j+1)}" + ], + "answer_type": "Expression", + "answer": "$J_{1}, ~ J_{2}$ satisfy the fundamental commutation relations of angular momentum operators\n\n\\begin{equation*}\n\\boldsymbol{J}_{1} \\times \\boldsymbol{J}_{1}=\\mathrm{i} \\boldsymbol{J}_{1}, \\quad \\boldsymbol{J}_{2} \\times \\boldsymbol{J}_{2}=\\mathrm{i} \\boldsymbol{J}_{2} \\tag{1}\n\\end{equation*}\n\n$\\boldsymbol{J}_{1}, ~ J_{2}$ belong to different degrees of freedom and commute with each other, so\n\n\\begin{align*}\n& {[J_{x}, J_{1 x}]=[J_{1 x}+J_{2 x}, J_{1 x}]=0} \\tag{2}\\\\\n& {[J_{x}, J_{1 y}]=[J_{1 x}+J_{2 x}, J_{1 y}]=\\mathrm{i} J_{1 z}}\n\\end{align*}\n\n$\\boldsymbol{J}$ and $\\boldsymbol{J}_{2}$ have similar relationships. In summary, $\\boldsymbol{J}_{1}$ or $\\boldsymbol{J}_{2}$ and $\\boldsymbol{J}$ satisfy all the relations between the vector operator $\\boldsymbol{A}$ and $\\boldsymbol{J}$ from the previous problem. Moreover,\n\n\\begin{align*}\n& \\boldsymbol{J} \\cdot \\boldsymbol{J}_{1}=\\boldsymbol{J}_{1}^{2}+\\boldsymbol{J}_{2} \\cdot \\boldsymbol{J}_{1}=\\frac{1}{2}(\\boldsymbol{J}^{2}+\\boldsymbol{J}_{1}^{2}-\\boldsymbol{J}_{2}^{2}) \\tag{3}\\\\\n& \\boldsymbol{J} \\cdot \\boldsymbol{J}_{2}=\\boldsymbol{J}_{2}^{2}+\\boldsymbol{J}_{1} \\cdot \\boldsymbol{J}_{2}=\\frac{1}{2}(\\boldsymbol{J}^{2}+\\boldsymbol{J}_{2}^{2}-\\boldsymbol{J}_{1}^{2}) \\tag{4}\n\\end{align*}\n\nUsing equation (5) from the previous problem, we have\n\n\\begin{align*}\n& j(j+1)\\langle\\boldsymbol{J}_{1}\\rangle_{j_{1} j_{2} j m}=\\frac{1}{2}[j(j+1)+j_{1}(j_{1}+1)-j_{2}(j_{2}+1)]\\langle\\boldsymbol{J}\\rangle_{j_{1} j_{2} j m} \\tag{5}\\\\\n& j(j+1)\\langle\\boldsymbol{J}_{2}\\rangle_{j_{1} j_{2} j m}=\\frac{1}{2}[j(j+1)+j_{2}(j_{2}+1)-j_{1}(j_{1}+1)]\\langle\\boldsymbol{J}\\rangle_{j_{1} j_{2} j m} \\tag{6}\n\\end{align*}\n\nSince in the state $|J_{z}=m\\rangle$\n\n$$ \\langle J_{x}\\rangle=\\langle J_{y}\\rangle=0, \\quad\\langle J_{z}\\rangle=m $$\n\nTherefore, equations (5) and (6) yield\n\n$$ \\langle J_{1 x}\\rangle_{j_{1} j_{2} j m}=\\langle J_{1 y}\\rangle_{j_{1} j_{2} j m}=\\langle J_{2 x}\\rangle_{j_{1} j_{2} j m}=\\langle J_{2 y}\\rangle_{j_{1} j_{2} j m}=0 $$\n\n\n\\begin{align*}\n\\langle J_{1 z}\\rangle_{j_{1} j_{2} j m} & =m \\frac{j(j+1)+j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}{2 j(j+1)} \\tag{7}\\\\\n\\langle J_{2 z}\\rangle_{j_{1} j_{2} j m} & =m \\frac{j(j+1)+j_{2}(j_{2}+1)-j_{1}(j_{1}+1)}{2 j(j+1)} \\\\\n& =m-\\langle J_{1 z}\\rangle_{j_{1} j_{2}>m}\n\\end{align*}", + "symbol": { + "$\\boldsymbol{J}_{1}$": "angular momentum for degree of freedom 1", + "$\\boldsymbol{J}_{2}$": "angular momentum for degree of freedom 2", + "$\\boldsymbol{J}$": "total angular momentum", + "$J_{1 z}$": "z-component of angular momentum for degree of freedom 1", + "$J_{2 z}$": "z-component of angular momentum for degree of freedom 2", + "$J_{x}$": "x-component of total angular momentum", + "$J_{1 x}$": "x-component of angular momentum for degree of freedom 1", + "$J_{2 x}$": "x-component of angular momentum for degree of freedom 2", + "$J_{y}$": "y-component of total angular momentum", + "$J_{1 y}$": "y-component of angular momentum for degree of freedom 1", + "$J_{2 y}$": "y-component of angular momentum for degree of freedom 2", + "$m$": "magnetic quantum number", + "$j_{1}$": "angular momentum quantum number for degree of freedom 1", + "$j_{2}$": "angular momentum quantum number for degree of freedom 2", + "$j$": "total angular momentum quantum number" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 34, + "topic": "Theoretical Foundations", + "question": "Two angular momenta $\\boldsymbol{J}_{1}$ and $\\boldsymbol{J}_{2}$, of equal magnitude but belonging to different degrees of freedom, couple to form a total angular momentum $\\boldsymbol{J}=\\boldsymbol{J}_{1}+\\boldsymbol{J}_{2}$, with $\\hbar=1$, and assume $\\boldsymbol{J}_{1}^{2}=\\boldsymbol{J}_{2}^{2}=j(j+1)$. In the state where the total angular momentum quantum number $J=0$, what is the probability when $J_{1z}$ takes the value $m$ (with $J_{2z}$ simultaneously taking the value $-m$)?", + "final_answer": [ + "1/(2j+1)" + ], + "answer_type": "Expression", + "answer": "The eigenstate of $(\\boldsymbol{J}_{1}^{2}, J_{1 z})$ is denoted by $|j m_{1}\\rangle_{1}$, and the eigenstate of $(\\boldsymbol{J}_{2}^{2}, J_{2 z})$ is denoted by $|j m_{2}\\rangle_{2}$. The common eigenstate of $(\\boldsymbol{J}_{1}^{2}, \\boldsymbol{J}_{2}^{2}$, $\\mathbf{J}^{2}, J_{z})$ is denoted by $|j j J M\\rangle$, where $M$ is the eigenvalue of $J_{z}$.\n$|j j J M\\rangle$ can also be abbreviated as $|J M\\rangle$. \nWhen $J=0$, $M=0, m_{1}=-m_{2}=m$, so the state under discussion can be expressed as \n\n\\begin{equation*}\n|j j 00\\rangle=\\sum_{m} C_{m}|j m\\rangle_{1}|j-m\\rangle_{2} \\tag{3}\n\\end{equation*}\n\n$C_{m}$ is the C.G. coefficient $\\langle j_{1} m_{1} j_{2} m_{2} \\mid J M\\rangle$ when $j_{1}=j_{2}=j, J=M=0, m_{1}=-m_{2}=m$. $|C_{m}|^{2}$ is the probability that $J_{1}$ takes the value $m$ (with $J_{2 z}$ taking the value $-m$ at the same time). Below, we solve for $C_{m}$.\n\nSince $\\boldsymbol{J}^{2}|j j 00\\rangle=0$, and $\\boldsymbol{J}^{2}$ is positive definite, it must be \n\n\\begin{equation*}\n\\boldsymbol{J}|j j 00\\rangle=0 \\tag{4}\n\\end{equation*}\n\n\nThus, \n\n\\begin{equation*}\n(J_{x}+\\mathrm{i} J_{y})|j j 00\\rangle=0 \\tag{5}\n\\end{equation*}\n\n\nWhere \n\n\\begin{equation*}\nJ_{x}+\\mathrm{i} J_{y}=(J_{1 x}+\\mathrm{i} J_{1 y})+(J_{2 x}+\\mathrm{i} J_{2 y})=J_{1+}+J_{2+} \\tag{6}\n\\end{equation*}\n\n\nAccording to the basic formula for angular momentum ladder operators, \n\n\\begin{align*}\nJ_{1+}|j m\\rangle_{1} & =(J_{1 x}+\\mathrm{i} J_{1 y})|j m\\rangle_{1}=a_{j m}|j m+1\\rangle_{1} \\\\\na_{j m} & =\\sqrt{(j-m)(j+m+1)} \\tag{7a}\\\\\nJ_{2+}|j,-m\\rangle_{2} & =(J_{2 x}+\\mathrm{i} J_{2 y})|j,-m\\rangle_{2}=a_{j,-m}|j, 1-m\\rangle_{2} \\\\\na_{j,-m} & =\\sqrt{(j+m)(j-m+1)}=a_{,, m-1} \\tag{7b}\n\\end{align*}\n\n\nSubstituting expressions (5) to (7) into (3), we obtain \n\n\\begin{equation*}\n\\sum_{m} C_{m}[a_{j m}|j, m+1\\rangle_{1}|j,-m\\rangle_{2}+a_{J, m-1}|j m\\rangle_{1}|j, 1-m\\rangle_{2}]=0 \\tag{8}\n\\end{equation*}\n\n\nSince \n\n\\sum_{m} C_{m} a_{j, m-1}|j m\\rangle_{1}|j, 1-m\\rangle_{2} \\xrightarrow{m \\rightarrow m+1} \\sum_{m} C_{m+1} a_{j m}|j m+1\\rangle_{1}|j,-m\\rangle_{2}\n\n\nTherefore, equation (8) becomes \n\n\\begin{equation*}\n\\sum_{m}(C_{m}+C_{m+1}) a_{j m}|j m+1\\rangle_{1}|j,-m\\rangle_{2}=0 \\tag{$\\prime$}\n\\end{equation*}\n\n\nSince each basis vector is linearly independent, it must be \n\n\\begin{equation*}\n(C_{m}+C_{m+1}) a_{j m}=0 \\tag{9}\n\\end{equation*}\n\n\nWhich implies \n\n\\begin{equation*}\nC_{m}=-C_{m+1}, \\quad m=j-1, j-2, \\cdots,(-j) \\tag{\\prime}\n\\end{equation*}\n\n\nIn equation (3), $|j j 00\\rangle, ~|j m\\rangle_{1}, ~|j,-m\\rangle_{2}$ etc. are all orthonormalized, and $m$ has a total of $(2 j+1)$ possible values. According to the normalization condition \n\n\\begin{equation*}\n\\sum_{m}|C_{m}|^{2}=1 \\tag{10}\n\\end{equation*}\n\n\nAnd equation ($9^{\\prime}$), we immediately have $|C_{m}|^{2}=1 /(2 j+1)$. If we take \n\nC_{j}=1 / \\sqrt{2 j+1}\n\n\nWe get \n\n\\begin{equation*}\nC_{m}=(-1)^{j-m} C_{j}=(-1)^{j-m} \\frac{1}{\\sqrt{2 j+1}} \\tag{11}\n\\end{equation*}\n\n\nSubstituting into equation (3), we obtain \n\n\\begin{equation*}\n|j j 00\\rangle=\\frac{1}{\\sqrt{2 j+1}} \\sum_{m}(-1)^{1^{-m}}|j m\\rangle_{1}|j,-m\\rangle_{2} \\tag{12}\n\\end{equation*}\n\n\nClearly, under the premise $J_{1 z}=-J_{2 z}$, the probabilities that $J_{1 z}$ and $J_{2 z}$ take each eigenvalue $(j, j-1, \\cdots,-j)$ are equal, both being $1 /(2 j+1)$.", + "symbol": { + "$\\boldsymbol{J}_{1}$": "angular momentum 1", + "$\\boldsymbol{J}_{2}$": "angular momentum 2", + "$\\boldsymbol{J}$": "total angular momentum", + "$\\hbar$": "reduced Planck's constant", + "$j$": "quantum number for angular momentum magnitude", + "$J$": "total angular momentum quantum number", + "$J_{1z}$": "z-component of angular momentum 1", + "$m$": "eigenvalue of $J_{1z}$", + "$J_{2z}$": "z-component of angular momentum 2", + "$m_1$": "eigenvalue of $J_{1z}$", + "$m_2$": "eigenvalue of $J_{2z}$", + "$M$": "eigenvalue of total angular momentum $J_{z}$", + "$C_{m}$": "Clebsch-Gordan coefficient for state $m$", + "$C_{m+1}$": "Clebsch-Gordan coefficient for state $m+1$", + "$a_{j m}$": "ladder operator coefficient for state $m$", + "$a_{j,-m}$": "ladder operator coefficient for state $-m$", + "$J_{x}$": "x-component of total angular momentum", + "$J_{y}$": "y-component of total angular momentum", + "$J_{1x}$": "x-component of angular momentum 1", + "$J_{1y}$": "y-component of angular momentum 1", + "$J_{2x}$": "x-component of angular momentum 2", + "$J_{2y}$": "y-component of angular momentum 2", + "$J_{1+}$": "ladder operator for angular momentum 1", + "$J_{2+}$": "ladder operator for angular momentum 2" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 35, + "topic": "Theoretical Foundations", + "question": "A particle with mass $\\mu$ and charge $q$ moves in a magnetic field $\\boldsymbol{B}=\\nabla \\times \\boldsymbol{A}$, where the Hamiltonian is $H = \\frac{1}{2} \\mu \\boldsymbol{v}^{2}$, with $\\boldsymbol{v}$ as the velocity operator. Calculate $\\mathrm{d} \\boldsymbol{v} / \\mathrm{d} t$.", + "final_answer": [ + "\\frac{q}{2 \\mu c}(\\boldsymbol{v} \\times \\boldsymbol{B}-\\boldsymbol{B} \\times \\boldsymbol{v})" + ], + "answer_type": "Expression", + "answer": "The Hamiltonian operator can be expressed as\n\n\\begin{equation*}\nH=\\frac{1}{2 \\mu}(\\boldsymbol{p}-\\frac{q}{c} \\boldsymbol{A})^{2}=\\frac{1}{2} \\mu \\boldsymbol{v}^{2} \\tag{2}\n\\end{equation*}\n\n\nUsing the commutation relations of $\\boldsymbol{v}$ and $v^{2}$, it can be easily demonstrated that\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{v}=\\frac{1}{\\mathrm{i} \\hbar}[\\boldsymbol{v}, H]=\\frac{q}{2 \\mu c}(\\boldsymbol{v} \\times \\boldsymbol{B}-\\boldsymbol{B} \\times \\boldsymbol{v}) \\tag{3}\n\\end{equation*}", + "symbol": { + "$\\mu$": "mass of the particle", + "$q$": "charge of the particle", + "$\\boldsymbol{B}$": "magnetic field", + "$\\boldsymbol{A}$": "magnetic vector potential", + "$H$": "Hamiltonian", + "$\\boldsymbol{v}$": "velocity operator", + "$c$": "speed of light in vacuum" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 36, + "topic": "Theoretical Foundations", + "question": "A particle with mass $\\mu$ and charge $q$ moves in a magnetic field $\\boldsymbol{B}=\\nabla \\times \\boldsymbol{A}$, where the Hamiltonian is $H = \\frac{1}{2} \\mu \\boldsymbol{v}^{2}$, with $\\boldsymbol{v}$ as the velocity operator. Let $\\boldsymbol{L}$ be the the angular momentum operator. Calculate $\\mathrm{d} \\boldsymbol{L} / \\mathrm{d} t$.", + "final_answer": [ + "\\frac{q}{2 c}[\\boldsymbol{r} \\times(\\boldsymbol{v} \\times \\boldsymbol{B})+(\\boldsymbol{B} \\times \\boldsymbol{v}) \\times \\boldsymbol{r}]" + ], + "answer_type": "Expression", + "answer": "The Hamiltonian operator can be expressed as\n\n\\begin{equation*}\nH=\\frac{1}{2 \\mu}(\\boldsymbol{p}-\\frac{q}{c} \\boldsymbol{A})^{2}=\\frac{1}{2} \\mu \\boldsymbol{v}^{2} \\tag{2}\n\\end{equation*}\n\n\nUsing the commutation relations of $\\boldsymbol{v}$ and $v^{2}$, it can be easily demonstrated that\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{v}=\\frac{1}{\\mathrm{i} \\hbar}[\\boldsymbol{v}, H]=\\frac{q}{2 \\mu c}(\\boldsymbol{v} \\times \\boldsymbol{B}-\\boldsymbol{B} \\times \\boldsymbol{v}) \\tag{3}\n\\end{equation*}\n\n\nIn classical electrodynamics, the Lorentz force is\n\n$$ \\boldsymbol{f}=\\frac{q}{c} \\boldsymbol{v} \\times \\boldsymbol{B}. $$\n\nThe equation of motion is\n\n\\begin{equation*}\n\\mu \\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{v}=\\boldsymbol{f}=\\frac{q}{c} \\boldsymbol{v} \\times \\boldsymbol{B} \\tag{$\\prime$}\n\\end{equation*}\n\n\nEquation (3) is the quantum mechanical extension of the classical equation of motion ($3^{\\prime}$). In equation (3),\n\n\\begin{equation*}\n\\boldsymbol{v} \\times \\boldsymbol{B}=\\frac{1}{\\mu}(\\boldsymbol{p}-\\frac{q}{c} \\boldsymbol{A}) \\times \\boldsymbol{B}=-\\boldsymbol{B} \\times \\boldsymbol{v}-\\frac{\\mathrm{i} \\hbar}{\\mu} \\nabla \\times \\boldsymbol{B} \\tag{4}\n\\end{equation*}\n\n\nFrom electrodynamics,\n\n$$\\nabla \\times \\boldsymbol{B}=\\frac{4 \\pi}{c} \\boldsymbol{j}$$\n\n$\\boldsymbol{j}$ is the source current that generates the magnetic field. Thus, if in the space where the particle moves, the source current $\\boldsymbol{j}=0$, then\n\n\\begin{equation*}\n\\frac{q^{v}}{c} \\boldsymbol{v} \\times \\boldsymbol{B}=-\\frac{q}{c} \\boldsymbol{B} \\times \\boldsymbol{v}=\\boldsymbol{f} \\tag{5}\n\\end{equation*}\n\n\nEquation (3) can still be rewritten in the form of equation ($3^{\\prime}$). For uniform fields, it is obvious that equations (3) and ($3^{\\prime}$) are equivalent.\nThe mechanical angular momentum can also be expressed as\n\n\\begin{equation*}\n\\boldsymbol{L}=\\mu \\boldsymbol{r} \\times \\boldsymbol{v}=\\frac{1}{2} \\mu(\\boldsymbol{r} \\times \\boldsymbol{v}-\\boldsymbol{v} \\times \\boldsymbol{r}) \\tag{$\\prime$}\n\\end{equation*}\n\n(Note that in $\\boldsymbol{r} \\times \\boldsymbol{v}$, the relevant components of $\\boldsymbol{r}$ and $\\boldsymbol{v}$ are commutative.) The time derivative of $\\boldsymbol{L}$ is\n\n$$\\frac{\\mathrm{d} \\boldsymbol{L}}{\\mathrm{~d} t}=\\frac{\\mu}{2}(\\frac{\\mathrm{~d} \\boldsymbol{r}}{\\mathrm{~d} t} \\times \\boldsymbol{v}+\\boldsymbol{r} \\times \\frac{\\mathrm{d} \\boldsymbol{v}}{\\mathrm{~d} t}-\\frac{\\mathrm{d} \\boldsymbol{v}}{\\mathrm{~d} t} \\times \\boldsymbol{r}-\\boldsymbol{v} \\times \\frac{\\mathrm{d} \\boldsymbol{r}}{\\mathrm{~d} t}).$$\n\nUsing equation (3) and $\\mathrm{d} \\boldsymbol{r} / \\mathrm{d} t=\\boldsymbol{v}$, we obtain\n\n\\begin{equation*}\n\\frac{\\mathrm{d} \\boldsymbol{L}}{\\mathrm{~d} t}=\\frac{q}{4 c}[\\boldsymbol{r} \\times(\\boldsymbol{v} \\times \\boldsymbol{B})+(\\boldsymbol{B} \\times \\boldsymbol{v}) \\times \\boldsymbol{r}-\\boldsymbol{r} \\times(\\boldsymbol{B} \\times \\boldsymbol{v})-(\\boldsymbol{v} \\times \\boldsymbol{B}) \\times \\boldsymbol{r}] \\tag{6}\n\\end{equation*}\n\nUnder the conditions where equation (5) holds, equation (6) can be simplified to\n\n\\begin{equation*}\n\\frac{\\mathrm{d} \\boldsymbol{L}}{\\mathrm{~d} t}=\\frac{q}{2 c}[\\boldsymbol{r} \\times(\\boldsymbol{v} \\times \\boldsymbol{B})+(\\boldsymbol{B} \\times \\boldsymbol{v}) \\times \\boldsymbol{r}]=\\frac{1}{2}(\\boldsymbol{r} \\times \\boldsymbol{f}-\\boldsymbol{f} \\times \\boldsymbol{r}) \\tag{7}\n\\end{equation*}\n\n\nNote that in $\\boldsymbol{r} \\times \\boldsymbol{f}$, the relevant components of $\\boldsymbol{r}$ and $\\boldsymbol{f}$ are non-commutative, and $\\boldsymbol{r} \\times \\boldsymbol{f} \\neq-\\boldsymbol{f} \\times \\boldsymbol{r}$, hence the right side of equation (7) is not equivalent to $\\boldsymbol{r} \\times \\boldsymbol{f}$.", + "symbol": { + "$\\mu$": "mass of the particle", + "$q$": "charge of the particle", + "$\\boldsymbol{B}$": "magnetic field", + "$\\boldsymbol{A}$": "vector potential", + "$H$": "Hamiltonian", + "$\\boldsymbol{v}$": "velocity operator", + "$\\boldsymbol{L}$": "angular momentum operator", + "$\\boldsymbol{f}$": "Lorentz force", + "$\\boldsymbol{r}$": "position vector", + "$\\boldsymbol{p}$": "momentum operator", + "$\\boldsymbol{j}$": "source current generating the magnetic field", + "$c$": "speed of light in vacuum", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 37, + "topic": "Theoretical Foundations", + "question": "A three-dimensional isotropic oscillator with mass $\\mu$, charge $q$, and natural frequency $\\omega_{0}$ is placed in a uniform external magnetic field $\\boldsymbol{B}$. Find the formula for energy levels.", + "final_answer": [ + "E_{n_{1} n_{2} m}=(n_{1}+\\frac{1}{2}) \\hbar \\omega_{0}+(2 n_{2}+1+|m|) \\hbar \\omega-m \\hbar \\omega_{\\mathrm{L}}" + ], + "answer_type": "Expression", + "answer": "Compared to the previous two questions, an additional harmonic oscillator potential $\\frac{1}{2} \\mu \\omega_{0}^{2}(x^{2}+y^{2}+z^{2})$ should be added to the Hamiltonian in this question, thus\n\n\\begin{align*}\nH & =\\frac{1}{2 \\mu} \\boldsymbol{p}^{2}+\\frac{1}{2} \\mu \\omega_{0}^{2}(x^{2}+y^{2}+z^{2})+\\frac{q^{2} B^{2}}{8 \\mu c^{2}}(x^{2}+y^{2})-\\frac{q B}{2 \\mu c} l_{z} \\\\\n& =H_{1}+H_{2}-\\omega_{\\mathrm{L}} l_{z} \\tag{1}\n\\end{align*}\n\n\nwhere\n\n\\begin{align*}\n& H_{1}=\\frac{1}{2 \\mu} p_{z}^{2}+\\frac{1}{2} \\mu \\omega_{0}^{2} z^{2} \\\\\n& H_{2}=\\frac{1}{2 \\mu}(p_{x}^{2}+p_{y}^{2})+\\frac{1}{2} \\mu(\\omega_{0}^{2}+\\omega_{\\mathrm{L}}^{2})(x^{2}+y^{2}) \\tag{2}\\\\\n& \\omega_{\\mathrm{L}}=\\frac{q B}{2 \\mu c}\n\\end{align*}\n\n$H_{1}, ~ H_{2}, ~ l_{z}$ are mutually commuting conserved quantities and can be chosen as a complete set of commuting observables. $H_{1}$ corresponds to a one-dimensional harmonic oscillator energy operator, with eigenvalues given by\n\n\\begin{equation*}\nE_{n_{1}}=(n_{1}+\\frac{1}{2}) \\hbar \\omega_{0}, \\quad n_{1}=0,1,2, \\cdots \\tag{3}\n\\end{equation*}\n\n$\\mathrm{H}_{2}$ corresponds to a two-dimensional isotropic oscillator total energy operator, with eigenvalues given by\n\n\\begin{align*}\n& E_{n_{2} m}=(2 n_{2}+1+|m|) \\hbar \\omega \\tag{4}\\\\\n& n_{2}=0,1,2, \\cdots, \\quad m=0, \\pm 1, \\pm 2, \\cdots\n\\end{align*}\n\n\nwhere\n\n\\begin{equation*}\n\\omega=\\sqrt{\\omega_{0}^{2}+\\omega_{L}^{2}} \\tag{5}\n\\end{equation*}\n\nThe eigenvalue of $l_{z}$ is $m \\hbar$. Therefore, the energy levels for this problem are\n\n\\begin{equation*}\nE_{n_{1} n_{2} m}=(n_{1}+\\frac{1}{2}) \\hbar \\omega_{0}+(2 n_{2}+1+|m|) \\hbar \\omega-m \\hbar \\omega_{\\mathrm{L}} \\tag{6}\n\\end{equation*}\n\n\nNote: $\\omega_{\\mathrm{L}}$ can be positive or negative depending on the sign of charge $q$; the magnetic quantum number $m$ can also be positive or negative. Therefore, the sign of $q$ does not affect the overall energy spectrum. For precision, only the case of $q>0(\\omega_{\\mathrm{L}}>0)$ is discussed below.\n\nThe relationship among the three frequencies $\\omega_{0}, ~ \\omega_{L}, ~ \\omega$ is given by:\n\n\\begin{equation*}\n0<(\\omega-\\omega_{\\mathrm{L}})<\\omega_{0} \\tag{7}\n\\end{equation*}\n\n\nThe dependence of the energy levels on the quantum numbers (listed in the order of their magnitude of change) is\n\n\\begin{array}{lll}\nm \\geqslant 0, & m \\text { increases by } 1, & E \\text { increases by } \\hbar(\\omega-\\omega_{\\mathrm{L}}) \\\\\n& n_{1} \\text { increases by } 1, & E \\text { increases by } \\hbar \\omega_{0} \\\\\nm \\leqslant 0, & |m| \\text { increases by } 1, & E \\text { increases by } \\hbar(\\omega+\\omega_{\\mathrm{L}}) \\\\\n& n_{2} \\text { increases by } 1, & E \\text { increases by } 2 \\hbar \\omega\n\\end{array}\n\n\nFor the ground state $E_{0}$, obviously, $n_{1}, ~ n_{2}, ~|m|$ all take their minimum values,\n\n\\begin{equation*}\nE_{0}=E_{000}=\\frac{1}{2} \\hbar \\omega_{0}+\\hbar \\omega \\tag{8}\n\\end{equation*}\n\n\nFor the first excited state $E_{1}, n_{1}=n_{2}=0, m=1$,\n\n\\begin{equation*}\nE_{1}=E_{001}=\\frac{1}{2} \\hbar \\omega_{0}+2 \\hbar \\omega-\\hbar \\omega_{\\mathrm{L}}=E_{0}+\\hbar(\\omega-\\omega_{\\mathrm{L}}) \\tag{9}\n\\end{equation*}", + "symbol": { + "$\\mu$": "mass of the oscillator", + "$q$": "charge", + "$\\omega_{0}$": "natural frequency of the oscillator", + "$\\boldsymbol{B}$": "external magnetic field", + "$H$": "Hamiltonian operator of the system", + "$\\boldsymbol{p}$": "momentum vector", + "$c$": "speed of light", + "$l_{z}$": "z-component of angular momentum", + "$H_{1}$": "one-dimensional harmonic oscillator Hamiltonian", + "$H_{2}$": "two-dimensional isotropic oscillator Hamiltonian", + "$p_{z}$": "momentum along the z-axis", + "$p_{x}$": "momentum along the x-axis", + "$p_{y}$": "momentum along the y-axis", + "$\\omega_{\\mathrm{L}}$": "Larmor frequency", + "$n_{1}$": "quantum number for z-axis oscillator", + "$n_{2}$": "quantum number for two-dimensional oscillator", + "$m$": "magnetic quantum number", + "$\\hbar$": "reduced Planck's constant", + "$\\omega$": "derived angular frequency" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 38, + "topic": "Theoretical Foundations", + "question": "A particle with mass $\\mu$ and charge $q$ moves in a uniform electric field $\\mathscr{E}$ (along the x-axis) and a uniform magnetic field $\\boldsymbol{B}$ (along the z-axis) that are perpendicular to each other. If the momentum of the particle in the y-direction is $p_y$ and in the z-direction is $p_z$, find the energy level expression of the system.", + "final_answer": [ + "E = (n+\\frac{1}{2}) \\frac{\\hbar B|q|}{\\mu c}-\\frac{c^{2} \\mathscr{E}^{2} \\mu}{2 B^{2}}-\\frac{c \\mathscr{E}}{B} p_{y}+\\frac{1}{2 \\mu} p_{z}^{2}" + ], + "answer_type": "Expression", + "answer": "With the electric field direction as the $x$ axis and the magnetic field direction as the $z$ axis, then\n\n\\begin{equation*}\n\\mathscr{L}=(\\mathscr{E}, 0,0), \\quad \\boldsymbol{B}=(0,0, B) \\tag{1}\n\\end{equation*}\n\n\nTaking the scalar and vector potentials of the electromagnetic field as\n\n\\begin{equation*}\n\\phi=-\\mathscr{E} x, \\quad \\boldsymbol{A}=(0, B x, 0) \\tag{2}\n\\end{equation*}\n\n\nSatisfying the relation\n\n$$\\mathscr{E}=-\\nabla \\phi, \\quad \\boldsymbol{B}=\\nabla \\times \\boldsymbol{A}.$$\n\n\nThe Hamiltonian of the particle is\n\n\\begin{equation*}\nH=\\frac{1}{2 \\mu}[p_{x}^{2}+(p_{y}-\\frac{q B}{c} x)^{2}+p_{z}^{2}]-q \\mathscr{E} x \\tag{3}\n\\end{equation*}\n\n\nWith the constants of motion set as $(H, p_{y}, p_{z})$, their common eigenfunction can be written as\n\n\\begin{equation*}\n\\psi(x, y, z)=\\psi(x) \\mathrm{e}^{\\mathrm{i}(p_{y} y+p_{z} z) / \\hbar} \\tag{4}\n\\end{equation*}\n\n\nwhere $p_{y}$ and $p_{z}$ are eigenvalues and can be any real numbers.\n$\\psi(x, y, z)$ satisfies the energy eigen-equation\n\n$$ H \\psi(x, y, z)=E \\psi(x, y, z) $$\n\n\nThus, $\\psi(x)$ satisfies the equation\n\n\\begin{equation*}\n\\frac{1}{2 \\mu}[p_{x}^{2}+(p_{y}-\\frac{q B}{c} x)^{2}+p_{z}^{2}] \\psi-q \\mathscr{E} x \\psi=E \\psi \\tag{5}\n\\end{equation*}\n\n\nThat is, for $\\psi(x)$, $H$ is equivalent to the following:\n\n\\begin{align*}\nH & \\Rightarrow-\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\partial^{2}}{\\partial x^{2}}+\\frac{q^{2} B^{2}}{2 \\mu c^{2}} x^{2}-(q \\mathscr{E}+\\frac{q B}{\\mu c} p_{y}) x+\\frac{1}{2 \\mu}(p_{y}^{2}+p_{x}^{2}) \\\\\n& =-\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\partial^{2}}{\\partial x^{2}}+\\frac{q^{2} B^{2}}{2 \\mu c^{2}}(x-x_{0})^{2}-\\frac{q^{2} B^{2}}{2 \\mu c^{2}} x_{0}^{2}+\\frac{1}{2 \\mu}(p_{y}^{2}+p_{z}^{2}) \\tag{6}\n\\end{align*}\n\n\nwhere\n\n\\begin{equation*}\nx_{0}=\\frac{\\mu c^{2}}{q^{2} B^{2}}(q \\mathscr{E}+\\frac{q B}{\\mu c} p_{y})=\\frac{\\mu c}{q B}(\\frac{c \\mathscr{E}}{B}+\\frac{p_{y}}{\\mu}) \\tag{7}\n\\end{equation*}\n\n\nEquation (6) corresponds to a one-dimensional harmonic oscillator energy operator\n\n$$-\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\partial^{2}}{\\partial x^{2}}+\\frac{1}{2} \\mu \\omega^{2}(x-x_{0})^{2}, \\quad \\omega=\\frac{|q| B}{\\mu c} $$\n\n\nPlus two constant terms. Therefore, the energy level of this problem is\n\n\\begin{align*}\nE & =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{q^{2} B^{2}}{2 \\mu c^{2}} x_{0}^{2}+\\frac{1}{2 \\mu}(p_{y}^{2}+p_{z}^{2}) \\\\\n& =(n+\\frac{1}{2}) \\frac{\\hbar B|q|}{\\mu c}-\\frac{c^{2} \\mathscr{E}^{2} \\mu}{2 B^{2}}-\\frac{c \\mathscr{E}}{B} p_{y}+\\frac{1}{2 \\mu} p_{z}^{2} \\tag{8}\n\\end{align*}\n\n\nwhere $p_{y}, ~ p_{z}$ are any real numbers, $n=0,1,2, \\cdots$.\nIn equation (4), $\\psi(x)$ is the one-dimensional harmonic oscillator energy eigenfunction with $(x-x_{0})$ as the variable, i.e.,\n\n\\begin{equation*}\n\\psi(x)=\\psi_{n}(x-x_{0})=H_{n}(\\xi) \\mathrm{e}^{-\\frac{1}{2} \\xi^{2}} \\tag{9}\n\\end{equation*}\n\n$H_{n}(\\xi)$ is the Hermite polynomial, $\\xi=(\\frac{|q| B}{\\hbar c})^{\\frac{1}{2}}(x-x_{0})$.", + "symbol": { + "$\\mu$": "mass of the particle", + "$q$": "charge of the particle", + "$\\mathscr{E}$": "electric field", + "$\\boldsymbol{B}$": "magnetic field", + "$p_y$": "momentum in the y-direction", + "$p_z$": "momentum in the z-direction", + "$B$": "magnitude of the magnetic field", + "$c$": "speed of light", + "$H$": "Hamiltonian", + "$\\hbar$": "reduced Planck's constant", + "$x_0$": "displacement term related to particle's motion", + "$n$": "quantum number" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 39, + "topic": "Theoretical Foundations", + "question": "A particle moves in one dimension. When the total energy operator is\n\n\\begin{equation*}\nH_{0}=\\frac{p^{2}}{2 m}+V(x) \\tag{1}\n\\end{equation*}\n\n\nthe energy level is $E_{n}^{(0)}$. If the total energy operator becomes\n\n\\begin{equation*}\nH=H_{0}+\\frac{\\lambda p}{m} \\tag{2}\n\\end{equation*}\n\n\nfind the energy level $E_{n}$.", + "final_answer": [ + "E_{n}=E_{n}^{(0)}-\\frac{\\lambda^{2}}{2 m}" + ], + "answer_type": "Expression", + "answer": "First, treat $\\lambda$ as a parameter, then\n\n\\begin{equation*}\n\\frac{\\partial H}{\\partial \\lambda}=\\frac{p}{m} \\tag{3}\n\\end{equation*}\n\n\nAccording to the Hellmann theorem, we have\n\n\\begin{equation*}\n\\frac{\\partial E_{n}}{\\partial \\lambda}=\\langle\\frac{\\partial H}{\\partial \\lambda}\\rangle_{n}=\\frac{1}{m}\\langle p\\rangle_{n} \\tag{4}\n\\end{equation*}\n\n\nHowever, since\n\n\\begin{equation*}\n\\frac{\\mathrm{d} x}{\\mathrm{~d} t}=\\frac{1}{\\mathrm{i} \\hbar}[x, H]=\\frac{1}{\\mathrm{i} \\hbar m}[x, \\frac{p^{2}}{2}+\\lambda p]=\\frac{1}{m}(p+\\lambda) \\tag{5}\n\\end{equation*}\n\n\nFor any bound state,\n\n$$ \\langle\\frac{\\mathrm{d} x}{\\mathrm{~d} t}\\rangle_{n}=\\frac{1}{\\mathrm{i} \\hbar}\\langle x H-H x\\rangle_{n}=0, $$\ntherefore\n\n\\begin{equation*}\n\\langle p\\rangle_{n}=-\\lambda \\tag{6}\n\\end{equation*}\n\n\nSubstitute into equation (4), obtaining\n\n\\begin{equation*}\n\\frac{\\partial E_{n}}{\\partial \\lambda}=-\\frac{\\lambda}{m} \\tag{7}\n\\end{equation*}\n\n\nIntegrate to get\n\n\\begin{equation*}\nE_{n}=-\\frac{\\lambda^{2}}{2 m}+C \\tag{8}\n\\end{equation*}\n\n$C$ is the integration constant. Since when $\\lambda=0$, $H=H_{0}, E_{n}=E_{n}^{(0)}$, so $C=E_{n}^{(0)}$. Substitute into equation (8), obtaining\n\n\\begin{equation*}\nE_{n}=E_{n}^{(0)}-\\frac{\\lambda^{2}}{2 m} \\tag{9}\n\\end{equation*}\n\n\nSolution Two: Write $H$ as\n\n\\begin{equation*}\nH=\\frac{p^{2}}{2 m}+\\frac{\\lambda p}{m}+V(x)=\\frac{P^{2}}{2 m}+V(x)-\\frac{\\lambda^{2}}{2 m} \\tag{10}\n\\end{equation*}\n\n\nwhere\n\n\\begin{equation*}\nP=p+\\lambda \\tag{11}\n\\end{equation*}\n\n\nIn the momentum representation,\n\n\\begin{equation*}\nx=\\mathrm{i} \\hbar \\frac{\\partial}{\\partial p}=\\mathrm{i} \\hbar \\frac{\\partial}{\\partial P} \\tag{12}\n\\end{equation*}\n\n\nTherefore,\n\n\\begin{align*}\nH_{0} & =\\frac{p^{2}}{2 m}+V(i \\hbar \\frac{\\partial}{\\partial p}) \\tag{13}\\\\\nH & =\\frac{P^{2}}{2 m}+V(i \\hbar \\frac{\\partial}{\\partial P})-\\frac{\\lambda^{2}}{2 m} \\tag{14}\n\\end{align*}\n\nThe difference between $H$ and $H_{0}$, aside from the constant term, is just replacing $p$ with $P$, which does not affect the energy levels. Therefore,\n\n\\begin{equation*}\nE_{n}=E_{n}^{(0)}-\\frac{\\lambda^{2}}{2 m} \\tag{15}\n\\end{equation*}", + "symbol": { + "$H_{0}$": "initial total energy operator", + "$p$": "momentum", + "$m$": "mass", + "$V(x)$": "potential energy as a function of position", + "$E_{n}^{(0)}$": "initial energy level", + "$H$": "total energy operator including perturbation", + "$\\lambda$": "perturbation parameter", + "$E_{n}$": "energy level after perturbation", + "$\\hbar$": "reduced Planck's constant", + "$x$": "position", + "$C$": "integration constant", + "$P$": "modified momentum", + "$i$": "imaginary unit" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 40, + "topic": "Theoretical Foundations", + "question": "A particle with mass $\\mu$ moves in a central force field,\n\n\\begin{equation*}\nV(r)=\\lambda r^{\\nu}, \\quad-2<\\nu, \\quad \\nu / \\lambda>0 \\tag{1}\n\\end{equation*}\n\n\nUse the Hellmann theorem and the virial theorem to analyze the dependence of the energy level structure on $\\hbar, ~ \\lambda, ~ \\mu$.", + "final_answer": [ + "E=C \\lambda^{2 /(2+\\nu)}(\\frac{\\hbar^{2}}{2 \\mu})^{\\nu /(2+\\nu)}" + ], + "answer_type": "Expression", + "answer": "The energy operator is\n\n\\begin{equation*}\nH=T+V=-\\frac{\\hbar^{2}}{2 \\mu} \\nabla^{2}+\\lambda r^{\\nu} \\tag{2}\n\\end{equation*}\n\n\nLet $\\beta=\\hbar^{2} / 2 \\mu, \\lambda$ and $\\beta$ be independent parameters. It is evident that\n\n\\begin{equation*}\n\\beta \\frac{\\partial H}{\\partial \\beta}=T, \\quad \\lambda \\frac{\\partial H}{\\partial \\lambda}=V \\tag{3}\n\\end{equation*}\n\n\nAccording to the Hellmann theorem, for any bound state,\n\n\\begin{align*}\n& \\langle T\\rangle=\\frac{\\beta \\partial E}{\\partial \\beta} \\tag{4}\\\\\n& \\langle V\\rangle=\\frac{\\lambda \\partial E}{\\partial \\lambda} \\tag{5}\n\\end{align*}\n\n\nAdding the two equations yields\n\n\\begin{equation*}\n\\beta \\frac{\\partial E}{\\partial \\beta}+\\lambda \\frac{\\partial E}{\\partial \\lambda}=\\langle T+V\\rangle=E \\tag{6}\n\\end{equation*}\n\n\nAnd from the virial theorem, we have\n\n$$ \\langle T\\rangle=\\frac{\\nu}{2}\\langle V\\rangle$$\n\n\nThat is\n\n\\begin{equation*}\n\\beta \\frac{\\partial E}{\\partial \\beta}=\\frac{\\nu}{2} \\lambda \\frac{\\partial E}{\\partial \\lambda} \\tag{7}\n\\end{equation*}\n\n\nSubstituting equation (7) into equation (6), we get\n\n\\begin{align*}\n& (1+\\frac{\\nu}{2}) \\lambda \\frac{\\partial E}{\\partial \\lambda}=E \\tag{8}\\\\\n& (1+\\frac{2}{\\nu}) \\beta \\frac{\\partial E}{\\partial \\beta}=E \\tag{9}\n\\end{align*}\n\n\nIntegrating equation (8), we obtain the construction relationship between $E$ and $\\lambda$\n\n\\begin{equation*}\nE=C_{1} \\lambda^{2 /(2+\\iota)} \\tag{10}\n\\end{equation*}\n\n$C_{1}$ is the \"integration constant\" and is independent of $\\lambda$. Integrating equation (9), we obtain the construction relationship between $E$ and $\\beta$\n\n\\begin{equation*}\nE=C_{2} \\beta^{\\nu /(2 \\downarrow \\imath)} \\tag{11}\n\\end{equation*}\n\n$C_{2}$ is independent of $\\nu$. Comparing equations (10) and (11), it follows that\n\n\\begin{equation*}\nE=C \\lambda^{2 /(2+\\tau)} \\beta^{\\nu /(2+\\nu)}=C \\lambda^{2 /(2+\\tau)}(\\frac{\\hbar^{2}}{2 \\mu})^{\\nu /(2+\\nu)} \\tag{12}\n\\end{equation*}\n\n$C$ is independent of $\\lambda, ~ \\beta$, and is a dimensionless pure number (related to $\\nu$ and quantum numbers). It is easy to verify that the above expression is the only possible energy construction that is dimensionally correct.\n\nFrom equation (12), it is evident (note $\\nu>-2$) that as the interaction strength $|\\lambda|$ increases, $|E|$ increases, and the energy level spacing increases. If $\\lambda$ is independent of the particle's mass, then when $\\nu>0, ~ \\mu$ increases, $|E|$ decreases; when $\\nu<0, ~ \\mu$ increases, $|E|$ increases.\n\nIf $\\lambda$ is independent of $\\mu$, from equations (2) and (4) it can also be seen that\n\n\\begin{equation*}\n\\mu \\frac{\\partial E}{\\partial \\mu}=-\\langle T\\rangle<0 \\tag{13}\n\\end{equation*}\n\n\nThat is, an increase in the particle's mass always leads to a decrease in the algebraic value of the energy levels.\nFrom equation (12) it can also be seen that if $\\lambda \\propto \\mu^{\\nu / 2}$, then $E$ is independent of $\\mu$. A famous example of this situation is the harmonic oscillator, i.e.,\n\n\\begin{align*}\n& V(\\boldsymbol{r})=\\lambda r^{2}=\\frac{1}{2} \\mu \\omega^{2} r^{2} \\\\\n& (\\nu=2, \\quad \\lambda=\\frac{1}{2} \\mu \\omega^{2})\n\\end{align*}\n\n\nThe energy levels are\n\n$$ E_{N}=(N+\\frac{3}{2}) \\hbar \\omega, \\quad N=0,1,2, \\cdots $$\n\n\nIf $\\omega$ remains constant, $E_{N}$ is independent of $\\mu$.", + "symbol": { + "$\\mu$": "mass of the particle", + "$\\nu$": "exponent in the potential function", + "$\\lambda$": "scaling factor of the potential", + "$\\hbar$": "reduced Planck's constant", + "$\\beta$": "parameter related to kinetic energy and mass", + "$E$": "energy", + "$T$": "kinetic energy", + "$V$": "potential energy", + "$C$": "dimensionless integration constant related to energy", + "$C_{1}$": "integration constant independent of $\\lambda$", + "$C_{2}$": "integration constant independent of $\\nu$", + "$\\omega$": "angular frequency", + "$E_{N}$": "energy level for quantum number $N$", + "$N$": "quantum number" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 41, + "topic": "Theoretical Foundations", + "question": "A particle moves in a potential field\n\\begin{equation*}\nV(x)=V_{0}|x / a|^{\\nu}, \\quad V_{0}, a>0 \\tag{1}.\n\\end{equation*}\n\nFind the dependence of energy levels on parameters as $\\nu \\rightarrow \\infty$.", + "final_answer": [ + "E_{n}=\\frac{n^{2} \\pi^{2} \\hbar^{2}}{8 \\mu a^{2}}" + ], + "answer_type": "Expression", + "answer": "The total energy operator is\n\n\\begin{equation*}\nH=T+V=-\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\mathrm{~d}^{2}}{\\mathrm{~d} x^{2}}+V_{0}|x / a|^{\\nu} \\tag{2}\n\\end{equation*}\n\n\nFrom dimensional analysis, if $x_{0}$ represents the characteristic length, we have\n\n\\begin{equation*}\nE \\sim \\frac{\\hbar^{2}}{\\mu x_{0}^{2}} \\sim \\frac{V_{0} x_{0}^{\\nu}}{a^{\\nu}} \\tag{3}\n\\end{equation*}\n\n\nIt is not difficult to solve for\n\n\\begin{equation*}\nx_{0} \\sim(\\frac{\\hbar^{2} a^{\\nu}}{\\mu V_{0}})^{\\frac{1}{\\nu+2}} \\xrightarrow{\\nu \\rightarrow \\infty} a \\tag{4}\n\\end{equation*}\n\n\n\\begin{equation*}\nE \\sim(\\frac{\\hbar^{2}}{\\mu})^{\\frac{1}{\\nu+2}} V_{0}^{2 /(\\nu+2)} a^{-2 \\nu /(\\nu+2)} \\xrightarrow{\\nu \\rightarrow \\infty} \\frac{\\hbar^{2}}{\\mu a^{2}} \\tag{5}\n\\end{equation*}\n\n\nThe construction of the characteristic length and energy levels is independent of $V_{0}$, \n\nIn fact, it is easy to see from Equation (1)\n\n$$ \\nu \\rightarrow \\infty, \\quad V(x) \\rightarrow \\begin{cases}0, & |x|a\\end{cases} $$\n\nThis is precisely an infinitely deep potential well of width $2 a$, with energy levels\n\n\\begin{equation*}\nE_{n}=\\frac{n^{2} \\pi^{2} \\hbar^{2}}{8 \\mu a^{2}} . \\quad n=1,2,3, \\cdots \\tag{7}\n\\end{equation*}\n\nWhen $\\nu$ is finite, according to the virial theorem, for any bound state, there is a relationship between the average kinetic energy and the average potential energy\n\n\\begin{equation*}\n\\frac{2}{\\nu}\\langle T\\rangle=\\langle V\\rangle \\tag{8}\n\\end{equation*}\n\n\nThus, as $u \\rightarrow \\infty$, it follows\n\n\\begin{equation*}\n\\langle V\\rangle \\rightarrow 0, \\quad\\langle T\\rangle \\rightarrow E \\tag{9}\n\\end{equation*}\n\n\nThis conclusion is also consistent with the infinite potential well problem.", + "symbol": { + "$V$": "potential energy", + "$x$": "position", + "$V_{0}$": "scaling factor of the potential", + "$a$": "characteristic length scale of the potential", + "$\\nu$": "exponent indicating the power to which the scaled position is raised in the potential", + "$H$": "total energy operator", + "$T$": "kinetic energy", + "$\\hbar$": "reduced Planck's constant", + "$\\mu$": "reduced mass", + "$E$": "energy", + "$x_{0}$": "characteristic length", + "$E_{n}$": "energy levels", + "$n$": "quantum number representing different energy levels" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 42, + "topic": "Theoretical Foundations", + "question": "A particle with mass $m$ moves in a uniform force field $V(x)=F x(F>0)$, with the motion constrained to the range $x \\geqslant 0$. Find its ground state energy level.", + "final_answer": [ + "E_{1}=1.8558(\\frac{\\hbar^{2} F^{2}}{m})^{1 / 3}" + ], + "answer_type": "Expression", + "answer": "The total energy operator is\n\n\\begin{equation*}\nH=T+V=\\frac{p^{2}}{2 m}+F x \\tag{1}\n\\end{equation*}\n\n\nIn momentum representation, the operator for $x$ is given by\n\n\\begin{equation*}\n\\hat{x}=\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} p} \\tag{2}\n\\end{equation*}\n\nThe operator for $H$ is given by\n\n\\begin{equation*}\n\\hat{H}=\\frac{p^{2}}{2 m}+\\mathrm{i} \\hbar F \\frac{\\mathrm{~d}}{\\mathrm{~d} p} \\tag{3}\n\\end{equation*}\n\n\nThe time-independent Schrödinger equation is\n\n\\begin{equation*}\n\\frac{p^{2}}{2 m} \\varphi(p)+\\mathrm{i} \\hbar F \\frac{\\mathrm{~d}}{\\mathrm{~d} p} \\varphi(p)=E \\varphi(p) \\tag{4}\n\\end{equation*}\n\n\nHere $\\varphi(p)$ is the wave function in the momentum representation. Equation (4) is a very simple first-order differential equation, whose solution is\n\n\\begin{equation*}\n\\varphi(p)=A \\exp [\\frac{\\mathrm{i}}{\\hbar F}(\\frac{p^{3}}{6 m}-E p)] \\tag{5}\n\\end{equation*}\n\n$A$ is the normalization constant.\nTransforming to the $x$ representation, the wave function is\n\n\\begin{align*}\n\\psi(x) & =(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\varphi(p) \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\mathrm{d} p \\\\\n& =A(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\exp \\frac{\\mathrm{i}}{\\hbar}[\\frac{p^{3}}{6 m F}+(x-\\frac{E}{F}) p] \\mathrm{d} p \\\\\n& =2 A(2 \\pi \\hbar)^{-1 / 2} \\int_{0}^{\\infty} \\cos [\\frac{p^{3}}{6 \\hbar m F}+\\frac{p}{\\hbar}(x-\\frac{E}{F})] \\mathrm{d} p \\\\\n& =\\frac{C}{\\sqrt{\\pi}} \\int_{0}^{\\infty} \\cos (\\frac{u^{3}}{3}+u \\xi) \\mathrm{d} u \\tag{6}\n\\end{align*}\n\n\nWhere\n\n\\begin{align*}\n& u=p(2 \\hbar m F)^{-1 / 3} \\tag{7}\\\\\n& \\xi=(x-\\frac{E}{F})(\\frac{2 m F}{\\hbar^{2}})^{1 / 3}=\\frac{x}{l}-\\lambda \\tag{8}\\\\\n& l=(\\frac{\\hbar^{2}}{2 m F})^{1 / 3} \\tag{9}\\\\\n& \\lambda=(\\frac{2 m}{\\hbar^{2} F^{2}})^{1 / 3} E=\\frac{2 m E}{\\hbar^{2}} l^{2} \\tag{10}\n\\end{align*}\n\n$l$ is the characteristic length of this problem.\nExcept for the normalization constant $C$, the right-hand side of equation (6) resembles the Airy function with $\\xi$ as the variable.\nWhen $\\xi>0(x>E / F$ , i.e., the classically forbidden region), the Airy function behaves like the modified Bessel function of the second kind, i.e.,\n\n\\begin{equation*}\n\\psi(x)=\\sqrt{\\xi} K \\frac{1}{3}(\\frac{2}{3} \\xi^{3 / 2}) \\xrightarrow{\\xi \\rightarrow \\infty} \\sqrt{\\xi}(\\frac{3 \\pi}{4 \\xi^{3 / 2}})^{1 / 2} \\mathrm{e}^{-\\frac{2}{3} \\xi^{3 / 2}} \\tag{11}\n\\end{equation*}\n\n\nWhen $\\xi<0$ ($x0$) through a $\\delta$ potential barrier $V(x)=V_{0} \\delta(x)$ in the momentum representation.", + "final_answer": [ + "\\frac{1}{1+(m V_{0} / \\hbar^{2} k)^{2}}" + ], + "answer_type": "Expression", + "answer": "The stationary Schrödinger equation in the $x$ representation is\n\n\\begin{equation*}\n\\psi^{\\prime \\prime}+k^{2} \\psi-\\frac{2 m V_{0}}{\\hbar^{2}} \\delta(x) \\psi=0, \\quad k=\\sqrt{2 m E} / \\hbar \\tag{1}\n\\end{equation*}\n\nLet\n\n\\begin{equation*}\n\\psi(x)=(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\mathrm{d} p \\varphi(p) \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\tag{2}\n\\end{equation*}\n\nwhere $\\varphi(p)$ is the wave function in the momentum representation, which should satisfy the equation\n\n\\begin{equation*}\n\\frac{p^{2}}{2 m} \\varphi(p)+\\int_{-\\infty}^{+\\infty} V_{p p^{\\prime}} \\varphi(p^{\\prime}) \\mathrm{d} p^{\\prime}=E \\varphi(p) \\tag{3}\n\\end{equation*}\n\nwhere\n\n\\begin{equation*}\nV_{p p^{\\prime}}=\\frac{1}{2 \\pi \\hbar} \\int_{-\\infty}^{+\\infty} \\mathrm{d} x V_{0} \\delta(x) \\mathrm{e}^{\\mathrm{i}(p^{\\prime}-p) x / \\hbar}=\\frac{V_{0}}{2 \\pi \\hbar} \\tag{4}\n\\end{equation*}\n\nThus, using equations (4) and (2), we can solve\n\n\\begin{equation*}\n\\int_{-\\infty}^{+\\infty} V_{p p^{\\prime}} \\varphi(p^{\\prime}) \\mathrm{d} p^{\\prime}=\\frac{V_{0}}{2 \\pi \\hbar} \\int_{-\\infty}^{+\\infty} \\varphi(p^{\\prime}) \\mathrm{d} p^{\\prime}=\\frac{V_{0}}{\\sqrt{2 \\pi \\hbar}} \\psi(0) \\tag{5}\n\\end{equation*}\n\nSubstituting equations (4) and (5) into equation (3), we obtain\n\n\\begin{equation*}\n(p^{2}-\\hbar^{2} k^{2}) \\varphi(p)+\\frac{2 m V_{0}}{\\sqrt{2 \\pi \\hbar}} \\psi(0)=0 \\tag{6}\n\\end{equation*}\n\nAccording to the fundamental formula of the $\\delta$ function\n\n\\begin{equation*}\n(\\xi-\\xi_{0}) \\delta(\\xi-\\xi_{0})=0 \\tag{7}\n\\end{equation*}\n\nThe general solution of equation (6) is\n\n\\begin{equation*}\n\\varphi(p)=C_{1} \\delta(p-\\hbar k)+C_{2} \\delta(p+\\hbar k)-\\frac{2 m V_{0}}{\\sqrt{2 \\pi \\hbar}} \\frac{\\psi(0)}{(p^{2}-\\hbar^{2} k^{2})} \\tag{8}\n\\end{equation*}\n\nwhere $C_{1}, ~ C_{2}, ~ \\psi(0)$ are to be determined. Substituting equation (8) into equation (2), we get the wave function in the $x$ representation\n\n\\begin{equation*}\n\\psi(x)=\\frac{C_{1}}{\\sqrt{2 \\pi \\hbar}} \\mathrm{e}^{\\mathrm{i} k x}+\\frac{C_{2}}{\\sqrt{2 \\pi \\hbar}} \\mathrm{e}^{-\\mathrm{i} k x}-\\frac{2 m V_{0}}{2 \\pi \\hbar} \\psi(0) \\int_{-\\infty}^{+\\infty} \\frac{\\mathrm{d} p}{p^{2}-\\hbar^{2} k^{2}} \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\tag{9}\n\\end{equation*}\n\nThe last integral in equation (9) should be taken as the principal value, which can be calculated using the contour integral method in the complex $p$ plane, resulting in\n\n\\int_{-\\infty}^{+\\infty} \\frac{\\mathrm{d} p}{p^{2}-\\hbar^{2} k^{2}} \\mathrm{e}^{\\mathrm{i} k x / \\hbar}= \\begin{cases}\\frac{\\mathrm{i} \\pi}{2 \\hbar k}(\\mathrm{e}^{\\mathrm{i} k x}-\\mathrm{e}^{-\\mathrm{i} k x}), & x>0 \\tag{10}\\\\ \\frac{i \\pi}{2 \\hbar k}(\\mathrm{e}^{-\\mathrm{i} k x}-\\mathrm{e}^{\\mathrm{i} k x}), & x<0\\end{cases}\n\nWhen $x \\rightarrow 0$, the right side of equation (10) becomes 0, and from equation (9) we get\n\n\\begin{equation*}\n\\psi(0)=(C_{1}+C_{2}) / \\sqrt{2 \\pi \\hbar} \\tag{11}\n\\end{equation*}\n\nSubstituting equations (10) and (11) into equation (9), we get\n\n\\sqrt{2 \\pi \\hbar} \\psi(x)=C_{1} \\mathrm{e}^{\\mathrm{i} k x}+C_{2} \\mathrm{e}^{-\\mathrm{i} k x}-\\frac{\\mathrm{i} m V_{0}}{2 \\hbar^{2} k}(C_{1}+C_{2})(\\mathrm{e}^{\\mathrm{i} k x}-\\mathrm{e}^{-\\mathrm{i} k x}), \\quad x>0\n\nGiven that the incident wave is $\\mathrm{e}^{\\mathrm{i} k x}$ (i.e., the incident momentum $p=\\hbar k$), in the region $x>0$ there should only be the transmitted wave, i.e., the $\\mathrm{e}^{\\mathrm{i} k x}$ term, and no $\\mathrm{e}^{-\\mathrm{i} k}$ term. Therefore, $C_{1}, ~ C_{2}$ must satisfy the following relation:\n\n\\begin{equation*}\nC_{2}=-\\mathrm{i} \\frac{m V_{0}}{2 \\hbar^{2} k}(C_{1}+C_{2}) \\tag{12}\n\\end{equation*}\n\nThus,\n\n\\begin{equation*}\n\\psi(x)=\\frac{C_{1}+C_{2}}{\\sqrt{2 \\pi \\hbar}} \\mathrm{e}^{\\mathrm{i} k x}, \\quad x>0 \\tag{13}\n\\end{equation*}\n\nSimilarly, we can get\n\n\\begin{equation*}\n\\psi(x)=\\frac{1}{\\sqrt{2 \\pi \\hbar}}[(C_{1}-C_{2}) \\mathrm{e}^{i k x}+2 C_{2} \\mathrm{e}^{-\\mathrm{i} k x}], \\quad x<0 \\tag{14}\n\\end{equation*}\n\nwhere the $\\mathrm{e}^{\\mathrm{ik} x}$ term is the incident wave, and the $\\mathrm{e}^{-\\mathrm{i} k x}$ term is the reflected wave. If the incident wave amplitude is set to 1, then equation (14) can be written as\n\n\\begin{equation*}\n\\psi(x)=\\mathrm{e}^{\\mathrm{i} k \\tau}+R \\mathrm{e}^{-\\mathrm{i} k x} \\tag{14'}\n\\end{equation*}\n\nThen we should take\n\n\\begin{equation*}\nC_{1}-C_{2}=\\sqrt{2 \\pi \\hbar} \\tag{15}\n\\end{equation*}\n\nFrom equations (13) and (14), it is easy to see that\n\n$$\\text { Transmission coefficient }=|\\frac{C_{1}+C_{2}}{C_{1}-C_{2}}|^{2}=\\frac{1}{|1-\\frac{2 C_{2}}{C_{1}+C_{2}}|^{2}}$$\n\nUsing equation (12), we obtain\n\n\\begin{equation*}\n\\text { Transmission coefficient }=\\frac{1}{|1+\\mathrm{i} m V_{0} / \\hbar^{2} k|^{2}}=\\frac{1}{1+(m V_{0} / \\hbar^{2} k)^{2}} \\tag{16}\n\\end{equation*}", + "symbol": { + "$E$": "energy of the particle", + "$m$": "mass of the particle", + "$V_{0}$": "strength of the delta potential barrier", + "$p$": "momentum", + "$\\hbar$": "reduced Planck's constant", + "$k$": "wave number related to the particle's energy", + "$C_{1}$": "coefficient related to the incident wave", + "$C_{2}$": "coefficient related to the reflected wave" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 44, + "topic": "Theoretical Foundations", + "question": "The energy operator of a one-dimensional harmonic oscillator is\n\n\\begin{equation*}\nH=\\frac{p_{x}^{2}}{2 \\mu}+\\frac{1}{2} \\mu \\omega^{2} x^{2} \\tag{1}\n\\end{equation*}\n\n\nTry to derive its energy level expression using the Heisenberg equation of motion for operators and the fundamental commutation relation.", + "final_answer": [ + "E_{n}=(n+\\frac{1}{2}) \\hbar \\omega" + ], + "answer_type": "Expression", + "answer": "Using the Heisenberg equation of motion, we obtain\n\n\\begin{equation*}\n\\frac{\\mathrm{d} p}{\\mathrm{~d} t}=\\frac{1}{\\mathrm{i} \\hbar}[p, H]=\\frac{\\mu \\omega^{2}}{2 \\mathrm{i} \\hbar}[p, x^{2}]=-\\mu \\omega^{2} x \\tag{2}\n\\end{equation*}\n\n\nTake the matrix element in the energy representation, yielding\n\n\\begin{equation*}\n\\mathrm{i} \\omega_{k n} p_{k n}=-\\mu \\omega^{2} x_{k n} \\tag{3}\n\\end{equation*}\n\n\nFrom the previous result,\n\n\\begin{equation*}\np_{k n}=\\mathrm{i} \\omega_{k n} \\mu x_{k n} \\tag{4}\n\\end{equation*}\n\n\nCombining equations (3) and (4), we obtain\n\n\\begin{equation*}\n(\\omega^{2}-\\omega_{k n}^{2}) x_{k n}=0 \\tag{5}\n\\end{equation*}\n\n\nwhere $k, ~ n$ can be understood as quantum state indices. From equation (5) it is evident\n\n\\begin{array}{ll}\n\\text { If } \\omega_{k n} \\neq \\pm \\omega, & \\text { then } x_{k n}=0 \\tag{6}\\\\\n\\text { If } x_{k n} \\neq 0, \\quad \\text { then } \\omega_{k n}= \\pm \\omega\n\\end{array}\n\n\nSince\n\n\\begin{equation*}\n(x^{2})_{k k}=\\sum_{n} x_{k n} x_{n k}=\\sum_{n}|x_{n k}|^{2}>0 \\tag{7}\n\\end{equation*}\n\n\nFor any chosen energy level $E_{k}$, there must exist some $n$ such that $x_{n k} \\neq 0$, then from equation (6), the energy level difference $E_{n}-$ $E_{k}= \\pm \\hbar \\omega$, that is, given any energy level, there must exist another energy level differing by $\\hbar \\omega$. Therefore, all energy levels are\n\n$$ E=E_{0}, \\quad E_{0}+\\hbar \\omega, \\quad E_{0}+2 \\hbar \\omega, \\cdots $$\n\nThat is\n\n\\begin{equation*}\nE_{n}=E_{0}+n \\hbar \\omega, \\quad n=0,1,2, \\cdots \\tag{8}\n\\end{equation*}\n\n\nTo find the ground state energy $E_{0}$, the equation proven in the previous problem can be used\n\n\\begin{equation*}\n\\sum_{n}(E_{n}-E_{k})|x_{n k}|^{2}=\\hbar^{2} / 2 \\mu \\tag{9}\n\\end{equation*}\n\n\nTaking $k$ as the ground state, from equations (6), (7), and (9), we get\n\n$$\\frac{\\hbar^{2}}{2 \\mu}=\\hbar \\omega \\sum_{n}|x_{n 0}|^{2}=\\hbar \\omega(x^{2})_{00}$$\n\n\nThus, the average potential energy of the ground state is\n\n\\begin{equation*}\n\\frac{1}{2} \\mu \\omega^{2}(x^{2})_{00}=\\frac{1}{4} \\hbar \\omega \\tag{10}\n\\end{equation*}\n\n\nAccording to the virial theorem, we have\n\n\\langle T\\rangle_{0}=\\langle V\\rangle_{0}=\\frac{1}{2} E_{0}\n\n\nComparing with equation (10), we obtain\n\n\\begin{equation*}\nE_{0}=\\frac{1}{2} \\hbar \\omega \\tag{11}\n\\end{equation*}\n\n\nSubstituting into equation (8), we get the energy level formula\n\n\\begin{equation*}\nE_{n}=(n+\\frac{1}{2}) \\hbar \\omega, \\quad n=0,1,2, \\cdots \\tag{$\\prime$}\n\\end{equation*}\n\n\nConsidering equation (6), equation (9) gives\n\n\\begin{equation*}\n|x_{k+1, k}|^{2}-|x_{k-1, k}|^{2}=\\frac{\\hbar}{2 \\mu \\omega} \\tag{12}\n\\end{equation*}\n\n\nBy appropriately choosing the phase factor $(\\mathrm{e}^{\\mathrm{i}})$ of each energy eigenfunction, all $x_{n k}$ can be made non-negative real numbers, and equation (12) can be rewritten as (changing $k$ to $n$)\n\n\\begin{equation*}\n(x_{n+1, n})^{2}-(x_{n, n-1})^{2}=\\frac{\\hbar}{2 \\mu \\omega} \\tag{$\\prime$}\n\\end{equation*}\n\n\nWhen $n=0$, the equation yields\n\n\\begin{equation*}\n(x_{10})^{2}=\\frac{\\hbar}{2 \\mu \\omega}, \\quad x_{10}=\\sqrt{\\frac{\\hbar}{2 \\mu \\omega}} \\tag{13}\n\\end{equation*}\n\n\nBy repeatedly using equation (12＇), we get\n\n\\begin{equation*}\nx_{n+1, n}=\\sqrt{\\frac{n+1}{2} \\frac{\\hbar}{\\mu \\omega}}, \\quad n=0,1,2, \\cdots \\tag{14}\n\\end{equation*}\n\n\nBy also using equation (4), we get\n\n\\begin{equation*}\np_{n+1, n}=\\mathrm{i} \\omega \\mu x_{n+1, n} \\tag{15}\n\\end{equation*}\n\n\nNote: The matrix elements of $x$ are real numbers, and the matrix elements of $p$ are purely imaginary numbers. Therefore\n\n\\begin{align*}\n& x_{n, n+1}=(x_{n+1, n})^{*}=x_{n+1, n} \\tag{16}\\\\\n& p_{n, n+1}=(p_{n+1, n})^{*}=-p_{n+1, n} \\tag{17}\n\\end{align*}", + "symbol": { + "$H$": "Hamiltonian or energy operator", + "$p_{x}$": "momentum in the x-direction", + "$\\mu$": "reduced mass", + "$\\omega$": "angular frequency", + "$x$": "position", + "$\\hbar$": "reduced Planck's constant", + "$p$": "momentum operator", + "$t$": "time", + "$k$": "quantum state index", + "$n$": "quantum state index", + "$E_{k}$": "energy of state k", + "$E_{n}$": "energy of state n", + "$E_{0}$": "ground state energy", + "$x_{k n}$": "position matrix element between states k and n", + "$\\omega_{k n}$": "angular frequency difference between states k and n", + "$x_{n 0}$": "position matrix element between state n and ground state", + "$(x^{2})_{00}$": "expectation value of position squared in the ground state", + "$\\langle T \\rangle_{0}$": "average kinetic energy in the ground state", + "$\\langle V \\rangle_{0}$": "average potential energy in the ground state", + "$|x_{k+1, k}|$": "magnitude of position matrix element between states k+1 and k", + "$|x_{k-1, k}|$": "magnitude of position matrix element between states k-1 and k", + "$x_{n+1, n}$": "position matrix element between states n+1 and n", + "$x_{10}$": "position matrix element between first excited state and ground state", + "$p_{n+1, n}$": "momentum matrix element between states n+1 and n" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 45, + "topic": "Theoretical Foundations", + "question": "The Hamiltonian of a one-dimensional harmonic oscillator is $H=\\frac{p^{2}}{2 m}+\\frac{1}{2} m \\omega^{2} x^{2}$. Compute $[x(t_1),p(t_2)]$ in the Heisenberg picture.", + "final_answer": [ + "\\mathrm{i} \\hbar \\cos \\omega(t_{2}-t_{1})" + ], + "answer_type": "Expression", + "answer": "It is easy to obtain\n\n\\begin{align*}\n{[x(t_{1}), x(t_{2})] } & =[x, p] \\frac{1}{m \\omega}(\\cos \\omega t_{1} \\sin \\omega t_{2}-\\sin \\omega t_{1} \\cos \\omega t_{2}) \\\\\n& =\\frac{i \\hbar}{m \\omega} \\sin \\omega(t_{2}-t_{1}). \\tag{1}\n\\end{align*}\n\nSimilarly, it can be obtained\n\n\\begin{align*}\n& {[p(t_{1}), p(t_{2})]=\\mathrm{i} m \\omega \\hbar \\sin \\omega(t_{2}-t_{1})} \\tag{2}\\\\\n& {[x(t_{1}), p(t_{2})]=\\mathrm{i} \\hbar \\cos \\omega(t_{2}-t_{1})} \\tag{3}\n\\end{align*}\n\nIn equation (3), when $t_{1}=t_{2}=t$, we obtain\n\n\\begin{equation*}\n[x(t), p(t)]=\\mathrm{i} \\hbar \\tag{4}\n\\end{equation*}\n\n\nThis commutation relation corresponds to $[x, p]=\\mathrm{i} \\hbar$ in the Schrödinger picture, which applies not only to the harmonic oscillator problem but also to any other problem. As for equations (1), (2), and (3), they are specific to the harmonic oscillator.", + "symbol": { + "$H$": "Hamiltonian", + "$p$": "momentum", + "$m$": "mass", + "$\\omega$": "angular frequency", + "$x$": "position", + "$t_1$": "time point 1", + "$t_2$": "time point 2", + "$\\mathrm{i}$": "imaginary unit", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 46, + "topic": "Theoretical Foundations", + "question": "Two localized non-identical particles with spin $1 / 2$ (ignoring orbital motion) have an interaction energy given by\n\n$$H=A \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2} $$\n\nwhere $\\boldsymbol{s}_{1}, \\boldsymbol{s}_{2}$ are spin operators (the eigenvalues of their $z$ components $s_{iz}$ being $\\pm 1/2$), and $A$ is a constant related to energy.\nAt time $t=0$, particle 1 has spin \"up\" (i.e., the measured value of $s_{1z}$ is $1/2$), and particle 2 has spin \"down\" (i.e., the measured value of $s_{2z}$ is $-1/2$). Determine the probability that at any time $t>0$, particle 1 has spin \"up\" (i.e., measuring $s_{1z}$ yields $1/2$) in the Heisenberg picture.", + "final_answer": [ + "w(t)=\\cos ^{2} \\frac{A t}{2}" + ], + "answer_type": "Expression", + "answer": "Solution 1: Starting from the Heisenberg equation of motion for the spin operators. It is easily observed from the construction of $H$ that the total spin $\\boldsymbol{S}$ commutes with $H$ and is a conserved quantity, hence the values and corresponding probabilities of the components of total $\\boldsymbol{S}$ remain constant, and the distribution probability of $\\boldsymbol{S}^{2}$ also does not change. At $t=0$, the spin state of the system $\\alpha(1) \\beta(2)$ is a common eigenstate of $s_{1 z}$ and $s_{2 z}$, and thus also an eigenstate of total $S_{z}$ with eigenvalue $S_{z}=0$. As $S_{z}$ is a conserved quantity, it can only have the eigenvalue 0 at any time point, and cannot take other eigenvalues. Hence, the probability that both particle 1 and 2 have spin \"up\" $(s_{1 z}=s_{2 z}=\\frac{1}{2}, S_{z}=1)$ is 0. At $t=0$, the total spin quantum numbers $S=1, ~ 0$, each with probabilities $1 / 2$. Since $\\boldsymbol{S}^{2}$ is conserved, this probability does not change over time. The above conclusions apply to parts (b) and (c).\n\nIn order to calculate $\\langle\\boldsymbol{s}_{1}\\rangle$ and $\\langle\\boldsymbol{s}_{2}\\rangle$, the Heisenberg equation of motion can be solved.\n\n\\begin{gather*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{s}_{1}=\\frac{1}{\\mathrm{i} \\hbar}[\\boldsymbol{s}_{1}, H]=\\mathrm{i} A[\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}, \\boldsymbol{s}_{1}]=-\\boldsymbol{A} \\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2} \\tag{1}\\\\\n\\frac{\\mathrm{~d}}{\\mathrm{~d} t} \\boldsymbol{s}_{2}=\\frac{1}{\\mathrm{i} \\hbar}[\\boldsymbol{s}_{2}, H]=\\mathrm{i} A[\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}, \\boldsymbol{s}_{2}]=\\boldsymbol{A} \\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2} \\tag{2}\n\\end{gather*}\n\n\nAdding and subtracting the two equations gives\n\n\\begin{align*}\n& \\frac{\\mathrm{d}}{\\mathrm{~d} t}(\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2})=\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{S}=0 \\tag{3}\\\\\n& \\frac{\\mathrm{~d}}{\\mathrm{~d} t}(\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2})=-2 A \\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2} \\tag{4}\n\\end{align*}\n\nMultiplying equation (2) by $s_{1}$ and equation (1) by $s_{2}$ gives\n\n\\begin{align*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}) & =A[s_{1} \\times(s_{1} \\times s_{2})-(s_{1} \\times s_{2}) \\times s_{2}] \\\\\n& =\\frac{A}{2}(s_{1}-s_{2}) \\tag{5}\n\\end{align*}\n\nMultiply equation (5) by $2i$, and add it to equation (4) to get\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(s_{1}-s_{2}+2 \\mathrm{i} s_{1} \\times s_{2})=\\mathrm{i} A(s_{1}-s_{2}+2 \\mathrm{i} s_{1} \\times s_{2}) \\tag{6}\n\\end{equation*}\n\n\nTake the average value and integrate with respect to $t$, then\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}+2 \\mathrm{i} \\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t}=\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}+2 \\mathrm{i} \\mathbf{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t=0} \\mathrm{e}^{\\mathrm{i} A t} \\tag{7}\n\\end{equation*}\n\n\nThe initial condition of the problem is\n\n\\begin{equation*}\n\\chi(t=0)=\\alpha(1) \\beta(2) \\tag{8}\n\\end{equation*}\n\n\nTherefore,\n\n\\begin{array}{l}\n\\langle\\boldsymbol{s}_{1}\\rangle_{t=0}=(0,0, \\frac{1}{2}), \\quad\\langle\\boldsymbol{s}_{2}\\rangle_{t=0}=(0,0,-\\frac{1}{2}) \\tag{$\\prime$}\\\\\n\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t=0}=0, \\quad\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}\\rangle_{t=0}=(0,0,1)=\\boldsymbol{e}_{3} \\\\\n\\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t=0}=\\langle\\boldsymbol{s}_{1}\\rangle_{t=0} \\times\\langle\\boldsymbol{s}_{2}\\rangle_{t=0}=0\n\\end{array}}\n\nSubstituting into equation (7), we get\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}\\rangle_{t}+2 \\mathrm{i}\\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t}=\\mathrm{e}^{\\mathrm{i} A t} \\boldsymbol{e}_{3}=(\\cos A t+\\mathrm{i} \\sin A t) \\boldsymbol{e}_{3} \\tag{9}\n\\end{equation*}\n\n\nSince $s_{1}, s_{2}, s_{1} \\times s_{2}$ are all Hermitian operators, their averages are real numbers, so\n\n\\begin{gather*}\n\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}\\rangle_{t}=\\boldsymbol{e}_{3} \\cos A t \\tag{10}\\\\\n\\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t}=\\frac{1}{2} \\boldsymbol{e}_{3} \\sin A t \\tag{11}\n\\end{gather*}\n\n\nEquation (3) shows that $\\boldsymbol{S}$ is conserved, and from the initial condition ($8^{\\prime}$):\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t}=\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t=0}=0 \\tag{12}\n\\end{equation*}\n\n\nAdding and subtracting equations (10) and (12), we get\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{1}\\rangle_{t}=-\\langle\\boldsymbol{s}_{2}\\rangle_{t}=\\frac{1}{2} e_{3} \\cos A t \\tag{11}\n\\end{equation*}\n\n\nThat is:\n\n\\begin{align*}\n& \\langle s_{1 x}\\rangle_{t}=\\langle s_{1 y}\\rangle_{t}=\\langle s_{2 x}\\rangle_{t}=\\langle s_{2 y}\\rangle_{t}=0 \\\\\n& \\langle s_{1 z}\\rangle_{t}=-\\langle s_{2 z}\\rangle_{t}=\\frac{1}{2} \\cos A t \\tag{13'}\n\\end{align*}\n\n\nThis result is consistent with equation (5) obtained in Exercise 6.47. Equation (11) corresponds to\n\n\\begin{align*}\n& \\langle s_{1 x} s_{2 y}-s_{1 y} s_{2 x}\\rangle_{t}=\\frac{1}{2} \\sin A t \\\\\n& \\langle s_{1 y} s_{2 z}-s_{1 z} s_{2 y}\\rangle_{t}=0 \\tag{11'}\\\\\n& \\langle s_{1 z} s_{2 x}-s_{1 x} s_{2 z}\\rangle_{t}=0\n\\end{align*}\n\n\nReaders can easily use equation ($4^{\\prime}$) obtained in Exercise 6.47 to verify this conclusion, although it is not easy to see this result there.\n\nAssume that at $t>0$, the probability of particle 1 having spin \"up\" $(s_{1 z}=\\frac{1}{2})$ is $w(t)$, then\n\n\\langle s_{1 z}\\rangle_{t}=\\frac{1}{2} \\cos A t=\\frac{1}{2} w(t)-\\frac{1}{2}[1-w(t)]=w(t)-\\frac{1}{2}\n\n\nThus,\n\n\\begin{equation*}\nw(t)=\\frac{1}{2}(1+\\cos A t)=\\cos ^{2} \\frac{A t}{2} \\tag{1}\n\\end{equation*}\n\n\nSolution 2: Using the Heisenberg picture of the average value formula\n\n\\begin{equation*}\n\\langle s_{1}\\rangle_{t}=\\langle\\chi(0)| s_{1}(t)|\\chi(0)\\rangle=\\langle\\chi(0)| U^{+}(t) s_{1} U(t)|\\chi(0)\\rangle \\tag{15}\n\\end{equation*}\n\n\nwhere\n\n\\begin{array}{c}\\nu(t)=\\mathrm{e}^{-\\mathrm{i} H t}=\\mathrm{e}^{-\\mathrm{i} A s_{1} \\cdot s_{2}} \\tag{16}\\\\\nU^{+}(t)=\\mathrm{e}^{\\mathrm{i} H t}=\\mathrm{e}^{\\mathrm{i} t s_{1} \\cdot s_{2}}\n\\end{array}}\n\n$s_{1}(t)$ can be expressed as (let $\\lambda=A t)$\n\n\\begin{align*}\n\\boldsymbol{s}_{1}(t) & =U^{+}(t) \\boldsymbol{s}_{1} U(t) \\\\\n& =\\frac{1}{2}(\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2})+\\frac{1}{2}(\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}) \\cos A t-\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2} \\sin A t \\tag{17}\n\\end{align*}\n\n\nSubstituting into equation (15), and using the initial averages\n\n\\begin{align*}\n\\langle\\boldsymbol{s}_{1}\\rangle_{t=0} & =\\langle\\chi(0)| \\boldsymbol{s}_{1}|\\chi(0)\\rangle=\\langle\\alpha(1)| \\boldsymbol{s}_{1}|\\alpha(1)\\rangle \\\\\n& =(0,0, \\frac{1}{2}) \\\\\n\\langle\\boldsymbol{s}_{2}\\rangle_{t=0} & =\\langle\\chi(0)| \\boldsymbol{s}_{2}|\\chi(0)\\rangle=\\langle\\beta(2)| \\boldsymbol{s}_{2}|\\beta(2)\\rangle \\\\\n& =(0,0,-\\frac{1}{2}) \\\\\n& \\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t=0}=\\langle\\boldsymbol{s}_{1}\\rangle_{t=0} \\times\\langle\\boldsymbol{s}_{2}\\rangle_{t=0}=0 \\tag{18}\n\\end{align*}\n\n\nwe get\n\n\\begin{align*}\n\\langle\\boldsymbol{s}_{1}\\rangle_{t} & =\\frac{1}{2}\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t=0}+\\frac{1}{2}\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}\\rangle_{t=0} \\cos A t-\\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t=0} \\sin A t \\\\\n& =\\frac{1}{2} \\boldsymbol{e}_{3} \\cos A t \\tag{19}\n\\end{align*}\n\n\nThe total spin $\\boldsymbol{S}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}$ commutes with both $\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}$ and thus also with $U$ and $U^{+}$, so\n\n\\begin{gather*}\n\\boldsymbol{S}(t)=\\boldsymbol{S}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2} \\tag{20}\\\\\n\\langle\\boldsymbol{S}\\rangle_{t}=\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t}=\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t=0}=0 \\tag{21}\n\\end{gather*}\n\n\nSo\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{2}\\rangle_{t}=\\langle\\boldsymbol{S}\\rangle_{t}-\\langle\\boldsymbol{s}_{1}\\rangle_{t}=-\\langle\\boldsymbol{s}_{1}\\rangle_{t}=-\\frac{1}{2} \\boldsymbol{e}_{3} \\cos A t \\tag{22}\n\\end{equation*}\n\n\nAll these results match those of Solution 1. The method to calculate the probability is the same as Solution 1 and thus the result is also the same, omitted here.", + "symbol": { + "$A$": "constant related to energy", + "$t$": "time", + "$\\boldsymbol{s}_{1}$": "spin operator for particle 1", + "$\\boldsymbol{s}_{2}$": "spin operator for particle 2", + "$s_{1z}$": "z-component of spin operator for particle 1", + "$s_{2z}$": "z-component of spin operator for particle 2", + "$\\boldsymbol{S}$": "total spin operator" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 47, + "topic": "Theoretical Foundations", + "question": "For a spin $1/2$ particle, $\\langle\\boldsymbol{\\sigma}\\rangle$ is often called the polarization vector, denoted as $\\boldsymbol{P}$. It is also the spatial orientation of the spin angular momentum. Assume the particle is localized and subject to a magnetic field $\\boldsymbol{B}(t)$ along the $z$ direction with time-varying intensity, with the potential given by\n\n$$H=-\\mu_{0} \\boldsymbol{\\sigma} \\cdot \\boldsymbol{B}(t)=-\\mu_{0} \\sigma_{z} B(t),$$\n\nIn the Heisenberg picture, find the time evolution of the polarization vector, i.e., find $P_x(t)=\\langle{\\sigma}_x\\rangle_{t}$. Let $\\boldsymbol{P}(t=0)$ point in the direction $(\\theta_{0}, \\varphi_{0})$, where $\\theta_{0}=2 \\delta, \\varphi_{0}=2 \\alpha$.", + "final_answer": [ + "\\langle\\sigma_{x}\\rangle_{t}=\\sin 2 \\delta \\cos \\varphi(t)" + ], + "answer_type": "Expression", + "answer": "According to the Heisenberg equation of motion.\n\n$$\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\sigma_{z}=\\frac{1}{\\mathrm{i} \\hbar}[\\sigma_{z}, H]=0,$$\n\nTherefore,\n\n\\begin{equation*}\n\\langle\\sigma_{z}\\rangle_{t}=\\langle\\sigma_{z}\\rangle_{t=0}=\\cos \\theta_{0}=\\cos 2 \\delta \\tag{1}\n\\end{equation*}\n\nMoreover,\n\n\\begin{align*}\n& \\frac{\\mathrm{d}}{\\mathrm{~d} t} \\sigma_{x}=\\frac{1}{\\mathrm{i} \\hbar}[\\sigma_{x}, H]=-\\frac{\\mu_{0}}{\\mathrm{i} \\hbar} B(t)[\\sigma_{x}, \\sigma_{x}]=\\frac{2 \\mu_{0}}{\\hbar} B(t)_{\\sigma_{y}} \\tag{2}\\\\\n& \\frac{\\mathrm{~d}}{\\mathrm{~d} t} \\sigma_{y}=\\frac{1}{\\mathrm{i} \\hbar}[\\sigma_{y}, H]=-\\frac{2 \\mu_{0}}{\\hbar} B(t) \\sigma_{x}\n\\end{align*}\n\n\nCombining the two equations, it is easy to obtain\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(\\sigma_{x}+\\mathrm{i} \\sigma_{y})=-\\mathrm{i} \\frac{2 \\mu_{0}}{\\hbar} B(t)(\\sigma_{x}+\\mathrm{i} \\sigma_{y}) \\tag{3}\n\\end{equation*}\n\n\nTaking the average value and integrating over $t$, we have\n\n\\begin{equation*}\n\\langle\\sigma_{x}+\\mathrm{i} \\sigma_{y}\\rangle_{t}=\\langle\\sigma_{x}+\\mathrm{i} \\sigma_{y}\\rangle_{t=0} \\exp [-\\mathrm{i} \\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau] \\tag{4}\n\\end{equation*}\n\nSince $\\langle\\sigma_{x}\\rangle$, $\\langle\\sigma_{y}\\rangle$ are real numbers, separate the real and imaginary parts in the above equation to get\n\n\\begin{align*}\n& \\langle\\sigma_{x}\\rangle_{t}=\\langle\\sigma_{x}\\rangle_{t=0} \\cos [\\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau]+\\langle\\sigma_{y}\\rangle_{t=0} \\sin [\\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau] \\tag{5}\\\\\n& \\langle\\sigma_{y}\\rangle_{t}=\\langle\\sigma_{y}\\rangle_{t=0} \\cos [\\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau]-\\langle\\sigma_{x}\\rangle_{t=0} \\sin [\\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau]\n\\end{align*}\n\n\nThe initial average values are\n\n\\begin{equation*}\n\\langle\\sigma_{x}\\rangle_{t=0}=\\sin 2 \\delta \\cos 2 \\alpha, \\quad\\langle\\sigma_{y}\\rangle_{t=0}=\\sin 2 \\delta \\sin 2 \\alpha \\tag{6}\n\\end{equation*}\n\n\nSubstituting into equation (5) and defining $\\varphi(t)$, we get\n\n\\begin{gather*}\n\\langle\\sigma_{x}\\rangle_{t}=\\sin 2 \\delta \\cos \\varphi(t), \\quad\\langle\\sigma_{y}\\rangle_{t}=\\sin 2 \\delta \\sin \\varphi(t) \\tag{7}\\\\\n\\boldsymbol{P}(t)=\\langle\\boldsymbol{\\sigma}\\rangle_{t}=(\\sin 2 \\delta \\cos \\varphi(t), \\sin 2 \\delta \\sin \\varphi(t), \\cos 2 \\delta) \\tag{8}\n\\end{gather*}", + "symbol": { + "$\\sigma_{x}$": "Pauli spin matrix for x direction", + "$\\sigma_{y}$": "Pauli spin matrix for y direction", + "$\\sigma_{z}$": "Pauli spin matrix for z direction", + "$\\langle\\sigma_{x}\\rangle_{t}$": "average value of sigma_x at time t", + "$\\langle\\sigma_{y}\\rangle_{t}$": "average value of sigma_y at time t", + "$\\langle\\sigma_{z}\\rangle_{t}$": "average value of sigma_z at time t", + "$\\langle\\sigma_{x}\\rangle_{t=0}$": "initial average value of sigma_x", + "$\\langle\\sigma_{y}\\rangle_{t=0}$": "initial average value of sigma_y", + "$\\langle\\sigma_{z}\\rangle_{t=0}$": "initial average value of sigma_z", + "$\\langle\\boldsymbol{\\sigma}\\rangle$": "polarization vector", + "$\\boldsymbol{P}$": "polarization vector denoted in problem", + "$\\boldsymbol{B}(t)$": "magnetic field at time t", + "$\\theta_{0}$": "initial polar angle", + "$\\varphi_{0}$": "initial azimuthal angle", + "$\\delta$": "parameter related to initial polar angle", + "$\\alpha$": "parameter related to initial azimuthal angle", + "$\\mu_{0}$": "magnetic moment", + "$\\hbar$": "reduced Planck constant" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 48, + "topic": "Others", + "question": "Evaluate the function\n\n$$ \\langle 0| \\phi(x) \\phi(y)|0\\rangle=D(x-y)=\\int \\frac{d^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{\\mathbf{p}}} e^{-i p \\cdot(x-y)} $$\n\nfor $(x-y)$ spacelike so that $(x-y)^{2}=-r^{2}$, explicitly in terms of Bessel functions.\n\n\\footnotetext{\n$\\ddagger$ With some additional work you can show that there are actually six conserved charges in the case of two complex fields, and $n(2 n-1)$ in the case of $n$ fields, corresponding to the generators of the rotation group in four and $2 n$ dimensions, respectively. The extra symmetries often do not survive when nonlinear interactions of the fields are included.\n}", + "final_answer": [ + "D(x-y) = \\frac{m}{4 \\pi^{2} r} K_{1}(m r)" + ], + "answer_type": "Expression", + "answer": "We evaluate the correlation function of a scalar field at two points,\n\n\\begin{equation*}\nD(x-y)=\\langle 0| \\phi(x) \\phi(y)|0\\rangle \\tag{2.28}\n\\end{equation*}\n\nwith $x-y$ being spacelike. Since any spacelike interval $x-y$ can be transformed to a form such that $x^{0}-y^{0}=0$, thus we will simply take:\n\n\\begin{equation*}\nx^{0}-y^{0}=0, \\quad \\text { and } \\quad|\\mathbf{x}-\\mathbf{y}|^{2}=r^{2}>0 . \\tag{2.29}\n\\end{equation*}\n\n\nNow:\n\n\\begin{align*}\nD(x-y) & =\\int \\frac{\\mathrm{d}^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{p}} e^{-i p \\cdot(x-y)}=\\int \\frac{\\mathrm{d}^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 \\sqrt{m^{2}+p^{2}}} e^{i \\mathbf{p} \\cdot(\\mathbf{x}-\\mathbf{y})} \\\\\n& =\\frac{1}{(2 \\pi)^{3}} \\int_{0}^{2 \\pi} \\mathrm{~d} \\varphi \\int_{-1}^{1} \\mathrm{~d} \\cos \\theta \\int_{0}^{\\infty} \\mathrm{d} p \\frac{p^{2}}{2 \\sqrt{m^{2}+p^{2}}} e^{i p r \\cos \\theta} \\\\\n& =\\frac{-\\mathrm{i}}{2(2 \\pi)^{2} r} \\int_{-\\infty}^{\\infty} \\mathrm{d} p \\frac{p e^{\\mathrm{i} p r}}{\\sqrt{m^{2}+p^{2}}} \\tag{2.30}\n\\end{align*}\n\n\nNow we make the path deformation on $p$-complex plane, as is shown in Figure 2.3 of Peskin \\& Daniel V. Schroeder. Then the integral becomes,\n\n\\begin{equation*}\nD(x-y)=\\frac{1}{4 \\pi^{2} r} \\int_{m}^{\\infty} \\mathrm{d} \\rho \\frac{\\rho e^{-\\rho r}}{\\sqrt{\\rho^{2}-m^{2}}}=\\frac{m}{4 \\pi^{2} r} \\mathrm{~K}_{1}(m r) . \\tag{2.31}\n\\end{equation*}", + "symbol": { + "$x$": "coordinate x", + "$y$": "coordinate y", + "$D(x-y)$": "correlation function of the scalar field", + "$p$": "momentum", + "$E_{\\mathbf{p}}$": "energy corresponding to momentum p", + "$m$": "mass", + "$\\phi(x)$": "scalar field at coordinate x", + "$\\phi(y)$": "scalar field at coordinate y", + "$r$": "spacelike interval", + "$\\theta$": "polar angle", + "$\\varphi$": "azimuthal angle", + "$\\rho$": "dummy variable for integration", + "$K_{1}$": "modified Bessel function of the second kind of order 1" + }, + "chapter": "The Klein-Gordon Field", + "section": "The spacelike correlation function" + }, + { + "id": 49, + "topic": "Others", + "question": "This problem concerns the discrete symmetries $P, C$, and $T$. Let $\\phi(x)$ be a complex-valued Klein-Gordon field. The current associated with this field is $J^{\\mu}=i(\\phi^{*} \\partial^{\\mu} \\phi-\\partial^{\\mu} \\phi^{*} \\phi)$. The Parity operator $P$ acts on the annihilation operator $a_{\\mathbf{p}}$ as $P a_{\\mathbf{p}} P=a_{-\\mathbf{p}}$, and on the field as $P \\phi(t, \\mathbf{x}) P =\\phi(t,-\\mathbf{x})$. Determine the transformation property of the current $J^{\\mu}(t, \\mathbf{x})$ under the Parity transformation $P$. Express your answer in terms of $J^{\\mu}(t,-\\mathbf{x})$ and the factor $(-1)^{s(\\mu)}$, where $s(\\mu)=0$ for $\\mu=0$ and $s(\\mu)=1$ for $\\mu=1,2,3$.", + "final_answer": [ + "P J^{\\mu}(t, \\mathbf{x}) P =(-1)^{s(\\mu)} J^{\\mu}(t,-\\mathbf{x})" + ], + "answer_type": "Expression", + "answer": "Now we work out the $C, P$ and $T$ transformation properties of a scalar field $\\phi$. Our starting point is\n\nP a_{\\mathbf{p}} P=a_{-\\mathbf{p}}, \\quad T a_{\\mathbf{p}} T=a_{-\\mathbf{p}}, \\quad C a_{\\mathbf{p}} C=b_{\\mathbf{p}}\n\n\nThen, for a complex scalar field\n\n\\begin{equation*}\n\\phi(x)=\\int \\frac{\\mathrm{d}^{3} k}{(2 \\pi)^{3}} \\frac{1}{\\sqrt{2 k^{0}}}[a_{\\mathbf{k}} e^{-\\mathrm{i} k \\cdot x}+b_{\\mathbf{k}}^{\\dagger} e^{\\mathrm{i} k \\cdot x}], \\tag{3.63}\n\\end{equation*}\n\nwe have\n\n\\begin{align*}\n& P \\phi(t, \\mathbf{x}) P=\\int \\frac{\\mathrm{d}^{3} k}{(2 \\pi)^{3}} \\frac{1}{\\sqrt{2 k^{0}}}[a_{-\\mathbf{k}} e^{-\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}+b_{-\\mathbf{k}}^{\\dagger} e^{\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}]=\\phi(t,-\\mathbf{x}) . \\tag{3.64a}\\\\\n& T \\phi(t, \\mathbf{x}) T=\\int \\frac{\\mathrm{d}^{3} k}{(2 \\pi)^{3}} \\frac{1}{\\sqrt{2 k^{0}}}[a_{-\\mathbf{k}} e^{\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}+b_{-\\mathbf{k}}^{\\dagger} e^{-\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}]=\\phi(-t, \\mathbf{x}) . \\tag{3.64b}\\\\\n& C \\phi(t, \\mathbf{x}) C=\\int \\frac{\\mathrm{d}^{3} k}{(2 \\pi)^{3}} \\frac{1}{\\sqrt{2 k^{0}}}[b_{\\mathbf{k}} e^{-\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}+a_{\\mathbf{k}}^{\\dagger} e^{\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}]=\\phi^{*}(t, \\mathbf{x}) . \\tag{3.64c}\n\\end{align*}\n\n\nAs a consequence, we can deduce the $C, P$, and $T$ transformation properties of the current $J^{\\mu}=\\mathrm{i}(\\phi^{*} \\partial^{\\mu} \\phi-(\\partial^{\\mu} \\phi^{*}) \\phi)$, as follows:\n\n\\begin{align*}\nP J^{\\mu}(t, \\mathbf{x}) P & =(-1)^{s(\\mu)} \\mathrm{i}[\\phi^{*}(t,-\\mathbf{x}) \\partial^{\\mu} \\phi(t,-\\mathbf{x})-(\\partial^{\\mu} \\phi^{*}(t,-\\mathbf{x})) \\phi(t,-\\mathbf{x})] \\\\\n& =(-1)^{s(\\mu)} J^{\\mu}(t,-\\mathbf{x}), \\tag{3.65a}\n\\end{align*}\n\nwhere $s(\\mu)$ is the label for space-time indices that equals to 0 when $\\mu=0$ and 1 when $\\mu=1,2,3$. In the similar way, we have\n\n\\begin{align*}\n& T J^{\\mu}(t, \\mathbf{x}) T=(-1)^{s(\\mu)} J^{\\mu}(-t, \\mathbf{x}) \\tag{3.65b}\\\\\n& C J^{\\mu}(t, \\mathbf{x}) C=-J^{\\mu}(t, \\mathbf{x}) \\tag{3.65c}\n\\end{align*}\n\n\nOne should be careful when playing with $T$ - it is antihermitian rather than hermitian, and anticommutes, rather than commutes, with $\\sqrt{-1}$.", + "symbol": { + "$\\phi(x)$": "complex-valued Klein-Gordon field", + "$J^{\\mu}$": "current associated with the Klein-Gordon field", + "$P$": "Parity operator", + "$a_{\\mathbf{p}}$": "annihilation operator for momentum $\\mathbf{p}$", + "$a_{-\\mathbf{p}}$": "annihilation operator for momentum $-\\mathbf{p}$", + "$s(\\mu)$": "space-time index label function, 0 if $\\mu=0$ and 1 if $\\mu=1,2,3$" + }, + "chapter": "The Dirac Field", + "section": "The discrete symmetries $P, C$ and $T$" + }, + { + "id": 50, + "topic": "Others", + "question": "Let us return to the problem of the creation of Klein-Gordon particles by a classical source. This process can be described by the Hamiltonian\n\n$$ H=H_{0}+\\int d^{3} x(-j(t, \\mathbf{x}) \\phi(x))$$\n\nwhere $H_{0}$ is the free Klein-Gordon Hamiltonian, $\\phi(x)$ is the Klein-Gordon field, and $j(x)$ is a c-number scalar function. We found that, if the system is in the vacuum state before the source is turned on, the source will create a mean number of particles\n\n$$ \\langle N\\rangle=\\int \\frac{d^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{\\mathbf{p}}}|\\tilde{\\jmath}(p)|^{2} .$$\n\n\nIn this problem we will verify that statement, and extract more detailed information, by using a perturbation expansion in the strength of the source. Compute the probability that the source creates one particle of momentum $k$ to all orders in the source $j$ by summing the perturbation series.", + "final_answer": [ + "P(k) = |\\tilde{j}(k)|^{2} e^{-|\\tilde{j}(k)|}" + ], + "answer_type": "Expression", + "answer": "The probability that the source creates one particle with momentum $\\mathbf{k}$ is given by,\n\n$$P(\\mathbf{k})=|\\langle\\mathbf{k}| T \\exp\\{i \\int \\mathrm{~d}^{4} x j(x) \\phi_{I}(x)\\}| 0\\rangle|^{2}. $$\nExpanding the amplitude to the first order in \\( j \\), we get:\n\n\\begin{equation}\n\\begin{split}\nP(k) &= \\left| \\langle \\mathbf{k} | 0 \\rangle + i \\int \\mathrm{d}^4x \\, j(x) \\int \\frac{\\mathrm{d}^3p}{(2\\pi)^3} \\frac{e^{i p \\cdot x}}{\\sqrt{2E_p}} \\langle \\mathbf{k} | a_p^\\dagger | 0 \\rangle + O(j^2) \\right|^2 \\\\ \n&= \\left| i \\int \\frac{\\mathrm{d}^3p}{(2\\pi)^3} \\frac{\\tilde{j}(p)}{\\sqrt{2E_p}} \\sqrt{2E_p (2\\pi)^3} \\, \\delta(\\mathbf{p} - \\mathbf{k}) \\right|^2 = |\\tilde{j}(k)|^2 + O(j^3).\n\\end{split}\n\\end{equation}\n\nIf we go on to work out all the terms in the perturbation expansion, we get:\n\n\\begin{equation*}\nP(k)=|\\sum_{n} \\frac{i(2 n+1)(2 n+1)(2 n-1) \\cdots 3 \\cdot 1}{(2 n+1)!} \\tilde{j}^{n+1}(k)|^{2}=|\\tilde{j}(k)|^{2} e^{-|\\tilde{j}(k)|} \n\\end{equation*}", + "symbol": { + "$\\mathbf{k}$": "momentum of the created particle", + "$j$": "source function in spacetime", + "$k$": "momentum", + "$x$": "spacetime coordinate", + "$\\phi_{I}(x)$": "interaction field at point x", + "$0$": "vacuum state", + "$P(k)$": "probability of creating a particle with momentum k", + "$\\tilde{j}(k)$": "Fourier transform of the source function at momentum k", + "$E_p$": "energy associated with momentum p", + "$\\mathbf{p}$": "momentum variable in integration", + "$a_p^\\dagger$": "creation operator for momentum p", + "$n$": "term index in perturbation series" + }, + "chapter": "Interacting Fields and Feynman Diagrams", + "section": "Scalar field with a classical source" + }, + { + "id": 51, + "topic": "Others", + "question": "Decay of a scalar particle. Consider the following Lagrangian, involving two real scalar fields $\\Phi$ and $\\phi$ :\n\n$$\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\Phi)^{2}-\\frac{1}{2} M^{2} \\Phi^{2}+\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{1}{2} m^{2} \\phi^{2}-\\mu \\Phi \\phi \\phi .$$\n\n\nThe last term is an interaction that allows a $\\Phi$ particle to decay into two $\\phi$ 's, provided that $M>2 m$. Assuming that this condition is met, calculate the lifetime of the $\\Phi$ to lowest order in $\\mu$.", + "final_answer": [ + "\\tau = \\frac{8\\pi M}{\\mu^2} (1 - \\frac{4m^2}{M^2})^{-1/2}" + ], + "answer_type": "Expression", + "answer": "This problem is based on the following Lagrangian,\n\n\\begin{equation*}\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\Phi)^{2}-\\frac{1}{2} M^{2} \\Phi^{2}+\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{1}{2} m^{2} \\phi^{2}-\\mu \\Phi \\phi \\phi . \\tag{4.17}\n\\end{equation*}\n\n\nWhen $M>2 m$, a $\\Phi$ particle can decay into two $\\phi$ particles. We want to calculate the lifetime of the $\\Phi$ particle to lowest order in $\\mu$.\n\nThe two-body decay rate is given in (4.86) of $\\mathrm{P} \\& \\mathrm{~S}$,\n\n\\begin{equation*}\n\\int \\mathrm{d} \\Gamma=\\frac{1}{2 M} \\int \\frac{\\mathrm{~d}^{3} p_{1} \\mathrm{~d}^{3} p_{2}}{(2 \\pi)^{6}} \\frac{1}{4 E_{\\mathbf{p}_{1}} E_{\\mathbf{p}_{2}}}|\\mathcal{M}(\\Phi(0) \\rightarrow \\phi(p_{1}) \\phi(p_{2}))|^{2}(2 \\pi)^{4} \\delta^{(4)}(p_{\\Phi}-p_{1}-p_{2}) . \\tag{4.18}\n\\end{equation*}\n\n\nTo lowest order in $\\mu$, the amplitude $\\mathcal{M}$ is given by,\n\n\\begin{equation*}\ni \\mathcal{M}=-2 i \\mu \\tag{4.19}\n\\end{equation*}\n\n\nThe delta function in our case reads,\n\n\\begin{equation*}\n\\delta^{(4)}(p_{\\Phi}-p_{1}-p_{2})=\\delta(M-E_{\\mathbf{p}_{1}}-E_{\\mathbf{p}_{2}}) \\delta^{(3)}(\\mathbf{p}_{1}+\\mathbf{p}_{2}), \\tag{4.20}\n\\end{equation*}\n\nthus,\n\n\\begin{equation*}\n\\Gamma=\\frac{1}{2} \\cdot \\frac{2 \\mu^{2}}{M} \\int \\frac{\\mathrm{~d}^{3} p_{1} \\mathrm{~d}^{3} p_{2}}{(2 \\pi)^{6}} \\frac{1}{4 E_{\\mathbf{p}_{1}} E_{\\mathbf{p}_{2}}}(2 \\pi)^{4} \\delta(M-E_{\\mathbf{p}_{1}}-E_{\\mathbf{p}_{2}}) \\delta^{(3)}(\\mathbf{p}_{1}+\\mathbf{p}_{2}), \\tag{4.21}\n\\end{equation*}\n\nwhere an additional factor of $1 / 2$ takes account of two identical $\\phi$ 's in final state. Furthermore, there are two mass-shell constraints,\n\n\\begin{equation*}\nm^{2}+\\mathbf{p}_{i}^{2}=E_{\\mathbf{p}_{i}}^{2} . \\quad(i=1,2) \\tag{4.22}\n\\end{equation*}\n\n\nHence,\n\n\\begin{equation*}\n\\Gamma=\\frac{\\mu^{2}}{M} \\int \\frac{\\mathrm{~d}^{3} p_{1}}{(2 \\pi)^{3}} \\frac{1}{4 E_{\\mathbf{p}_{1}}^{2}}(2 \\pi) \\delta(M-2 E_{\\mathbf{p}_{1}})=\\frac{\\mu^{2}}{8 \\pi M}(1-\\frac{4 m^{2}}{M^{2}})^{1 / 2} \\tag{4.23}\n\\end{equation*}\n\n\nThen the lifetime $\\tau$ of $\\Phi$ is,\n\n\\begin{equation*}\n\\tau=\\Gamma^{-1}=\\frac{8 \\pi M}{\\mu^{2}}(1-\\frac{4 m^{2}}{M^{2}})^{-1 / 2} \\tag{4.24}\n\\end{equation*}", + "symbol": { + "$\\Phi$": "real scalar field (parent particle)", + "$\\phi$": "real scalar field (decay particles)", + "$M$": "mass of the $\\Phi$ particle", + "$m$": "mass of the $\\phi$ particle", + "$\\mu$": "coupling constant", + "$\\mathcal{M}$": "decay amplitude", + "$\\mathcal{L}$": "Lagrangian density", + "$E_{\\mathbf{p}_{1}}$": "energy of the first $\\phi$ particle", + "$E_{\\mathbf{p}_{2}}$": "energy of the second $\\phi$ particle", + "$p_{\\Phi}$": "momentum of the $\\Phi$ particle", + "$p_{1}$": "momentum of the first $\\phi$ particle", + "$p_{2}$": "momentum of the second $\\phi$ particle", + "$\\Gamma$": "decay rate", + "$\\tau$": "lifetime of the $\\Phi$ particle" + }, + "chapter": "Interacting Fields and Feynman Diagrams", + "section": "Decay of a scalar particle" + }, + { + "id": 52, + "topic": "Others", + "question": "Consider a theory with two real scalar fields, $\\Phi$ with mass $M$ and $\\phi$ with mass $m$, described by the Lagrangian $\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\Phi)^{2}-\\frac{1}{2} M^{2} \\Phi^{2}+\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{1}{2} m^{2} \\phi^{2}-\\mu \\Phi \\phi \\phi$. If $M > 2m$, the $\\Phi$ particle can decay into two $\\phi$ particles. To the lowest order in the coupling constant $\\mu$, the decay rate $\\Gamma$ for this process is given by $\\Gamma = \\frac{\\mu^2}{8\\pi M} \\sqrt{1 - \\frac{4m^2}{M^2}}$. Calculate the value of the product $\\Gamma M / \\mu^2$ if $M=3m$.", + "final_answer": [ + "\\frac{\\sqrt{5}}{24\\pi}" + ], + "answer_type": "Expression", + "answer": "This problem is based on the following Lagrangian,\n\n\\begin{equation*}\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\Phi)^{2}-\\frac{1}{2} M^{2} \\Phi^{2}+\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{1}{2} m^{2} \\phi^{2}-\\mu \\Phi \\phi \\phi . \n\\end{equation*}\n\n\nWhen $M>2 m$, a $\\Phi$ particle can decay into two $\\phi$ particles. We want to calculate the lifetime of the $\\Phi$ particle to lowest order in $\\mu$.\n\nThe two-body decay rate is given in (4.86) of $\\mathrm{P} \\& \\mathrm{~S}$,\n\n\\begin{equation*}\n\\int \\mathrm{d} \\Gamma=\\frac{1}{2 M} \\int \\frac{\\mathrm{~d}^{3} p_{1} \\mathrm{~d}^{3} p_{2}}{(2 \\pi)^{6}} \\frac{1}{4 E_{\\mathbf{p}_{1}} E_{\\mathbf{p}_{2}}}|\\mathcal{M}(\\Phi(0) \\rightarrow \\phi(p_{1}) \\phi(p_{2}))|^{2}(2 \\pi)^{4} \\delta^{(4)}(p_{\\Phi}-p_{1}-p_{2}) . \n\\end{equation*}\n\n\nTo lowest order in $\\mu$, the amplitude $\\mathcal{M}$ is given by,\n\n\\begin{equation*}\ni \\mathcal{M}=-2 i \\mu \n\\end{equation*}\n\n\nThe delta function in our case reads,\n\n\\begin{equation*}\n\\delta^{(4)}(p_{\\Phi}-p_{1}-p_{2})=\\delta(M-E_{\\mathbf{p}_{1}}-E_{\\mathbf{p}_{2}}) \\delta^{(3)}(\\mathbf{p}_{1}+\\mathbf{p}_{2}), \n\\end{equation*}\n\nthus,\n\n\\begin{equation*}\n\\Gamma=\\frac{1}{2} \\cdot \\frac{2 \\mu^{2}}{M} \\int \\frac{\\mathrm{~d}^{3} p_{1} \\mathrm{~d}^{3} p_{2}}{(2 \\pi)^{6}} \\frac{1}{4 E_{\\mathbf{p}_{1}} E_{\\mathbf{p}_{2}}}(2 \\pi)^{4} \\delta(M-E_{\\mathbf{p}_{1}}-E_{\\mathbf{p}_{2}}) \\delta^{(3)}(\\mathbf{p}_{1}+\\mathbf{p}_{2}), \n\\end{equation*}\n\nwhere an additional factor of $1 / 2$ takes account of two identical $\\phi$ 's in final state. Furthermore, there are two mass-shell constraints,\n\n\\begin{equation*}\nm^{2}+\\mathbf{p}_{i}^{2}=E_{\\mathbf{p}_{i}}^{2} . \\quad(i=1,2) \n\\end{equation*}\n\n\nHence,\n\n\\begin{equation*}\n\\Gamma=\\frac{\\mu^{2}}{M} \\int \\frac{\\mathrm{~d}^{3} p_{1}}{(2 \\pi)^{3}} \\frac{1}{4 E_{\\mathbf{p}_{1}}^{2}}(2 \\pi) \\delta(M-2 E_{\\mathbf{p}_{1}})=\\frac{\\mu^{2}}{8 \\pi M}(1-\\frac{4 m^{2}}{M^{2}})^{1 / 2}\n\\end{equation*}\n\n\nThen the lifetime $\\tau$ of $\\Phi$ is,\n\n\\begin{equation*}\n\\tau=\\Gamma^{-1}=\\frac{8 \\pi M}{\\mu^{2}}(1-\\frac{4 m^{2}}{M^{2}})^{-1 / 2}.\n\\end{equation*}\n\nWe thus have:\n\\begin{equation}\n \\Gamma M / \\mu^2 = \\frac{1}{8\\pi}\\sqrt{1-\\frac{4m^2}{M^2}} = \\frac{1}{8\\pi}\\sqrt{1-\\frac{4}{9}} = \\frac{\\sqrt{5}}{24\\pi}\n\\end{equation}", + "symbol": { + "$\\Phi$": "real scalar field with mass $M$", + "$M$": "mass of the real scalar field $\\Phi$", + "$\\phi$": "real scalar field with mass $m$", + "$m$": "mass of the real scalar field $\\phi$", + "$\\mu$": "coupling constant", + "$\\Gamma$": "decay rate of $\\Phi$ particle" + }, + "chapter": "Interacting Fields and Feynman Diagrams", + "section": "Decay of a scalar particle" + }, + { + "id": 53, + "topic": "Others", + "question": "Equivalent photon approximation. Consider the process in which electrons of very high energy scatter from a target. In leading order in $\\alpha$, the electron is connected to the target by one photon propagator. If the initial and final energies of the electron are $E$ and $E^{\\prime}$, the photon will carry momentum $q$ such that $q^{2} \\approx-2 E E^{\\prime}(1-\\cos \\theta)$. In the limit of forward scattering, whatever the energy loss, the photon momentum approaches $q^{2}=0$; thus the reaction is highly peaked in the forward direction. It is tempting to guess that, in this limit, the virtual photon becomes a real photon. Let us investigate in what sense that is true. Working in the frame where $p=(E, 0,0, E)$, compute explicitly the quantity $\\bar{u}_+(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{1} u_+(p)$ using massless electrons, where $u_+(p)$ and $u_+(p^{\\prime})$ are spinors of positive helicity, and $\\boldsymbol{\\epsilon}_{1}$ is a unit vector parallel to the plane of scattering. This quantity is needed only for scattering near the forward direction, and you need only provide the term of order $\\theta$. Note that for $\\boldsymbol{\\epsilon}_{1}$ (in the plane of scattering), the small $\\hat{3}$ component of $\\boldsymbol{\\epsilon}_{1}$ also gives a term of order $\\theta$ which must be taken into account.", + "final_answer": [ + "-\\sqrt{E E^{\\prime}} \\frac{E+E^{\\prime}}{|E-E^{\\prime}|} \\theta" + ], + "answer_type": "Expression", + "answer": "It is easy to find that\n\n\\epsilon_{1}^{\\mu}=N(0, p^{\\prime} \\cos \\theta-p, 0,-p^{\\prime} \\sin \\theta), \\quad \\epsilon_{2}^{\\mu}=(0,0,1,0),\n\nwhere $N=(E^{2}+E^{\\prime 2}-2 E E^{\\prime} \\cos \\theta)^{-1 / 2}$ is the normalization constant. Then, for the righthanded electron with spinor $u_{+}(p)=\\sqrt{2 E}(0,0,1,0)^{T}$ and left-handed electron with $u_{-}(p)=$ $\\sqrt{2 E}(0,1,0,0)^{T}$, it is straightforward to show that\n\n\\begin{equation*}\\nu_{+}(p^{\\prime})=\\sqrt{2 E^{\\prime}}(0,0, \\cos \\frac{\\theta}{2}, \\sin \\frac{\\theta}{2})^{T}, \\quad u_{-}(p^{\\prime})=\\sqrt{2 E^{\\prime}}(-\\sin \\frac{\\theta}{2}, \\cos \\frac{\\theta}{2}, 0,0) \\tag{6.15}\n\\end{equation*}\n\nand,\n\n\\begin{align*}\n& \\bar{u}_{ \\pm}(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{1} u_{ \\pm}(p) \\simeq-\\sqrt{E E^{\\prime}} \\frac{E+E^{\\prime}}{|E-E^{\\prime}|} \\theta, \\tag{6.16}\\\\\n& \\bar{u}_{ \\pm}(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{2} u_{ \\pm}(p) \\simeq \\pm \\mathrm{i} \\sqrt{E E^{\\prime}} \\theta \\tag{6.17}\\\\\n& \\bar{u}_{ \\pm}(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{1} u_{\\mp}(p)=\\bar{u}_{ \\pm}(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{2} u_{\\mp}(p)=0 . \\tag{6.18}\n\\end{align*}\n\n\nThat is to say, we have,\n\n\\begin{equation*}\nC_{ \\pm}=-\\sqrt{E E^{\\prime}} \\frac{E+E^{\\prime}}{|E-E^{\\prime}|} \\theta, \\quad \\quad D_{ \\pm}= \\pm \\mathrm{i} \\sqrt{E E^{\\prime}} \\theta \\tag{6.19}\n\\end{equation*}", + "symbol": { + "$p$": "momentum vector of the initial particle", + "$E$": "energy of the initial particle", + "$p^{\\prime}$": "momentum vector of the scattered particle", + "$E^{\\prime}$": "energy of the scattered particle", + "$\\bar{u}_+(p^{\\prime})$": "conjugate spinor of positive helicity for scattered particle", + "$u_+(p)$": "spinor of positive helicity for initial particle", + "$\\boldsymbol{\\gamma}$": "gamma matrices in the Dirac equation", + "$\\boldsymbol{\\epsilon}_{1}$": "unit vector in the plane of scattering", + "$\\boldsymbol{\\epsilon}_{2}$": "unit vector perpendicular to the plane of scattering", + "$\\theta$": "scattering angle", + "$N$": "normalization constant", + "$\\epsilon_{1}^{\\mu}$": "components of the unit vector parallel to the scattering plane", + "$\\epsilon_{2}^{\\mu}$": "components of the unit vector perpendicular to the scattering plane", + "$C_{ \\pm}$": "coefficient related to the scattering amplitude", + "$D_{ \\pm}$": "coefficient related to the imaginary part of the scattering amplitude" + }, + "chapter": "Radiative Corrections: Introduction", + "section": "Equivalent photon approximation" + }, + { + "id": 54, + "topic": "Others", + "question": "Exotic contributions to $\\boldsymbol{g} \\mathbf{- 2}$. Any particle that couples to the electron can produce a correction to the electron-photon form factors and, in particular, a correction to $g-2$. Because the electron $g-2$ agrees with QED to high accuracy, these corrections allow us to constrain the properties of hypothetical new particles. The unified theory of weak and electromagnetic interactions contains a scalar particle $h$ called the Higgs boson, which couples to the electron according to\n\n$$ H_{\\mathrm{int}}=\\int d^{3} x \\frac{\\lambda}{\\sqrt{2}} h \\bar{\\psi} \\psi $$\n\n\nCompute the contribution of a virtual Higgs boson to the electron $(g-2)$, in terms of $\\lambda$ and the mass $m_{h}$ of the Higgs boson.", + "final_answer": [ + "\\frac{\\lambda^{2} m^2}{8 \\pi^{2} m_h^2}[\\log (\\frac{m_h^2}{m^2})-\\frac{7}{6}]" + ], + "answer_type": "Expression", + "answer": "The 1-loop vertex correction from Higgs boson is,\n\n\\begin{align*}\n\\bar{u}(p^{\\prime}) \\delta \\Gamma^{\\mu} u(p) & =(\\frac{\\mathrm{i} \\lambda}{\\sqrt{2}})^{2} \\int \\frac{\\mathrm{~d}^{d} k}{(2 \\pi)^{d}} \\frac{\\mathrm{i}}{(k-p)^{2}-m_{h}^{2}} \\bar{u}(p^{\\prime}) \\frac{\\mathrm{i}}{\\not k+q-m} \\gamma^{\\mu} \\frac{\\mathrm{i}}{\\not k-m} u(p) \\\\\n& =\\frac{\\mathrm{i} \\lambda^{2}}{2} \\int_{0}^{1} \\mathrm{~d} x \\int_{0}^{1-x} \\mathrm{~d} y \\int \\frac{\\mathrm{~d}^{d} k^{\\prime}}{(2 \\pi)^{d}} \\frac{2 \\bar{u}(p^{\\prime}) N^{\\mu} u(p)}{(k^{\\prime 2}-\\Delta)^{3}}, \\tag{6.26}\n\\end{align*}\n\nwith\n\n\\begin{align*}\n& N^{\\mu}=(\\not k+q+m) \\gamma^{\\mu}(k+m), \\tag{6.27}\\\\\n& k^{\\prime}=k-x p+y q, \\tag{6.28}\\\\\n& \\Delta=(1-x) m^{2}+x m_{h}^{2}-x(1-x) p^{2}-y(1-y) q^{2}+2 x y p \\cdot q . \\tag{6.29}\n\\end{align*}\n\n\nTo put this correction into the following form,\n\n\\begin{equation*}\n\\Gamma^{\\mu}=\\gamma^{\\mu} F_{1}(q)+\\frac{i \\sigma^{\\mu \\nu} q_{\\nu}}{2 m} F_{2}(q) \\tag{6.30}\n\\end{equation*}\n\nwe first rewrite $N^{\\mu}$ as,\n\n\\begin{equation*}\nN^{\\mu}=A \\gamma^{\\mu}+B(p^{\\prime}+p)^{\\mu}+C(p^{\\prime}-p)^{\\mu} \\tag{6.31}\n\\end{equation*}\n\nwhere term proportional to $(p^{\\prime}-p)$ can be thrown away by Ward identity $q_{\\mu} \\Gamma^{\\mu}(q)=0$. This can be done by gamma matrix calculations, leading to the following result,\n\n\\begin{equation*}\nN^{\\mu}=[(\\frac{2}{d}-1) k^{\\prime 2}+(3+2 x-x^{2}) m^{2}+(y-x y-y^{2}) q^{2}] \\gamma^{\\mu}+(x^{2}-1) m(p^{\\prime}+p)^{\\mu} . \\tag{6.32}\n\\end{equation*}\n\n\nThen, using Gordon identity, we find,\n\n\\begin{equation*}\nN^{\\mu}=[(\\frac{2}{d}-1) k^{\\prime 2}+(x+1)^{2} m^{2}+(y-y^{2}-x y) q^{2}] \\gamma^{\\mu}+\\frac{\\mathrm{i} \\sigma^{\\mu \\nu} q_{\\nu}}{2 m} \\cdot 2 m^{2}(1-x^{2}) \\tag{6.33}\n\\end{equation*}\n\n\nComparing this with (6.30), we see that\n\n\\begin{align*}\n\\delta F_{2}(q=0) & =2 \\mathrm{i} \\lambda^{2} m^{2} \\int_{0}^{1} \\mathrm{~d} x \\int_{0}^{1-x} \\mathrm{~d} y \\int \\frac{\\mathrm{~d}^{4} k^{\\prime}}{(2 \\pi)^{4}} \\frac{1-x^{2}}{(k^{\\prime 2}-\\Delta)^{3}} \\\\\n& =\\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x \\frac{(1-x)^{2}(1+x)}{(1-x)^{2}+x(m_{h} / m)^{2}} . \\tag{6.34}\n\\end{align*}\n\n\nTo carry out the integration over $x$, we use the approximation that $m_{h} \\gg m$. Then,\n\n\\begin{align*}\n\\delta F_{2}(q=0) & \\simeq \\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x[\\frac{1}{1+x(m_{h} / m)^{2}}-\\frac{1+x-x^{2}}{(m_{h} / m)^{2}}] \\\\\n& \\simeq \\frac{\\lambda^{2}}{(4 \\pi)^{2}(m_{h} / m)^{2}}[\\log (m_{h}^{2} / m^{2})-\\frac{7}{6}] . \\tag{6.35}\n\\end{align*}", + "symbol": { + "$h$": "Higgs boson", + "$m_{h}$": "mass of the Higgs boson", + "$\\lambda$": "coupling constant", + "$m$": "mass of the electron" + }, + "chapter": "Radiative Corrections: Introduction", + "section": "Exotic contributions to $g-2$" + }, + { + "id": 55, + "topic": "Others", + "question": "Exotic contributions to $\\boldsymbol{g} \\mathbf{- 2}$. Any particle that couples to the electron can produce a correction to the electron-photon form factors and, in particular, a correction to $g-2$. Because the electron $g-2$ agrees with QED to high accuracy, these corrections allow us to constrain the properties of hypothetical new particles. Some more complex versions of this theory contain a pseudoscalar particle called the axion, which couples to the electron according to\n\n$$ H_{\\mathrm{int}}=\\int d^{3} x \\frac{i \\lambda}{\\sqrt{2}} a \\bar{\\psi} \\gamma^{5} \\psi $$\n\n\nThe axion may be as light as the electron, or lighter, and may couple more strongly than the Higgs boson. Compute the contribution of a virtual axion to the $g-2$ of the electron, and work out the excluded values of $\\lambda$ and $m_{a}$.", + "final_answer": [ + "-\\frac{\\lambda^{2}}{(4 \\pi)^{2}(m_{a}^{2} / m^{2})}[\\log (m_{a}^{2} / m^{2})-\\frac{11}{6}]" + ], + "answer_type": "Expression", + "answer": "The 1-loop correction from the axion is given by,\n\n\\begin{align*}\n\\bar{u}(p^{\\prime}) \\delta \\Gamma^{\\mu} u(p) & =(\\frac{-\\lambda}{\\sqrt{2}})^{2} \\int \\frac{\\mathrm{~d}^{d} k}{(2 \\pi)^{d}} \\frac{\\mathrm{i}}{(k-p)^{2}-m_{h}^{2}} \\bar{u}(p^{\\prime}) \\gamma^{5} \\frac{\\mathrm{i}}{\\not k+\\not q-m} \\gamma^{\\mu} \\frac{\\mathrm{i}}{\\nVdash-m} \\gamma^{5} u(p) \\\\\n& =-\\frac{\\mathrm{i} \\lambda^{2}}{2} \\int_{0}^{1} \\mathrm{~d} x \\int_{0}^{1-x} \\frac{\\mathrm{~d} y \\int \\frac{\\mathrm{~d}^{d} k^{\\prime}}{(2 \\pi)^{d}} \\frac{2 \\bar{u}(p^{\\prime}) N^{\\mu} u(p)}{(k^{\\prime 2}-\\Delta)^{3}},}{} \\tag{6.37}\n\\end{align*}\n\nin which $k^{\\prime}$ and $\\Delta$ are still defined as in (a) except the replacement $m_{h} \\rightarrow m_{a}$, while $N^{\\mu}$ is now given by,\n\n\\begin{equation*}\nN^{\\mu}=\\gamma^{5}(\\not k+q q+m) \\gamma^{\\mu}(\\not k+m) \\gamma^{5}=-(\\nless q+q-m) \\gamma^{\\mu}(\\not k-m) . \\tag{6.38}\n\\end{equation*}\n\n\nRepeating the same derivation as was done in (a), we get,\n\n\\begin{equation*}\nN^{\\mu}=[-(\\frac{2}{d}-1) k^{\\prime 2}-(1-x-y) y q^{2}+(1-x)^{2} m^{2}] \\gamma^{\\mu}-(1-x)^{2} m(p^{\\prime}+p)^{2} . \\tag{6.39}\n\\end{equation*}\n\n\nAgain, using Gordon identity, we get,\n\n\\begin{align*}\n\\delta F_{2}(q=0) & =-2 \\mathrm{i} \\lambda^{2} m^{2} \\int_{0}^{1} \\mathrm{~d} x \\int_{0}^{1-x} \\mathrm{~d} y \\int \\frac{\\mathrm{~d}^{4} k^{\\prime}}{(2 \\pi)^{4}} \\frac{(1-x)^{2}}{(k^{\\prime 2}-\\Delta)^{3}} \\\\\n& =-\\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x \\frac{(1-x)^{3}}{(1-x)^{2}+x m_{a}^{2} / m^{2}} \\\\\n& \\simeq-\\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x[\\frac{1}{1+x m_{a}^{2} / m^{2}}-\\frac{3-3 x+x^{2}}{m_{a}^{2} / m^{2}}] \\\\\n& =-\\frac{\\lambda^{2}}{(4 \\pi)^{2}(m_{a}^{2} / m^{2})}[\\log (m_{a}^{2} / m^{2})-\\frac{11}{6}] . \\tag{6.40}\n\\end{align*}\n\n\nFor order-of-magnitude estimation, it's easy to see that $\\lambda m / m_{a} \\gtrsim 10^{-5}$ is excluded.", + "symbol": { + "$\\lambda$": "axion-electron coupling constant", + "$a$": "axion field", + "$\\psi$": "electron field", + "$\\gamma^{5}$": "gamma matrix in Dirac theory", + "$m$": "electron mass", + "$m_{a}$": "axion mass", + "$p$": "initial momentum of the electron", + "$p^{\\prime}$": "final momentum of the electron", + "$q$": "momentum transfer in the process", + "$k$": "loop momentum", + "$k^{\\prime}$": "shifted loop momentum", + "$\\Delta$": "parameter related to the shifted loop momentum" + }, + "chapter": "Radiative Corrections: Introduction", + "section": "Exotic contributions to $g-2$" + }, + { + "id": 56, + "topic": "Others", + "question": "Although we have discussed QED radiative corrections at length in the last two chapters, so far we have made no attempt to compute a full radiatively corrected cross section. The reason is of course that such calculations are quite lengthy. Nevertheless it would be dishonest to pretend that one understands radiative corrections after computing only isolated effects as we have done. This \"final project\" is an attempt to remedy this situation. The project is the computation of one of the simplest, but most important, radiatively corrected cross sections. \n\nStrongly interacting particles-pions, kaons, and protons-are produced in $e^{+} e^{-}$annihilation when the virtual photon creates a pair of quarks. If one ignores the effects of the strong interactions, it is easy to calculate the total cross section for quark pair production. In this final project, we will analyze the first corrections to this formula due to the strong interactions.\n\nLet us represent the strong interactions by the following simple model: Introduce a new massless vector particle, the gluon, which couples universally to quarks:\n\n$$\\Delta H=\\int d^{3} x g \\bar{\\psi}_{f i} \\gamma^{\\mu} \\psi_{f i} B_{\\mu} $$\n\n\nHere $f$ labels the type (\"flavor\") of the quark ( $u, d, s, c$, etc.) and $i=1,2,3$ labels the color. The strong coupling constant $g$ is independent of flavor and color. The electromagnetic coupling of quarks depends on the flavor, since the $u$ and $c$ quarks have charge $Q_{f}=+2 / 3$ while the $d$ and $s$ quarks have charge $Q_{f}=-1 / 3$. By analogy to $\\alpha$, let us define\n\n\\alpha_{g}=\\frac{g^{2}}{4 \\pi}\n\n\nIn this exercise, we will compute the radiative corrections to quark pair production proportional to $\\alpha_{g}$.\n\nThis model of the strong interactions of quarks does not quite agree with the currently accepted theory of the strong interactions, quantum chromodynamics (QCD). However, all of the results that we will derive here are also\ncorrect in QCD with the replacement\n\n$$ \\alpha_{g} \\rightarrow \\frac{4}{3} \\alpha_{s} . $$\n\n\nThroughout this exercise, you may ignore the masses of quarks. You may also ignore the mass of the electron, and average over electron and positron polarizations. To control infrared divergences, it will be necessary to assume that the gluons have a small nonzero mass $\\mu$, which can be taken to zero only at the end of the calculation. However (as we discussed in Problem 5.5), it is consistent to sum over polarization states of this massive boson by the replacement:\n\n\\sum \\epsilon^{\\mu} \\epsilon^{\\nu *} \\rightarrow-g^{\\mu \\nu}\n\nthis also implies that we may use the propagator\n\n\\widehat{B^{\\mu} B^{\\nu}}=\\frac{-i g^{\\mu \\nu}}{k^{2}-\\mu^{2}+i \\epsilon} Draw the Feynman diagrams for the process $e^{+} e^{-} \\rightarrow \\bar{q} q g$, to leading order in $\\alpha$ and $\\alpha_{g}$, and compute the differential cross section. You may throw away the information concerning the correlation between the initial beam axis and the directions of the final particles. This is conveniently done as follows: The usual trace tricks for evaluating the square of the matrix element give for this process a result of the structure\n\n\\int d \\Pi_{3} \\frac{1}{4} \\sum|\\mathcal{M}|^{2}=L_{\\mu \\nu} \\int d \\Pi_{3} H^{\\mu \\nu}\n\nwhere $L_{\\mu \\nu}$ represents the electron trace and $H^{\\mu \\nu}$ represents the quark trace. If we integrate over all parameters of the final state except $x_{1}$ and $x_{2}$, which are scalars, the only preferred 4 -vector characterizing the final state is $q^{\\mu}$. On the other hand, $H_{\\mu \\nu}$ satisfies\n\nq^{\\mu} H_{\\mu \\nu}=H_{\\mu \\nu} q^{\\nu}=0\n\n\nWhy is this true? (There is an argument based on general principles; however, you might find it a useful check on your calculation to verify this property explicitly.) Since, after integrating over final-state vectors, $\\int H^{\\mu \\nu}$ depends only on $q^{\\mu}$ and scalars, it can only have the form\n\n$$ \\int d \\Pi_{3} H^{\\mu \\nu}=(g^{\\mu \\nu}-\\frac{q^{\\mu} q^{\\nu}}{q^{2}}) \\cdot H $$\n\nwhere $H$ is a scalar. With this information, show that\n\n$$ L_{\\mu \\nu} \\int d \\Pi_{3} H^{\\mu \\nu}=\\frac{1}{3}(g^{\\mu \\nu} L_{\\mu \\nu}) \\cdot \\int d \\Pi_{3}(g^{\\rho \\sigma} H_{\\rho \\sigma})$$\n\n\nUsing this trick, derive the differential cross section\n\n$$ \\frac{d \\sigma}{d x_{1} d x_{2}}(e^{+} e^{-} \\rightarrow \\bar{q} q g)=\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi} \\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}$$\n\nin the limit $\\mu \\rightarrow 0$. If we assume that each original final-state particle is realized physically as a jet of strongly interacting particles, this formula gives the probability for observing three-jet events in $e^{+} e^{-}$annihilation and the kinematic distribution of these events. The form of the distribution in the $x_{i}$ is an absolute prediction, and it agrees with experiment. The\nnormalization of this distribution is a measure of the strong-interaction coupling constant.", + "final_answer": [ + "\\frac{d \\sigma}{d x_{1} d x_{2}}(e^{+} e^{-} \\rightarrow \\bar{q} q g)=\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi} \\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}" + ], + "answer_type": "Expression", + "answer": "Now we calculate the differential cross section for the process $e^{+} e^{-} \\rightarrow q \\bar{g} g$ to lowest order in $\\alpha$ and $\\alpha_{g}$. First, the amplitude is\n\n\\begin{equation*}\n\\mathrm{i} \\mathcal{M}=Q_{f}(-\\mathrm{i} e)^{2}(-\\mathrm{i} g) \\epsilon_{\\nu}^{*}(k_{3}) \\bar{u}(k_{1})[\\gamma^{\\nu} \\frac{\\mathrm{i}}{\\not k_{1}+\\not k_{3}} \\gamma^{\\mu}-\\gamma^{\\mu} \\frac{\\mathrm{i}}{\\not k_{2}+\\not k_{3}} \\gamma^{\\nu}] v(k_{2}) \\frac{-\\mathrm{i}}{q^{2}} \\bar{v}(p_{2}) \\gamma_{\\mu} u(p_{1}) . \\tag{7.54}\n\\end{equation*}\n\n\nThen, the squared amplitude is\n\n\\begin{align*}\n\\frac{1}{4} \\sum|\\mathrm{i} \\mathcal{M}|^{2}= & \\frac{Q_{f}^{2} g^{2} e^{4}}{4 s^{2}}(-g_{\\nu \\sigma}) \\operatorname{tr}(\\gamma_{\\mu} \\not p_{1} \\gamma_{\\rho} \\not \\not_{2}) \\\\\n& \\times \\operatorname{tr}[(\\gamma^{\\nu} \\frac{1}{\\not k_{1}+\\not k_{3}} \\gamma^{\\mu}-\\gamma^{\\mu} \\frac{1}{\\not k_{2}+\\not k_{3}} \\gamma^{\\nu}) \\not k_{2}(\\gamma^{\\rho} \\frac{1}{\\not k_{1}+\\not k_{3}} \\gamma^{\\sigma}-\\gamma^{\\sigma} \\frac{1}{\\not k_{2}+\\not k_{3}} \\gamma^{\\rho}) \\not k_{1}] \\\\\n= & \\frac{4 Q_{f}^{2} g^{2} e^{4}}{3 s^{2}}(8 p_{1} \\cdot p_{2})[\\frac{4(k_{1} \\cdot k_{2})(k_{1} \\cdot k_{2}+q \\cdot k_{3})}{(k_{1}+k_{3})^{2}(k_{2}+k_{3})^{2}} \\\\\n&+(\\frac{1}{(k_{1}+k_{3})^{4}}+\\frac{1}{(k_{2}+k_{3})^{4}})(2(k_{1} \\cdot k_{3})(k_{2} \\cdot k_{3})-\\mu^{2}(k_{1} \\cdot k_{2}))] . \\tag{7.55}\n\\end{align*}\n\n\nWe have used the trick described in Peskin's book (P261) when getting through the last equal sign. Now rewrite the quantities of final-state kinematics in terms of $x_{i}$, and set $\\mu \\rightarrow 0$, we obtain\n\n\\begin{align*}\n\\frac{1}{4} \\sum|\\mathrm{i} \\mathcal{M}|^{2} & =\\frac{2 Q_{f}^{2} g^{2} e^{4}}{3 s^{2}}(8 p_{1} \\cdot p_{2})[\\frac{2(1-x_{3})}{(1-x_{1})(1-x_{2})}+\\frac{1-x_{1}}{1-x_{2}}+\\frac{1-x_{2}}{1-x_{1}}] \\\\\n& =\\frac{8 Q_{f}^{2} g^{2} e^{4}}{3 s} \\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})} . \\tag{7.56}\n\\end{align*}\n\n\nThus the differential cross section, with 3 colors counted, reads\n\n\\begin{align*}\n\\frac{\\mathrm{d} \\sigma}{\\mathrm{~d} x_{1} \\mathrm{~d} x_{2}}|_{\\mathrm{COM}} & =\\frac{1}{2 E_{\\mathbf{p}_{1}} 2 E_{\\mathbf{p}_{2}}|v_{\\mathbf{p}_{1}}-v_{\\mathbf{p}_{2}}|} \\frac{s}{128 \\pi^{3}}(\\frac{1}{4} \\sum|\\mathcal{M}|^{2}) \\\\\n& =\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi} \\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})} \\tag{7.57}\n\\end{align*}\n\nwhere we have used the fact that the initial electron and positron are massless, which implies that $2 E_{\\mathbf{p}_{1}}=2 E_{\\mathbf{p}_{2}}=\\sqrt{s}$ and $|v_{\\mathbf{p}_{1}}-v_{\\mathbf{p}_{2}}|=2$ in COM frame.", + "symbol": { + "$e$": "electron", + "$\\alpha$": "fine-structure constant", + "$\\alpha_{g}$": "strong-interaction coupling constant", + "$q$": "quark four-momentum", + "$L_{\\mu \\nu}$": "electron trace", + "$H^{\\mu \\nu}$": "quark trace", + "$s$": "center-of-mass energy squared", + "$Q_{f}$": "quark charge factor", + "$x_{1}$": "final state scalar 1", + "$x_{2}$": "final state scalar 2", + "$p_{1}$": "electron four-momentum", + "$p_{2}$": "positron four-momentum", + "$k_{1}$": "quark four-momentum", + "$k_{2}$": "anti-quark four-momentum", + "$k_{3}$": "gluon four-momentum", + "$\\mu$": "small parameter tending to zero" + }, + "chapter": "Final Project I", + "section": "Radiation of Gluon Jets" + }, + { + "id": 57, + "topic": "Others", + "question": "Although we have discussed QED radiative corrections at length in the last two chapters, so far we have made no attempt to compute a full radiatively corrected cross section. The reason is of course that such calculations are quite lengthy. Nevertheless it would be dishonest to pretend that one understands radiative corrections after computing only isolated effects as we have done. This \"final project\" is an attempt to remedy this situation. The project is the computation of one of the simplest, but most important, radiatively corrected cross sections. \n\nStrongly interacting particles-pions, kaons, and protons-are produced in $e^{+} e^{-}$annihilation when the virtual photon creates a pair of quarks. If one ignores the effects of the strong interactions, it is easy to calculate the total cross section for quark pair production. In this final project, we will analyze the first corrections to this formula due to the strong interactions.\n\nLet us represent the strong interactions by the following simple model: Introduce a new massless vector particle, the gluon, which couples universally to quarks:\n\n$$\\Delta H=\\int d^{3} x g \\bar{\\psi}_{f i} \\gamma^{\\mu} \\psi_{f i} B_{\\mu}$$\n\n\nHere $f$ labels the type (\"flavor\") of the quark ( $u, d, s, c$, etc.) and $i=1,2,3$ labels the color. The strong coupling constant $g$ is independent of flavor and color. The electromagnetic coupling of quarks depends on the flavor, since the $u$ and $c$ quarks have charge $Q_{f}=+2 / 3$ while the $d$ and $s$ quarks have charge $Q_{f}=-1 / 3$. By analogy to $\\alpha$, let us define\n\n$$\\alpha_{g}=\\frac{g^{2}}{4 \\pi}$$\n\n\nIn this exercise, we will compute the radiative corrections to quark pair production proportional to $\\alpha_{g}$.\n\nThis model of the strong interactions of quarks does not quite agree with the currently accepted theory of the strong interactions, quantum chromodynamics (QCD). However, all of the results that we will derive here are also\ncorrect in QCD with the replacement\n\n$$\\alpha_{g} \\rightarrow \\frac{4}{3} \\alpha_{s} .$$\n\nThroughout this exercise, you may ignore the masses of quarks. You may also ignore the mass of the electron, and average over electron and positron polarizations. To control infrared divergences, it will be necessary to assume that the gluons have a small nonzero mass $\\mu$, which can be taken to zero only at the end of the calculation. However (as we discussed in Problem 5.5), it is consistent to sum over polarization states of this massive boson by the replacement:\n\n\\sum \\epsilon^{\\mu} \\epsilon^{\\nu *} \\rightarrow-g^{\\mu \\nu}\n\nthis also implies that we may use the propagator\n\n\\widehat{B^{\\mu} B^{\\nu}}=\\frac{-i g^{\\mu \\nu}}{k^{2}-\\mu^{2}+i \\epsilon} Now replace $\\mu \\neq 0$ in the formula of the differential cross section, and carefully integrate over the region. You may assume $\\mu^{2} \\ll q^{2}$. In this limit, you will find infrared-divergent terms of order $\\log (q^{2} / \\mu^{2})$ and also $\\log ^{2}(q^{2} / \\mu^{2})$, finite terms of order 1 , and terms explicitly suppressed by powers of $(\\mu^{2} / q^{2})$. You may drop terms of the last type throughout this calculation. For the moment, collect and evaluate only the infrared-divergent terms.", + "final_answer": [ + "\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi}[\\log ^{2} \\frac{\\mu^{2}}{s}+3 \\log \\frac{\\mu^{2}}{s}]" + ], + "answer_type": "Expression", + "answer": "Now we reevaluate the averaged squared amplitude, with $\\mu$ kept nonzero in the formula\n\\begin{align*}\n\\frac{1}{4} \\sum|\\mathrm{i} \\mathcal{M}|^{2}= & \\frac{4 Q_{f}^{2} g^{2} e^{4}}{3 s^{2}}(8 p_{1} \\cdot p_{2})[\\frac{4(k_{1} \\cdot k_{2})(k_{1} \\cdot k_{2}+q \\cdot k_{3})}{(k_{1}+k_{3})^{2}(k_{2}+k_{3})^{2}} \\\\\n&+(\\frac{1}{(k_{1}+k_{3})^{4}}+\\frac{1}{(k_{2}+k_{3})^{4}})(2(k_{1} \\cdot k_{3})(k_{2} \\cdot k_{3})-\\mu^{2}(k_{1} \\cdot k_{2}))].\n\\end{align*}\n\nThe result is\n\n\\begin{equation*}\n\\frac{1}{4} \\sum|\\mathrm{i} \\mathcal{M}|^{2}=\\frac{8 Q_{f}^{2} g^{2} e^{4}}{3 s} F(x_{1}, x_{2}, \\mu^{2} / s) \\tag{7.58}\n\\end{equation*}\n\nwhere\n\n\\begin{align*}\nF(x_{1}, x_{2}, \\frac{\\mu^{2}}{s})= & \\frac{2(x_{1}+x_{2}-1+\\frac{\\mu^{2}}{s})(1+\\frac{\\mu^{2}}{s})}{(1-x_{1})(1-x_{2})} \\\\\n& +[\\frac{1}{(1-x_{1})^{2}}+\\frac{1}{(1-x_{2})^{2}}]((1-x_{1})(1-x_{2})-\\frac{\\mu^{2}}{s}) \\tag{7.59}\n\\end{align*}\n\n\nThe cross section, then, can be got by integrating over $\\mathrm{d} x_{1} \\mathrm{~d} x_{2}$ :\n\n\\begin{align*}\n\\sigma(e^{+} e^{-} \\rightarrow q \\bar{q} g) & =\\frac{1}{2 E_{\\mathbf{p}_{1}} 2 E_{\\mathbf{p}_{2}}|v_{\\mathbf{p}_{1}}-v_{\\mathbf{p}_{2}}|} \\frac{s}{128 \\pi^{3}} \\int \\mathrm{~d} x_{1} \\mathrm{~d} x_{2}(\\frac{1}{4} \\sum|\\mathcal{M}|^{2}) \\\\\n& =\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi} \\int_{0}^{1-\\frac{\\mu^{2}}{s}} \\frac{\\mathrm{~d} x_{1} \\int_{1-x_{1}-\\frac{\\mu^{2}}{s}}^{1-\\frac{t}{s(1-x_{1})}} \\mathrm{d} x_{2} F(x_{1}, x_{2}, \\frac{\\mu^{2}}{s})}{} \\\\\n& =\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi}[\\log ^{2} \\frac{\\mu^{2}}{s}+3 \\log \\frac{\\mu^{2}}{s}+5-\\frac{1}{3} \\pi^{2}+\\mathcal{O}(\\mu^{2})] \\tag{7.60}\n\\end{align*}", + "symbol": { + "$\\mu$": "mass scale parameter", + "$q$": "momentum transfer", + "$Q_{f}$": "quark charge factor", + "$g$": "strong coupling constant", + "$e$": "electric charge", + "$s$": "Mandelstam variable, center-of-mass energy squared", + "$x_{1}$": "energy fraction variable (1)", + "$x_{2}$": "energy fraction variable (2)", + "$E_{\\mathbf{p}_{1}}$": "energy of particle 1", + "$E_{\\mathbf{p}_{2}}$": "energy of particle 2", + "$v_{\\mathbf{p}_{1}}$": "velocity of particle 1", + "$v_{\\mathbf{p}_{2}}$": "velocity of particle 2", + "$\\alpha$": "fine structure constant", + "$\\alpha_{g}$": "strong coupling constant alpha", + "$\\mathcal{M}$": "amplitude", + "$F(x_{1}, x_{2}, \\frac{\\mu^{2}}{s})$": "dimensionless function related to energy fraction variables", + "$t$": "momentum transfer squared" + }, + "chapter": "Final Project I", + "section": "Radiation of Gluon Jets" + }, + { + "id": 58, + "topic": "Others", + "question": "Scalar QED. This problem concerns the theory of a complex scalar field $\\phi$ interacting with the electromagnetic field $A^{\\mu}$. The Lagrangian is\n\n$$\\mathcal{L}=-\\frac{1}{4} F_{\\mu \\nu}^{2}+(D_{\\mu} \\phi)^{*}(D^{\\mu} \\phi)-m^{2} \\phi^{*} \\phi$$\n\nwhere $D_{\\mu}=\\partial_{\\mu}+i e A_{\\mu}$ is the usual gauge-covariant derivative. Compute, to lowest order, the differential cross section for $e^{+} e^{-} \\rightarrow \\phi \\phi^{*}$. Ignore the electron mass (but not the scalar particle's mass), and average over the electron and positron polarizations. Find the asymptotic angular dependence and total cross section. Compare your results to the corresponding formulae for $e^{+} e^{-} \\rightarrow \\mu^{+} \\mu^{-}$.\n\nCompute the contribution of the charged scalar to the photon vacuum polarization, using dimensional regularization. Note that there are two diagrams. To put the answer into the expected form,\n\n$$\\Pi^{\\mu \\nu}(q^{2})=(g^{\\mu \\nu} q^{2}-q^{\\mu} q^{\\nu}) \\Pi(q^{2})$$\n\nit is useful to add the two diagrams at the beginning, putting both terms over a common denominator before introducing a Feynman parameter. Show that, for $-q^{2} \\gg m^{2}$, the charged boson contribution to $\\Pi(q^{2})$ is exactly $1 / 4$ that of a virtual electron-positron pair.", + "final_answer": [ + "(\\frac{d\\sigma}{d\\Omega})_{CM} = \\frac{\\alpha^2}{8s} (1 - \\frac{m^2}{E^2})^{3/2} \\sin^2{\\theta}" + ], + "answer_type": "Expression", + "answer": "Now we calculate the spin-averaged differential cross section for the process $e^{+} e^{-} \\rightarrow$ $\\phi^{*} \\phi$. The scattering amplitude is given by\n\n\\begin{equation*}\n\\mathrm{i} \\mathcal{M}=(-\\mathrm{i} e)^{2} \\bar{v}(k_{2}) \\gamma^{\\mu} u(k_{1}) \\frac{-\\mathrm{i}}{s}(p_{1}-p_{2})_{\\mu} . \\tag{9.3}\n\\end{equation*}\n\n\nThen the spin-averaged and squared amplitude is\n\n\\frac{1}{4} \\sum_{\\text {spins }}|\\mathcal{M}|^{2}=\\frac{e^{4}}{4 s^{2}} \\operatorname{tr}[(\\not p_{1}-\\not p_{2}) \\not k_{1}(\\not p_{1}-\\not p_{2}) \\not k_{2}]\n\n\n\\begin{equation*}\n=\\frac{e^{4}}{4 s^{2}}[8(k_{1} \\cdot p_{1}-k_{1} \\cdot p_{2})(k_{2} \\cdot p_{1}-k_{2} \\cdot p_{2})-4(k_{1} \\cdot k_{2})(p_{1}-p_{2})^{2}] . \\tag{9.4}\n\\end{equation*}\n\n\nWe may parameterize the momenta as\n\n\\begin{array}{ll}\nk_{1}=(E, 0,0, E), & p_{1}=(E, p \\sin \\theta, 0, p \\cos \\theta), \\\\\nk_{2}=(E, 0,0,-E), & p_{2}=(E,-p \\sin \\theta, 0,-p \\cos \\theta),\n\\end{array}\n\nwith $p=\\sqrt{E^{2}-m^{2}}$. Then we have\n\n\\begin{equation*}\n\\frac{1}{4} \\sum_{\\text {spins }}|\\mathcal{M}|^{2}=\\frac{e^{4} p^{2}}{2 E^{2}} \\sin ^{2} \\theta \\tag{9.5}\n\\end{equation*}\n\n\nThus the differential cross section is:\n\n\\begin{equation*}\n(\\frac{\\mathrm{d} \\sigma}{\\mathrm{~d} \\Omega})_{\\mathrm{CM}}=\\frac{1}{2(2 E)^{2}} \\frac{p}{8(2 \\pi)^{2} E}(\\frac{1}{4} \\sum|\\mathcal{M}|^{2})=\\frac{\\alpha^{2}}{8 s}(1-\\frac{m^{2}}{E^{2}})^{3 / 2} \\sin ^{2} \\theta \\tag{9.6}\n\\end{equation*}", + "symbol": { + "$e^{+}$": "positron", + "$e^{-}$": "electron", + "$\\phi$": "scalar particle", + "$\\phi^{*}$": "conjugate scalar particle", + "$m$": "scalar particle mass", + "$\\Pi^{\\mu \\nu}$": "photon polarization tensor", + "$q$": "momentum transfer", + "$\\Pi(q^{2})$": "polarization function", + "$s$": "center of mass energy squared", + "$\\mathcal{M}$": "scattering amplitude", + "$e$": "elementary charge", + "$k_{1}$": "momentum of incoming electron", + "$k_{2}$": "momentum of incoming positron", + "$p_{1}$": "momentum of outgoing scalar particle", + "$p_{2}$": "momentum of outgoing conjugate scalar particle", + "$E$": "energy of incoming particles", + "$p$": "three-momentum magnitude of outgoing particles", + "$\\sigma$": "differential cross section", + "$\\alpha$": "fine-structure constant", + "$\\theta$": "scattering angle" + }, + "chapter": "Functional Methods", + "section": "Scalar QED" + }, + { + "id": 59, + "topic": "Others", + "question": "We discussed the effective potential for an $O(N)$-symmetric $\\phi^{4}$ theory in four dimensions. We computed the perturbative corrections to this effective potential, and used the renormalization group to clarify the behavior of the potential for small values of the scalar field mass. After all this work, however, we found that the qualitative dependence of the theory on the mass parameter was unchanged by perturbative corrections. The theory still possessed a second-order phase transition as a function of the mass. The loop corrections affected this picture only in providing some logarithmic corrections to the scaling behavior near the phase transition.\n\nHowever, loop corrections are not always so innocuous. For some systems, they can change the structure of the phase transition qualitatively. This Final Project treats the simplest example of such a system, the Coleman-Weinberg model. The analysis of this model draws on a broad variety of topics discussed in Part II; it provides a quite nontrivial application of the effective potential formalism and the use of the renormalization group equation. The phenomenon displayed in this exercise reappears in many contexts, from displacive phase transitions in solids to the thermodynamics of the early universe.\n\nThis problem makes use of material in starred sections of the book Peskin \\& Schroeder, in particular, Sections 11.3, 11.4, and 13.2. Parts (a) and (e), however, depend only on the unstarred material of Part II. We recommend part (e) as excellent practice in the computation of renormalization group functions.\n\nThe Coleman-Weinberg model is the quantum electrodynamics of a scalar field in four dimensions, considered for small values of the scalar field mass. The Lagrangian is\n\n$$\\mathcal{L}=-\\frac{1}{4}(F_{\\mu \\nu})^{2}+(D_{\\mu} \\phi)^{\\dagger} D^{\\mu} \\phi-m^{2} \\phi^{\\dagger} \\phi-\\frac{\\lambda}{6}(\\phi^{\\dagger} \\phi)^{2},$$\n\nwhere $\\phi(x)$ is a complex-valued scalar field and $D_{\\mu} \\phi=(\\partial_{\\mu}+i e A_{\\mu}) \\phi$. Working in Landau gauge ( $\\partial^{\\mu} A_{\\mu}=0$ ), compute the one-loop correction to the effective potential $V(\\phi_{\\mathrm{cl}})$. Show that it is renormalized by counterterms for $m^{2}$ and $\\lambda$. Renormalize by minimal subtraction, introducing a renormalization scale $M$.", + "final_answer": [ + " V_{\\mathrm{eff}}[\\phi_{\\mathrm{cl}}]= m^{2} \\phi_{\\mathrm{cl}}^{2}+\\frac{\\lambda}{6} \\phi_{\\mathrm{cl}}^{4}-\\frac{1}{4(4 \\pi)^{2}}[3(2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{2 e^{2} \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})+(m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})+(m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})]" + ], + "answer_type": "Expression", + "answer": "Now we calculate the 1-loop effective potential of the model. We know that 1-loop correction of the effective Lagrangian is given by,\n\n\\begin{equation*}\n\\Delta \\mathcal{L}=\\frac{\\mathrm{i}}{2} \\log \\operatorname{det}[-\\frac{\\delta^{2} \\mathcal{L}}{\\delta \\varphi \\delta \\varphi}]_{\\varphi=0}+\\delta \\mathcal{L} \\tag{13.26}\n\\end{equation*}\n\nwhere $\\varphi$ is the fluctuating fields and $\\delta \\mathcal{L}$ denotes counterterms.\nLet the background value of the complex scalar be $\\phi_{\\mathrm{cl}}$. By the assumption of Poincaré symmetry, $\\phi_{\\mathrm{cl}}$ must be a constant. For the same reason, the background value of the vector field $A_{\\mu}$ must vanish. In addition, we can set $\\phi_{\\mathrm{cl}}$ to be real without loss of generality. Then we have,\n\n\\phi(x)=\\phi_{\\mathrm{cl}}+\\varphi_{1}(x)+\\mathrm{i} \\varphi_{2}(x),\n\nwhere $\\varphi_{1}(x), \\varphi_{2}(x)$, together with $A_{\\mu}(x)$, now serve as fluctuating fields. Expanding the Lagrangian around the background fields and keeping terms quadratic in fluctuating fields only, we get,\n\n\\begin{align*}\n\\mathcal{L}= & -\\frac{1}{2} F_{\\mu \\nu} F^{\\mu \\nu}+|(\\partial_{\\mu}+\\mathrm{i} e A_{\\mu})(\\phi_{\\mathrm{cl}}+\\varphi_{1}+\\mathrm{i} \\varphi_{2})|^{2} \\\\\n& -m^{2}|\\phi_{\\mathrm{cl}}+\\varphi_{1}+\\mathrm{i} \\varphi_{2}|^{2}-\\frac{\\lambda}{6}|\\phi_{\\mathrm{cl}}+\\varphi_{1}+\\mathrm{i} \\varphi_{2}|^{4} \\\\\n= & \\frac{1}{2} A_{\\mu}[g^{\\mu \\nu}(\\partial^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})-\\partial^{\\mu} \\partial^{\\nu}] A_{\\nu}+\\frac{1}{2} \\varphi_{1}(-\\partial^{2}-m^{2}-\\lambda \\phi_{\\mathrm{cl}}^{2}) \\varphi_{1} \\\\\n& +\\frac{1}{2} \\varphi_{2}(-\\partial^{2}-m^{2}-\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2}) \\varphi_{2}-2 e \\phi_{\\mathrm{cl}} A_{\\mu} \\partial^{\\mu} \\varphi_{2}+\\cdots, \\tag{13.27}\n\\end{align*}\n\nwhere \". . .\" denotes terms other than being quadratic in fluctuating fields. Now we impose the Landau gauge condition $\\partial_{\\mu} A^{\\mu}=0$ to the Lagrangian, which removes the off-diagonal term $-2 e \\phi_{\\mathrm{cl}} A_{\\mu} \\partial^{\\mu} \\varphi_{2}$. Then, according to (13.26), the 1-loop effective Lagrangian can be evaluated as,\n\n\\begin{align*}\n\\frac{\\mathrm{i}}{2} \\log \\operatorname{det}[- & \\frac{\\delta^{2} \\mathcal{L}}{\\delta \\varphi \\delta \\varphi}]_{\\varphi=0}=\\frac{\\mathrm{i}}{2}[\\log \\operatorname{det}(-\\eta^{\\mu \\nu}(\\partial^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})+\\partial^{\\mu} \\partial^{\\nu}) \\\\\n& +\\log \\operatorname{det}(\\partial^{2}+m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})+\\log \\operatorname{det}(\\partial^{2}+m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})] \\\\\n= & \\frac{\\mathrm{i}}{2}\n\\end{align*} \\quad \\frac{\\mathrm{~d}^{d} k}{(2 \\pi)^{d}}[\\operatorname{tr} \\log (-k^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{3} .\n\n\nIn the second equality we use the following identity,\n\n\\begin{equation*}\n\\operatorname{det}(\\lambda I+A B)=\\lambda^{n-1}(\\lambda+B A) \\tag{13.29}\n\\end{equation*}\n\nwhere $A$ and $B$ are matrices of $n \\times 1$ and $1 \\times n$, respectively, $\\lambda$ is an arbitrary complex number and $I$ is the $n \\times n$ identity matrix. In our case, this gives,\n\n\\begin{equation*}\n\\operatorname{det}(-\\eta^{\\mu \\nu}(\\partial^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})+\\partial^{\\mu} \\partial^{\\nu})=-2 e^{2} \\phi_{\\mathrm{cl}}^{2}(\\partial^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{3} \\tag{13.30}\n\\end{equation*}\n\n\nThen the second equality follows up to an irrelevant constant term. The third equality makes use of the trick in (11.72) of P\\&S. Then, for $d=4-\\epsilon$ and $\\epsilon \\rightarrow 0$, we have,\n\n\\frac{\\mathrm{i}}{2} \\log \\operatorname{det}[-\\frac{\\delta^{2} \\mathcal{L}}{\\delta \\varphi \\delta \\varphi}]_{\\varphi=0}=\\frac{1}{4(4 \\pi)^{2}}[3(2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}(\\Delta-\\log (2 e^{2} \\phi_{\\mathrm{cl}}^{2}))\n\n\n\\begin{align*}\n& +(m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})^{2}(\\Delta-\\log (m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})) \\\\\n& +(m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})^{2}(\\Delta-\\log (m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2}))], \\tag{13.31}\n\\end{align*}\n\nwhere we define $\\Delta \\equiv \\frac{2}{\\epsilon}-\\gamma+\\log 4 \\pi+\\frac{3}{2}$ for brevity.\nNow, with $\\overline{M S}$ scheme, we can determine the counterterms in (13.26) to be\n\n\\begin{equation*}\n\\delta \\mathcal{L}=\\frac{-1}{4(4 \\pi)^{2}}[\\frac{2}{\\epsilon}-\\gamma+\\log 4 \\pi-\\log M^{2}](3(2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}+(m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})^{2}+(m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})^{2}) . \\tag{13.32}\n\\end{equation*}\n\nwhere $M$ is the renormalization scale. Now the effective potential follows directly from (13.26), (13.31) and (13.32),\n\n\\begin{align*}\n& V_{\\mathrm{eff}}[\\phi_{\\mathrm{cl}}]=m^{2} \\phi_{\\mathrm{cl}}^{2}+\\frac{\\lambda}{6} \\phi_{\\mathrm{cl}}^{4}-\\frac{1}{4(4 \\pi)^{2}}[3(2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{2 e^{2} \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2}) \\\\\n& \\quad+(m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})+(m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})] . \\tag{13.33}\n\\end{align*}", + "symbol": { + "$A_{\\mu}$": "vector field", + "$\\phi_{\\mathrm{cl}}$": "classical field value", + "$m$": "mass parameter", + "$\\lambda$": "coupling constant", + "$M$": "renormalization scale", + "$e$": "electric charge", + "$\\gamma$": "Euler-Mascheroni constant", + "$\\varphi_{1}$": "fluctuating field component 1", + "$\\varphi_{2}$": "fluctuating field component 2", + "$d$": "spacetime dimensionality" + }, + "chapter": "Final Project II", + "section": "The Coleman-Weinberg Potential" + }, + { + "id": 60, + "topic": "Others", + "question": "We discussed the effective potential for an $O(N)$-symmetric $\\phi^{4}$ theory in four dimensions. We computed the perturbative corrections to this effective potential, and used the renormalization group to clarify the behavior of the potential for small values of the scalar field mass. After all this work, however, we found that the qualitative dependence of the theory on the mass parameter was unchanged by perturbative corrections. The theory still possessed a second-order phase transition as a function of the mass. The loop corrections affected this picture only in providing some logarithmic corrections to the scaling behavior near the phase transition.\n\nHowever, loop corrections are not always so innocuous. For some systems, they can change the structure of the phase transition qualitatively. This Final Project treats the simplest example of such a system, the ColemanWeinberg model. The analysis of this model draws on a broad variety of topics discussed in Part II; it provides a quite nontrivial application of the effective potential formalism and the use of the renormalization group equation. The phenomenon displayed in this exercise reappears in many contexts, from displacive phase transitions in solids to the thermodynamics of the early universe.\n\nThis problem makes use of material in starred sections of the book Peskin \\% Schroeder, in particular, Sections 11.3, 11.4, and 13.2. Parts (a) and (e), however, depend only on the unstarred material of Part II. We recommend part (e) as excellent practice in the computation of renormalization group functions.\n\nThe Coleman-Weinberg model is the quantum electrodynamics of a scalar field in four dimensions, considered for small values of the scalar field mass. The Lagrangian is\n\n$$\\mathcal{L}=-\\frac{1}{4}(F_{\\mu \\nu})^{2}+(D_{\\mu} \\phi)^{\\dagger} D^{\\mu} \\phi-m^{2} \\phi^{\\dagger} \\phi-\\frac{\\lambda}{6}(\\phi^{\\dagger} \\phi)^{2},\n$$\nwhere $\\phi(x)$ is a complex-valued scalar field and $D_{\\mu} \\phi=(\\partial_{\\mu}+i e A_{\\mu}) \\phi$. Construct the renormalization-group-improved effective potential at $\\mu^{2}=$ 0 by applying the results of part (e) to the calculation of part (c). Compute $\\langle\\phi\\rangle$ and the mass of the $\\sigma$ particle as a function of $\\lambda, e^{2}, M$. Compute the ratio $m_{\\sigma} / m_{A}$ to leading order in $e^{2}$, for $\\lambda \\ll e^{2}$.", + "final_answer": [ + "m_{\\sigma}/m_A = \\frac{\\sqrt{6}e}{4\\pi}" + ], + "answer_type": "Expression", + "answer": "The effective potential obtained in (c) is not a solution to the renormalization group equation, since it is only a first order result in perturbation expansion. However, it is possible to find an effective potential as a solution to the RG equation, with the result in (c) serving as a sort of \"initial condition\". The effective potential obtained in this way is said to be RG improved.\n\nThe Callan-Symansik equation for the effective potential reads\n\n\\begin{equation*}\n(M \\frac{\\partial}{\\partial M}+\\beta_{\\lambda} \\frac{\\partial}{\\partial \\lambda}+\\beta_{e} \\frac{\\partial}{\\partial e}-\\gamma_{\\phi} \\phi_{\\mathrm{cl}} \\frac{\\partial}{\\partial \\phi_{\\mathrm{cl}}}) V_{\\mathrm{eff}}(\\phi_{\\mathrm{cl}}, \\lambda, e ; M)=0 . \\tag{13.53}\n\\end{equation*}\n\n\nThe solution to this equation is well known, that is, the dependence of the sliding energy scale $M$ is described totally by running parameters,\n\n\\begin{equation*}\nV_{\\mathrm{eff}}(\\phi_{\\mathrm{cl}}, \\lambda, e ; M)=V_{\\mathrm{eff}}(\\bar{\\phi}_{\\mathrm{cl}}(M^{\\prime}), \\bar{\\lambda}(M^{\\prime}), \\bar{e}(M^{\\prime}) ; M^{\\prime}), \\tag{13.54}\n\\end{equation*}\n\nwhere barred quantities satisfy\n\n\\begin{equation*}\nM \\frac{\\partial \\bar{\\lambda}}{\\partial M}=\\beta_{\\lambda}(\\bar{\\lambda}, \\bar{e}), \\quad M \\frac{\\partial \\bar{e}}{\\partial M}=\\beta_{e}(\\bar{\\lambda}, \\bar{e}), \\quad M \\frac{\\partial \\bar{\\phi}_{\\mathrm{cl}}}{\\partial M}=-\\gamma_{\\phi}(\\bar{\\lambda}, \\bar{e}) \\bar{\\phi}_{\\mathrm{cl}} \\tag{13.55}\n\\end{equation*}\n\n\nThe RG-improved effective potential should be such that when expanded in terms of coupling constants $\\lambda$ and $e$, it will recover the result in (c) at the given order. For simplicity here we work under the assumption that $\\lambda \\sim e^{4}$, so that all terms of higher orders of coupling constants than $\\lambda$ and $e^{4}$ can be ignored. In this case, the perturbative calculation in (c) gives\n\n\\begin{equation*}\nV_{\\mathrm{eff}}=\\frac{\\lambda}{6} \\phi_{\\mathrm{cl}}^{4}+\\frac{3 e^{4} \\phi_{\\mathrm{cl}}^{4}}{(4 \\pi)^{2}}(\\log \\frac{2 e^{2} \\phi_{\\mathrm{cl}}^{2}}{M^{2}}-\\frac{3}{2}) . \\tag{13.56}\n\\end{equation*}\n\n\nNow we claim that the RG-improved edition of this result reads\n\n\\begin{equation*}\nV_{\\mathrm{eff}}=\\frac{\\bar{\\lambda}}{6} \\bar{\\phi}_{\\mathrm{cl}}^{4}+\\frac{3 \\bar{e}^{4} \\bar{\\phi}_{\\mathrm{cl}}^{4}}{(4 \\pi)^{2}}(\\log 2 \\bar{e}^{2}-\\frac{3}{2}) . \\tag{13.57}\n\\end{equation*}\n\n\nTo see this, we firstly solve the renormalization group equations (13.55),\n\n\\begin{align*}\n\\bar{\\lambda}(M^{\\prime}) & =\\bar{e}^{4}(\\frac{\\lambda}{e^{4}}+\\frac{9}{4 \\pi^{2}} \\log \\frac{M^{\\prime}}{M}) \\tag{13.58}\\\\\n\\bar{e}^{2}(M^{\\prime}) & =\\frac{e^{2}}{1-(e^{2} / 24 \\pi^{2}) \\log (M^{\\prime} / M)} \\tag{13.59}\\\\\n\\bar{\\phi}_{\\mathrm{cl}}(M^{\\prime}) & =\\phi_{\\mathrm{cl}}(\\frac{M^{\\prime}}{M})^{2 e^{2} /(4 \\pi)^{2}} \\tag{13.60}\n\\end{align*}\n\nwhere the unbarred quantities $\\lambda, e$ and $\\phi_{\\mathrm{cl}}$ are evaluated at scale $M$. Now we substitute these results back into the RG-improved effective potential (13.57) and expand in terms of coupling constants. Then it is straightforward to see that the result recovers (13.56). To see the spontaneous symmetry breaking still occurs, we note that the running coupling $\\bar{\\lambda}(M^{\\prime})$ flows to negative value rapidly for small $M^{\\prime}=\\phi_{\\mathrm{cl}}$, while $\\bar{e}(M^{\\prime})$ changes mildly along the $\\phi_{\\mathrm{cl}}$ scale, as can be seen directly from Figure 13.2. Therefore the the coefficient before $\\phi_{\\mathrm{cl}}^{4}$ is negative for small $\\phi_{\\mathrm{cl}}$ and positive for large $\\phi_{\\mathrm{cl}}$. As a consequence, the minimum of this effective potential should be away from $\\phi_{\\mathrm{cl}}=0$, namely the $U(1)$ symmetry is spontaneously broken.\n\nTo find the scalar mass $m_{\\sigma}$ in this case (with $\\mu=0$ ), we calculate the second derivative of the effective potential $V_{\\text {eff }}$ with respect to $\\phi_{\\mathrm{cl}}$. Since the renormalization scale $M$ can be arbitrarily chosen, we set it to be $M^{2}=2 e^{2}\\langle\\phi_{\\mathrm{cl}}^{2}\\rangle$ to simplify the calculation. Then the vanishing of the first derivative of $V_{\\mathrm{eff}}$ at $\\phi_{\\mathrm{cl}}=\\langle\\phi_{\\mathrm{cl}}\\rangle$ implies that $\\lambda=9 e^{4} / 8 \\pi^{2}$. Insert this back to $V_{\\text {eff }}$ in (13.56), we find that\n\n\\begin{equation*}\nV_{\\mathrm{eff}}=\\frac{3 e^{4} \\phi_{\\mathrm{cl}}^{4}}{16 \\pi^{2}}(\\log \\frac{\\phi_{\\mathrm{cl}}^{2}}{\\langle\\phi_{\\mathrm{cl}}^{2}\\rangle}-\\frac{1}{2}) . \\tag{13.61}\n\\end{equation*}\n\n\nThen, taking the second derivative of this expression with respect to $\\phi_{\\mathrm{cl}}$, we get the scalar mass $m_{\\sigma}^{2}=3 e^{4}\\langle\\phi_{\\mathrm{cl}}^{2}\\rangle / 4 \\pi^{2}=3 e^{4} v^{2} / 8 \\pi^{2}$. Recall that the gauge boson's mass $m_{A}$ is given by $m_{A}=e^{2} v^{2}$ at the leading order, thus we conclude that $m_{\\sigma}^{2} / m_{A}^{2}=3 e^{2} / 8 \\pi^{2}$ at the leading order in $e^{2}$.", + "symbol": { + "$\\mu$": "scale parameter", + "$\\phi$": "field value", + "$\\sigma$": "scalar particle", + "$\\lambda$": "coupling constant", + "$e$": "electric charge", + "$M$": "energy scale", + "$\\langle\\phi\\rangle$": "vacuum expectation value of the field", + "$m_{\\sigma}$": "mass of the sigma particle", + "$m_{A}$": "mass of the gauge boson A", + "$\\beta_{\\lambda}$": "beta function for the coupling constant $\\lambda$", + "$\\beta_{e}$": "beta function for the electric charge $e$", + "$\\gamma_{\\phi}$": "anomalous dimension of the field", + "$\\phi_{\\mathrm{cl}}$": "classical field", + "$V_{\\mathrm{eff}}$": "effective potential", + "$\\bar{\\lambda}$": "running coupling constant $\\lambda$", + "$\\bar{e}$": "running electric charge", + "$\\bar{\\phi}_{\\mathrm{cl}}$": "running classical field", + "$\\langle\\phi_{\\mathrm{cl}}^{2}\\rangle$": "vacuum expectation value of the squared classical field", + "$v$": "vacuum expectation value of the field" + }, + "chapter": "Final Project II", + "section": "The Coleman-Weinberg Potential" + }, + { + "id": 61, + "topic": "Others", + "question": "Deep inelastic scattering from a photon. Consider the problem of deepinelastic scattering of an electron from a photon. This process can actually be measured by analyzing the reaction $e^{+} e^{-} \\rightarrow e^{+} e^{-}+X$ in the regime where the positron goes forward, with emission of a collinear photon, which then has a hard reaction with the electron. Let us analyze this process to leading order in QED and to leading-log order in QCD. To predict the photon structure functions, it is reasonable to integrate the renormalization group equations with the initial condition that the parton distribution for photons in the photon is $\\delta(x-1)$ at $Q^{2}=(\\frac{1}{2} \\mathrm{GeV})^{2}$. Take $\\Lambda=150 \\mathrm{MeV}$. Assume for simplicity that there are four flavors of quarks, $u, d, c$, and $s$, with charges $2 / 3$, $-1 / 3,2 / 3,-1 / 3$, respectively, and that it is always possible to ignore the masses of these quarks. Use the Altarelli-Parisi equations to compute the parton distributions for quarks and antiquarks in the photon, to leading order in QED and to zeroth order in QCD.", + "final_answer": [ + "f_{q}(x, Q)=f_{\\bar{q}}(x, Q)=\\frac{3 Q_{q}^{2} \\alpha}{2 \\pi} \\log \\frac{Q^{2}}{Q_{0}^{2}}[x^{2}+(1-x)^{2}]" + ], + "answer_type": "Expression", + "answer": "The A-P equation for parton distributions in the photon can be easily written down by using the QED splitting functions listed in (17.121) of Peskin \\& Schroeder. Taking account of quarks' electric charge properly, we have,\n\n\\begin{align*}\n& \\frac{\\mathrm{d}}{\\mathrm{~d} \\log Q} f_{q}(x, Q)= \\frac{3 Q_{q}^{2} \\alpha}{\\pi} \\int_{x}^{1} \\frac{\\mathrm{d} z}{z}{P_{e \\leftarrow e}(z) f_{q}(\\frac{x}{z}, Q)+P_{e \\leftarrow \\gamma}(z) f_{\\gamma}(\\frac{x}{z}, Q)}, \\tag{18.68}\\\\\n& \\frac{\\mathrm{d}}{\\mathrm{~d} \\log Q} f_{\\bar{q}}(x, Q)= \\frac{3 Q_{q}^{2} \\alpha}{\\pi} \\int_{x}^{1} \\frac{\\mathrm{d} z}{z}{P_{e \\leftarrow e}(z) f_{\\bar{q}}(\\frac{x}{z}, Q)+P_{e \\leftarrow \\gamma}(z) f_{\\gamma}(\\frac{x}{z}, Q)}, \\tag{18.69}\\\\\n& \\frac{\\mathrm{d}}{\\mathrm{~d} \\log Q} f_{\\gamma}(x, Q)= \\sum_{q} \\frac{3 Q_{q}^{2} \\alpha}{\\pi} \\int_{x}^{1} \\frac{\\mathrm{d} z}{z}{P_{\\gamma \\leftarrow e}(z)[f_{q}(\\frac{x}{z}, Q)+f_{\\bar{q}}(\\frac{x}{z}, Q)] \\\\\n&\\quad+P_{\\gamma \\leftarrow \\gamma}(z) f_{\\gamma}(\\frac{x}{z}, Q)}, \\tag{18.70}\n\\end{align*}\n\nwhere the splitting functions are\n\n\\begin{align*}\n& P_{e \\leftarrow e}(z)=\\frac{1+z^{2}}{(1-z)_{+}}+\\frac{3}{2} \\delta(1-z), \\tag{18.71}\\\\\n& P_{\\gamma \\leftarrow e}(z)=\\frac{1+(1-z)^{2}}{z}, \\tag{18.72}\\\\\n& P_{e \\leftarrow \\gamma}(z)=z^{2}+(1-z)^{2} \\tag{18.73}\\\\\n& P_{\\gamma \\leftarrow \\gamma}(z)=-\\frac{2}{3} \\delta(1-z) . \\tag{18.74}\n\\end{align*}\n\n\nWe take $q=u, d, c, s$, and $Q_{u, c}=+2 / 3, Q_{d, s}=-1 / 3$. The factor 3 in the A-P equations above takes account of 3 colors. Since no more leptons appear in final states other than original $e^{+} e^{-}$, they are not included in the photon structure. With the initial condition\n$f_{\\gamma}(x, Q_{0})=\\delta(1-x)$ and $f_{q, \\bar{q}}(x, Q_{0})=0$ where $Q_{0}=0.5 \\mathrm{GeV}$, these distribution functions can be solved from the equations above to the first order in $\\alpha$, to be\n\n\\begin{align*}\n& f_{q}(x, Q)=f_{\\bar{q}}(x, Q)=\\frac{3 Q_{q}^{2} \\alpha}{2 \\pi} \\log \\frac{Q^{2}}{Q_{0}^{2}}[x^{2}+(1-x)^{2}] \\tag{18.75}\\\\\n& f_{\\gamma}(x, Q)=(1-\\sum_{q} \\frac{Q_{q}^{2} \\alpha}{\\pi} \\log \\frac{Q^{2}}{Q_{0}^{2}}) \\delta(1-x) \\tag{18.76}\n\\end{align*}", + "symbol": { + "$f_{q}$": "parton distribution for quarks", + "$f_{\\bar{q}}$": "parton distribution for antiquarks", + "$Q_{q}$": "electric charge of quark", + "$\\alpha$": "fine-structure constant (QED coupling constant)", + "$Q$": "momentum transfer or energy scale", + "$x$": "Bjorken scaling variable", + "$f_{\\gamma}$": "parton distribution for the photon", + "$Q_{0}$": "initial energy scale", + "$u$": "up quark", + "$d$": "down quark", + "$c$": "charm quark", + "$s$": "strange quark" + }, + "chapter": "Operator Products and Effective Vertices", + "section": "Deep inelastic scattering from a photon" + }, + { + "id": 62, + "topic": "Others", + "question": "Neutral-current deep inelastic scattering. In Eq. (17.35) of Peskin \\& Schroeder, we wrote formulae for neutrino and antineutrino deep inelastic scattering with $W^{\\pm}$ exchange. Neutrinos and antineutrinos can also scatter by exchanging a $Z^{0}$. This process, which leads to a hadronic jet but no observable outgoing lepton, is called the neutral current reaction. Compute $\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{d} x \\mathrm{d} y}(\\nu p \\rightarrow \\nu X)$ for neutral current deep inelastic scattering of neutrinos from protons, accounting for scattering from $u$ and $d$ quarks and antiquarks.", + "final_answer": [ + "\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{d} x \\mathrm{d} y}(\\nu p \\rightarrow \\nu X)=\\frac{G_{F}^{2} s x}{4 \\pi} {[(1-\\frac{4}{3} s_{w}^{2})^{2}+\\frac{16}{9} s_{w}^{4}(1-y^{2})] f_{u}(x) + [(1-\\frac{2}{3} s_{w}^{2})^{2}+\\frac{4}{9} s_{w}^{4}(1-y^{2})] f_{d}(x) +[\\frac{16}{9} s_{w}^{4}+(1-\\frac{4}{3} s_{w}^{2})^{2}(1-y^{2})] f_{\\bar{u}}(x) +[\\frac{4}{9} s_{w}^{4}+(1-\\frac{2}{3} s_{w}^{2})^{2}(1-y^{2})] f_{\\bar{d}}(x)}" + ], + "answer_type": "Expression", + "answer": "In this problem we study the neutral-current deep inelastic scattering. The process is mediated by $Z^{0}$ boson. Assuming $m_{Z}$ is much larger than the energy scale of the scattering process, we can write down the corresponding effective operators, from the neutral-current Feynman rules in electroweak theory,\n\n\\begin{align*}\n\\Delta \\mathcal{L}=\\frac{g^{2}}{4 m_{W}^{2}}(\\bar{\\nu} \\gamma^{\\mu}) P_{L} \\nu & {[\\bar{u} \\gamma_{\\mu}((1-\\frac{4}{3} s_{w}^{2}) P_{L}-\\frac{4}{3} s_{w}^{2} P_{R}) u} \\\\\n& +\\bar{d} \\gamma_{\\mu}((1-\\frac{2}{3} s_{w}^{2}) P_{L}-\\frac{2}{3} s_{w}^{2} P_{R}) d]+ \\text { h.c. } \\tag{20.30}\n\\end{align*}\n\nwhere $P_{L}=(1-\\gamma^{5}) / 2$ and $P_{R}=(1+\\gamma^{5}) / 2$ are left- and right-handed projectors, respectively. Compare the effective operator with the charged-operator in (17.31) of Peskin \\& Schroeder, we can write down directly the differential cross section for neutrino scattering by modifying (17.35) in Peskin \\& Schroeder properly, as\n\n\\begin{align*}\n\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{~d} x \\mathrm{~d} y}(\\nu p \\rightarrow \\nu X)=\\frac{G_{F}^{2} s x}{4 \\pi} & {[(1-\\frac{4}{3} s_{w}^{2})^{2}+\\frac{16}{9} s_{w}^{4}(1-y^{2})] f_{u}(x) \\\\\n+ & {[(1-\\frac{2}{3} s_{w}^{2})^{2}+\\frac{4}{9} s_{w}^{4}(1-y^{2})] f_{d}(x) } \\\\\n& +[\\frac{16}{9} s_{w}^{4}+(1-\\frac{4}{3} s_{w}^{2})^{2}(1-y^{2})] f_{\\bar{u}}(x) \\\\\n& +[\\frac{4}{9} s_{w}^{4}+(1-\\frac{2}{3} s_{w}^{2})^{2}(1-y^{2})] f_{\\bar{d}}(x)}. \\tag{20.31}\n\\end{align*}", + "symbol": { + "$W^{\\pm}$": "charged W boson", + "$Z^{0}$": "neutral Z boson", + "$m_{Z}$": "mass of the Z boson", + "$g$": "electroweak coupling constant", + "$m_{W}$": "mass of the W boson", + "$P_{L}$": "left-handed projector", + "$P_{R}$": "right-handed projector", + "$G_{F}$": "Fermi coupling constant", + "$s$": "center-of-mass energy squared", + "$x$": "Bjorken scaling variable", + "$y$": "inelasticity", + "$s_{w}$": "sine of the Weinberg angle", + "$u$": "up quark", + "$d$": "down quark", + "$\\nu$": "neutrino", + "$f_{u}(x)$": "parton distribution function for up quark", + "$f_{d}(x)$": "parton distribution function for down quark", + "$f_{\\bar{u}}(x)$": "parton distribution function for up antiquark", + "$f_{\\bar{d}}(x)$": "parton distribution function for down antiquark" + }, + "chapter": "Gauge Theories with Spontaneous Symmetry Breaking", + "section": "Neutral-current deep inelastic scattering" + }, + { + "id": 63, + "topic": "Others", + "question": "Neutral-current deep inelastic scattering. In Eq. (17.35) of Peskin \\& Schroeder, we wrote formulae for neutrino and antineutrino deep inelastic scattering with $W^{ \\pm}$exchange. Neutrinos and antineutrinos can also scatter by exchanging a $Z^{0}$. This process, which leads to a hadronic jet but no observable outgoing lepton, is called the neutral current reaction. Compute $\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{d} x \\mathrm{d} y}(\\bar{\\nu} p \\rightarrow \\bar{\\nu} X)$ for neutral current deep inelastic scattering of antineutrinos from protons, accounting for scattering from $u$ and $d$ quarks and antiquarks.", + "final_answer": [ + "\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{d} x \\mathrm{d} y}(\\bar{\\nu} p \\rightarrow \\bar{\\nu} X)=\\frac{G_{F}^{2} s x}{4 \\pi} {[\\frac{16}{9} s_{w}^{4}+(1-\\frac{4}{3} s_{w}^{2})^{2}(1-y^{2})] f_{u}(x) +[\\frac{4}{9} s_{w}^{4}+(1-\\frac{2}{3} s_{w}^{2})^{2}(1-y^{2})] f_{d}(x) +[(1-\\frac{4}{3} s_{w}^{2})^{2}+\\frac{16}{9} s_{w}^{4}(1-y^{2})] f_{\\bar{u}}(x) +[(1-\\frac{2}{3} s_{w}^{2})^{2}+\\frac{4}{9} s_{w}^{4}(1-y^{2})] f_{\\bar{d}}(x)}" + ], + "answer_type": "Expression", + "answer": "In this problem we study the neutral-current deep inelastic scattering. The process is mediated by $Z^{0}$ boson. Assuming $m_{Z}$ is much larger than the energy scale of the scattering process, we can write down the corresponding effective operators, from the neutral-current Feynman rules in electroweak theory,\n\n\\begin{align*}\n\\Delta \\mathcal{L}=\\frac{g^{2}}{4 m_{W}^{2}}(\\bar{\\nu} \\gamma^{\\mu}) P_{L} \\nu & {[\\bar{u} \\gamma_{\\mu}((1-\\frac{4}{3} s_{w}^{2}) P_{L}-\\frac{4}{3} s_{w}^{2} P_{R}) u} \\\\\n& +\\bar{d} \\gamma_{\\mu}((1-\\frac{2}{3} s_{w}^{2}) P_{L}-\\frac{2}{3} s_{w}^{2} P_{R}) d]+ \\text { h.c. } \\tag{20.30}\n\\end{align*}\n\nwhere $P_{L}=(1-\\gamma^{5}) / 2$ and $P_{R}=(1+\\gamma^{5}) / 2$ are left- and right-handed projectors, respectively. Compare the effective operator with the charged-operator in (17.31) of Peskin \\& Schroeder, we can write down directly the differential cross section for antineutrino scattering by modifying (17.35) in Peskin \\& Schroeder properly, as\n\n\\begin{align*}\n\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{~d} x \\mathrm{~d} y}(\\bar{\\nu} p \\rightarrow \\bar{\\nu} X)=\\frac{G_{F}^{2} s x}{4 \\pi} & {[\\frac{16}{9} s_{w}^{4}+(1-\\frac{4}{3} s_{w}^{2})^{2}(1-y^{2})] f_{u}(x) \\\\\n& +[\\frac{4}{9} s_{w}^{4}+(1-\\frac{2}{3} s_{w}^{2})^{2}(1-y^{2})] f_{d}(x) \\\\\n& +[(1-\\frac{4}{3} s_{w}^{2})^{2}+\\frac{16}{9} s_{w}^{4}(1-y^{2})] f_{\\bar{u}}(x) \\\\\n& +[(1-\\frac{2}{3} s_{w}^{2})^{2}+\\frac{4}{9} s_{w}^{4}(1-y^{2})] f_{\\bar{d}}(x)} . \\tag{20.32}\n\\end{align*}", + "symbol": { + "$W^{\\pm}$": "charged W boson", + "$Z^{0}$": "neutral Z boson", + "$m_{Z}$": "mass of the Z boson", + "$g$": "electroweak coupling constant", + "$m_{W}$": "mass of the W boson", + "$\\nu$": "neutrino", + "$\\bar{\\nu}$": "antineutrino", + "$\\bar{u}$": "up antiquark", + "$\\bar{d}$": "down antiquark", + "$u$": "up quark", + "$d$": "down quark", + "$s_{w}$": "sine of the Weinberg angle", + "$P_{L}$": "left-handed projector", + "$P_{R}$": "right-handed projector", + "$G_{F}$": "Fermi coupling constant", + "$s$": "Mandelstam variable, center-of-mass energy squared", + "$x$": "Bjorken scaling variable", + "$y$": "inelasticity", + "$f_{u}(x)$": "parton distribution function for up quarks", + "$f_{d}(x)$": "parton distribution function for down quarks", + "$f_{\\bar{u}}(x)$": "parton distribution function for up antiquarks", + "$f_{\\bar{d}}(x)$": "parton distribution function for down antiquarks" + }, + "chapter": "Gauge Theories with Spontaneous Symmetry Breaking", + "section": "Neutral-current deep inelastic scattering" + }, + { + "id": 64, + "topic": "Others", + "question": "A model with two Higgs fields. Find the couplings of the physical charged Higgs boson $\\phi^{+}$ to the quark mass eigenstates, given the Lagrangian $$\\mathcal{L}_{m} = -\\lambda_{d}^{ij} \\bar{Q}_{L}^{i} \\cdot \\phi_{1} d_{R}^{j} - \\lambda_{u}^{ij} \\epsilon^{ab} \\bar{Q}_{La}^{i} \\phi_{2b}^{\\dagger} u_{R}^{j} + \\text{h.c.}$$", + "final_answer": [ + "\\frac{\\sqrt{2}}{v} \\left( \\bar{u}_L V_{\\text{CKM}} m_d d_R \\phi^+ \\tan\\beta + \\bar{d}_L V_{\\text{CKM}}^\\dagger m_u u_R \\phi^- \\cot\\beta \\right) + \\text{h.c.}" + ], + "answer_type": "Expression", + "answer": "Assuming that the Yukawa interactions between quarks and scalars take the following form:\n\n\\begin{align}\n\\mathcal{L}_m &= -\\left(\\bar{u}_L \\quad \\bar{d}_L\\right)\\left[\\lambda_d\\left(\\frac{\\pi_1^+}{\\sqrt{2}v_1}\\right)d_R + \\lambda_u\\left(\\frac{\\frac{1}{\\sqrt{2}}v_2}{\\pi^-}\\right)u_R\\right] + \\text{h.c.}, \\tag{20.44}\n\\end{align}\nwhere we have suppressed flavor indices and neglected neutral scalar components. We focus on charged components only. Using Peskin \\& Schroeder's notation, we perform the replacements \\( u_L \\to U_u u_L \\), \\( d_L \\to U_d d_L \\), \\( u_R \\to W_u u_R \\), and \\( d_R \\to W_d d_R \\). With the diagonalization \\(\\lambda_d = U_d D_d W_d^\\dagger\\) and \\(\\lambda_u = U_u D_u W_u^\\dagger\\) (where \\(D_d\\) and \\(D_u\\) are diagonal matrices), we derive:\n\n\\begin{align}\n\\mathcal{L}_m &= -\\frac{1}{\\sqrt{2}}\\left(v_1\\bar{d}_L D_d d_R + v_2\\bar{u}_L D_u u_R\\right) \\nonumber \\\\\n&\\quad - \\bar{u}_L V_{\\text{CKM}} D_d d_R \\pi_1^+ + \\bar{d}_L V_{\\text{CKM}}^\\dagger D_u u_R \\pi_2^- + \\text{h.c.}. \\tag{20.45}\n\\end{align}\n\nFrom the first term, the diagonal quark mass matrices are given by \\( m_u = \\frac{v_1}{\\sqrt{2}}D_u \\) and \\( m_d = \\frac{v_2}{\\sqrt{2}}D_d \\). Defining \\( v = \\sqrt{v_1^2 + v_2^2} \\), and using the relations \\( \\pi_1^+ = -\\phi^+ \\sin\\beta + \\cdots \\), \\( \\pi_2^+ = \\phi^+ \\cos\\beta + \\cdots \\), the Yukawa interactions with charged bosons become:\n\n\\begin{align}\n\\mathcal{L}_m &\\Rightarrow -\\frac{\\sqrt{2}}{v_1} \\left( \\bar{u}_L V_{\\text{CKM}} m_d d_R \\pi_1^+ + \\bar{d}_L V_{\\text{CKM}}^\\dagger m_u u_R \\pi_2^- \\right) + \\text{h.c.} \\nonumber \\\\\n&\\Rightarrow \\frac{\\sqrt{2}}{v} \\left( \\bar{u}_L V_{\\text{CKM}} m_d d_R \\phi^+ \\tan\\beta + \\bar{d}_L V_{\\text{CKM}}^\\dagger m_u u_R \\phi^- \\cot\\beta \\right) + \\text{h.c.}. \\tag{20.46}\n\\end{align}", + "symbol": { + "$\\phi^{+}$": "charged Higgs boson", + "$\\mathcal{L}_{m}$": "mass term in the Lagrangian", + "$\\lambda_{d}^{ij}$": "Yukawa coupling matrix for down-type quarks", + "$\\lambda_{u}^{ij}$": "Yukawa coupling matrix for up-type quarks", + "$\\bar{Q}_{L}^{i}$": "left-handed quark doublet", + "$d_{R}^{j}$": "right-handed down-type quark", + "$\\phi_{1}$": "scalar field 1", + "$\\epsilon^{ab}$": "antisymmetric tensor", + "$\\bar{Q}_{La}^{i}$": "component of the left-handed quark doublet", + "$\\phi_{2b}^{\\dagger}$": "conjugate of the component of scalar field 2", + "$u_{R}^{j}$": "right-handed up-type quark", + "$u_L$": "left-handed up-type quark", + "$d_L$": "left-handed down-type quark", + "$v_1$": "vacuum expectation value component 1", + "$v_2$": "vacuum expectation value component 2", + "$v$": "Higgs vacuum expectation value", + "$\\pi_1^+$": "Goldstone boson component for charged Higgs interactions", + "$\\pi_2^-$": "Goldstone boson component for charged Higgs interactions", + "$V_{\\text{CKM}}$": "CKM matrix", + "$m_u$": "up-type quark mass matrix", + "$m_d$": "down-type quark mass matrix", + "$\\tan\\beta$": "ratio of two Higgs doublet vacuum expectation values", + "$\\cot\\beta$": "cotangent of the ratio of the two Higgs doublet vacuum expectation values" + }, + "chapter": "Gauge Theories with Spontaneous Symmetry Breaking", + "section": "A model with two Higgs fields" + }, + { + "id": 65, + "topic": "Others", + "question": "We discussed the mystery of the origin of spontaneous symmetry breaking in the weak interactions. The simplest hypothesis is that the $S U(2) \\times U(1)$ gauge symmetry of the weak interactions is broken by the expectation value of a two-component scalar field $\\phi$. However, since we have almost no experimental information about the mechanism of this symmetry breaking, many other possibilities can be suggested.\n\nEventually, this problem should be resolved by experimental observation of the particles associated with the symmetry breaking. To form incisive experimental tests, we should compute the properties expected for these particles. We saw in Section 20.2 of Peskin \\& Schroeder that, if the symmetry is indeed broken by a single scalar field $\\phi$, the symmetry-breaking sector contributes only one new particle, a scalar $h^{0}$ called the Higgs boson. The mass $m_{h}$ of this particle is unknown. However, the couplings of the $h^{0}$ to known fermions and bosons are completely determined by the masses of those particles and the weak interaction coupling constants. Thus, it is possible to compute the amplitudes for production and decay of the $h^{0}$ in some detail. More complicated models of $S U(2) \\times U(1)$ symmetry breaking typically contain one or more particles that share some properties with the $h^{0}$. Thus, this study is a useful starting point for the more general analysis of experimental tests of these models.\n\nIn this Final Project you will compute the amplitudes for the Higgs boson $h^{0}$ to decay to pairs of quarks, leptons, and gauge bosons. The computations beautifully illustrate the working of perturbation theory for non-Abelian gauge fields. Those decays of the Higgs boson that involve quarks and gluons bring in aspects of QCD. Thus, this exercise reviews all of the important technical methods of Part III. Except for a question raised at the end of part (a), the problem relies only on material from unstarred sections of Part III. The material in Chapter 20 plays an essential role. Material from Chapter 21 enters the analysis only in parts (b) and (f), and the other parts of the problem (except for the final summary in part (h)) do not rely on these. If you have studied Section 19.5, you will have some additional insight into the results of parts (c) and (f), but this insight is not necessary to work the problem.\n\nConsider, then, the minimal form of the Glashow-Weinberg-Salam electroweak theory with one Higgs scalar field $\\phi$. The physical Higgs boson $h^{0}$ of\nthis theory was discussed in Section 20.2, and we listed there the couplings of this particle to quarks, leptons, and gauge bosons. You can now use that information to compute the amplitudes for the various possible decays of the $h^{0}$ as a function of its mass $m_{h}$. You will discover that the decay pattern has a complicated dependence on the mass of the Higgs boson, with different decay modes dominating in different mass ranges. The dependences of the various decay rates on $m_{h}$ illustrate many aspects of the physics of gauge theories that we have discussed in Part III.\n\nIn working this exercise, you should consider $m_{h}$ as a free parameter. For the other parameters of weak-interaction theory, you might use the following values: $m_{b}=5 \\mathrm{GeV}, m_{t}=175 \\mathrm{GeV}, m_{W}=80 \\mathrm{GeV}, m_{Z}=91 \\mathrm{GeV}, \\sin ^{2} \\theta_{w}$ $=0.23, \\alpha_{s}(m_{Z})=0.12$. Provide the tree-level decay width $\\Gamma(h^0 \\rightarrow f\\bar{f})$ for a Higgs boson $h^0$ decaying into a fermion-antifermion pair $f\\bar{f}$ (where $f$ is a quark or lepton of the Standard Model), expressed in terms of the fine-structure constant $\\alpha$, the Higgs mass $m_h$, the fermion mass $m_f$, the W boson mass $m_W$, the weak mixing angle $\\theta_w$, and the color factor $N_c(f)$ (1 for leptons, 3 for quarks).", + "final_answer": [ + "\\Gamma(h^0 \\rightarrow f\\bar{f}) = (\\frac{\\alpha m_h}{8\\sin^2\\theta_w}) \\cdot \\frac{m_f^2}{m_W^2} (1 - \\frac{4m_f^2}{m_h^2})^{3/2}" + ], + "answer_type": "Expression", + "answer": "In this final project, we calculate partial widths of various decay channels of the standard model Higgs boson. Although a standard-model-Higgs-like boson has been found at the LHC with mass around 125 GeV , it is still instructive to treat the mass of the Higgs boson as a free parameter in the following calculation.\n\nThe main decay modes of Higgs boson include $h^{0} \\rightarrow f \\bar{f}$ with $f$ the standard model fermions, $h^{0} \\rightarrow W^{+} W^{-}, h^{0} \\rightarrow Z^{0} Z^{0}, h^{0} \\rightarrow g g$ and $h^{0} \\rightarrow \\gamma \\gamma$. The former three processes appear at the tree level, while the leading order contributions to the latter two processes are at one-loop level. We will work out the decay widths of these processes in the following.\n\nIn this problem we only consider the two-body final states. The calculation of decay width needs the integral over the phase space of the two-body final states. By momentum conservation and rotational symmetry, we can always parameterize the momenta of two final particles in CM frame to be $p_{1}=(E, 0,0, p)$ and $p_{2}=(E, 0,0,-p)$, where $E=\\frac{1}{2} m_{h}$ by energy conservation. Then the amplitude $\\mathcal{M}$ will have no angular dependence. Then the phase space integral reads\n\n\\begin{equation*}\n\\int \\mathrm{d} \\Pi_{2}|\\mathcal{M}|^{2}=\\frac{1}{4 \\pi} \\frac{p}{m_{h}}|\\mathcal{M}|^{2} . \\tag{21.50}\n\\end{equation*}\n\n\nThen the decay width is given by\n\n\\begin{equation*}\n\\Gamma=\\frac{1}{2 m_{h}} \\int \\mathrm{~d} \\Pi_{2}|\\mathcal{M}|^{2}=\\frac{1}{8 \\pi} \\frac{p}{m_{h}^{2}}|\\mathcal{M}|^{2} . \\tag{21.51}\n\\end{equation*}\n\n\nIn part (d) of this problem, we will also be dealing with the production of the Higgs boson from two-gluon initial state, thus we also write down the formula here for the cross section of the one-body final state from two identical initial particle. This time, the two ingoing particles have momenta $k_{1}=(E, 0,0, k)$ and $k_{2}=(E, 0,0,-k)$, with $E^{2}=k^{2}+m_{i}^{2}$ and $2 E=m_{f}$ where $m_{i}$ and $m_{f}$ are masses of initial particles and final particle, respectively. The final particle has momentum $p=(m_{f}, 0,0,0)$. Then, the cross section is given by\n\n\\begin{align*}\n\\sigma & =\\frac{1}{2 \\beta s} \\int \\frac{\\mathrm{~d}^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{p}}|\\mathcal{M}|^{2}(2 \\pi)^{4} \\delta^{(4)}(p-k_{1}-k_{2}) \\\\\n& =\\frac{1}{4 m_{f} \\beta s}|\\mathcal{M}|^{2}(2 \\pi) \\delta(2 k-m_{f})=\\frac{\\pi}{\\beta m_{f}^{2}}|\\mathcal{M}|^{2} \\delta(s-m_{f}^{2}), \\tag{21.52}\n\\end{align*}\n\nwhere $\\beta=\\sqrt{1-(4 m_{i} / m_{f})^{2}}$ is the magnitude of the velocity of the initial particle in the center-of-mass frame.\n(a) The easiest calculation of above processes is $h^{0} \\rightarrow f \\bar{f}$, where $f$ represents all quarks and charged leptons. The tree level contribution to this process involves a single Yukawa vertex only. The corresponding amplitude is given by\n\n\\begin{equation*}\n\\mathrm{i} \\mathcal{M}(h^{0} \\rightarrow f \\bar{f})=-\\frac{\\mathrm{i} m_{f}}{v} \\bar{u}^{*}(p_{1}) v(p_{2}) . \\tag{21.53}\n\\end{equation*}\n\n\nThen it is straightforward to get the squared amplitude with final spins summed to be\n\n\\begin{equation*}\n\\sum|\\mathcal{M}(h^{0} \\rightarrow f \\bar{f})|^{2}=\\frac{m_{f}^{2}}{v^{2}} \\operatorname{tr}[(\\not p_{1}+m_{f})(\\not p_{2}-m_{f})]=\\frac{2 m_{f}^{2}}{v^{2}}(m_{h}^{2}-4 m_{f}^{2}) \\tag{21.54}\n\\end{equation*}\n\n\nIn CM frame, the final states momenta can be taken to be $p_{1}=(E, 0,0, p)$ and $p_{2}=$ $(E, 0,0,-p)$, with $E=\\frac{1}{2} m_{h}$ and $p^{2}=E^{2}-m_{f}^{2}$. Then the decay width is given by\n\n\\begin{equation*}\n\\Gamma(h^{0} \\rightarrow f \\bar{f})=\\frac{1}{8 \\pi} \\frac{p}{m_{h}^{2}}|\\mathcal{M}|^{2}=\\frac{m_{h} m_{f}^{2}}{8 v^{2}}(1-\\frac{4 m_{f}^{2}}{m_{h}^{2}})^{3 / 2} \\tag{21.55}\n\\end{equation*}\n\n\nThis expression can be expressed in terms of the fine structure constant $\\alpha$, the mass of $W$ boson $m_{w}$ and Weinberg angle $\\sin \\theta_{w}$, as\n\n\\begin{equation*}\n\\Gamma(h^{0} \\rightarrow f \\bar{f})=\\frac{\\alpha m_{h}}{8 \\sin ^{2} \\theta_{w}} \\frac{m_{f}^{2}}{m_{W}^{2}}(1-\\frac{4 m_{f}^{2}}{m_{h}^{2}})^{3 / 2} \\tag{21.56}\n\\end{equation*}", + "symbol": { + "$\\Gamma$": "decay width", + "$h^0$": "Higgs boson", + "$f$": "fermion", + "$\\bar{f}$": "antifermion", + "$\\alpha$": "fine-structure constant", + "$m_h$": "Higgs mass", + "$m_f$": "fermion mass", + "$m_W$": "W boson mass", + "$\\theta_w$": "weak mixing angle", + "$N_c(f)$": "color factor" + }, + "chapter": "Final Project III", + "section": "Decays of the Higgs Boson" + }, + { + "id": 66, + "topic": "Others", + "question": "We discussed the mystery of the origin of spontaneous symmetry breaking in the weak interactions. The simplest hypothesis is that the $S U(2) \\times U(1)$ gauge symmetry of the weak interactions is broken by the expectation value of a two-component scalar field $\\phi$. However, since we have almost no experimental information about the mechanism of this symmetry breaking, many other possibilities can be suggested.\n\nEventually, this problem should be resolved by experimental observation of the particles associated with the symmetry breaking. To form incisive experimental tests, we should compute the properties expected for these particles. We saw in Section 20.2 that, if the symmetry is indeed broken by a single scalar field $\\phi$, the symmetry-breaking sector contributes only one new particle, a scalar $h^{0}$ called the Higgs boson. The mass $m_{h}$ of this particle is unknown. However, the couplings of the $h^{0}$ to known fermions and bosons are completely determined by the masses of those particles and the weak interaction coupling constants. Thus, it is possible to compute the amplitudes for production and decay of the $h^{0}$ in some detail. More complicated models of $S U(2) \\times U(1)$ symmetry breaking typically contain one or more particles that share some properties with the $h^{0}$. Thus, this study is a useful starting point for the more general analysis of experimental tests of these models.\n\nIn this Final Project you will compute the amplitudes for the Higgs boson $h^{0}$ to decay to pairs of quarks, leptons, and gauge bosons. The computations beautifully illustrate the working of perturbation theory for non-Abelian gauge fields. Those decays of the Higgs boson that involve quarks and gluons bring in aspects of QCD. Thus, this exercise reviews all of the important technical methods of Part III. Except for a question raised at the end of part (a), the problem relies only on material from unstarred sections of Part III. The material in Chapter 20 plays an essential role. Material from Chapter 21 enters the analysis only in parts (b) and (f), and the other parts of the problem (except for the final summary in part (h)) do not rely on these. If you have studied Section 19.5, you will have some additional insight into the results of parts (c) and (f), but this insight is not necessary to work the problem.\n\nConsider, then, the minimal form of the Glashow-Weinberg-Salam electroweak theory with one Higgs scalar field $\\phi$. The physical Higgs boson $h^{0}$ of\nthis theory was discussed in Section 20.2, and we listed there the couplings of this particle to quarks, leptons, and gauge bosons. You can now use that information to compute the amplitudes for the various possible decays of the $h^{0}$ as a function of its mass $m_{h}$. You will discover that the decay pattern has a complicated dependence on the mass of the Higgs boson, with different decay modes dominating in different mass ranges. The dependences of the various decay rates on $m_{h}$ illustrate many aspects of the physics of gauge theories that we have discussed in Part III.\n\nIn working this exercise, you should consider $m_{h}$ as a free parameter. For the other parameters of weak-interaction theory, you might use the following values: $m_{b}=5 \\mathrm{GeV}, m_{t}=175 \\mathrm{GeV}, m_{W}=80 \\mathrm{GeV}, m_{Z}=91 \\mathrm{GeV}, \\sin ^{2} \\theta_{w}$ $=0.23, \\alpha_{s}(m_{Z})=0.12$. If the Higgs boson mass $m_h$ is sufficiently large (specifically, if $m_h > 2m_Z$), it can also decay to $Z^{0} Z^{0}$. Compute the decay width $\\Gamma(h^0 \\rightarrow Z^0Z^0)$. As context from the original problem, if $m_h \\gg m_Z$, this decay width can be approximated by $\\Gamma(h^0 \\rightarrow Z^0Z^0) \\approx \\Gamma(h^0 \\rightarrow \\phi^3\\phi^3)$, where $\\phi^3$ is a Goldstone boson, and an explanation and verification of this approximation was requested.", + "final_answer": [ + "\\Gamma(h^{0} \\rightarrow Z^{0} Z^{0})=\\frac{\\alpha m_{h}^{3}}{32 \\pi m_{Z}^{2} \\sin ^{2} \\theta_{w}}(1-4 \\tau_{Z}^{-1}+12 \\tau_{Z}^{-2})(1-4 \\tau_{Z}^{-1})^{1 / 2}" + ], + "answer_type": "Expression", + "answer": "For $h^{0} \\rightarrow Z^{0} Z^{0}$ process, it can be easily checked that nothing gets changed in the calculation for $h^0 \\rightarrow W^+W^-$ decay (described elsewhere) except that all $m_{W}$ should be replaced with $m_{Z}$, while an additional factor $1 / 2$ is needed to account for the identical particles in final state. Therefore we have\n\n\\begin{equation*}\n\\Gamma(h^{0} \\rightarrow Z^{0} Z^{0})=\\frac{\\alpha m_{h}^{3}}{32 \\pi m_{Z}^{2} \\sin ^{2} \\theta_{w}}(1-4 \\tau_{Z}^{-1}+12 \\tau_{Z}^{-2})(1-4 \\tau_{Z}^{-1})^{1 / 2} \\tag{21.60}\n\\end{equation*}\n\nwhere $\\tau_{Z} \\equiv(m_{h} / m_{Z})^{2}$.", + "symbol": { + "$m_h$": "Higgs boson mass", + "$m_Z$": "Z boson mass", + "$\\Gamma$": "decay width", + "$h^0$": "Higgs boson in the decay process", + "$Z^0$": "Z boson", + "$\\phi^3$": "Goldstone boson", + "$\\alpha$": "fine-structure constant", + "$\\theta_{w}$": "Weinberg angle", + "$\\tau_{Z}$": "dimensionless parameter related to Z boson" + }, + "chapter": "Final Project III", + "section": "Decays of the Higgs Boson" + }, + { + "id": 67, + "topic": "Others", + "question": "Derive the commutator $[M^{\\mu \\nu}, M^{\\rho \\sigma}]$. Hints :\n- Denote $\\Lambda^{\\prime} \\mu_{\\nu} \\approx \\delta^{\\mu}{ }_{v}+\\chi^{\\mu}{ }_{v}$. Check $(\\Lambda^{-1} \\Lambda^{\\prime} \\Lambda)_{\\rho \\sigma} \\approx \\delta_{\\rho \\sigma}+\\chi_{\\mu \\nu} \\Lambda^{\\mu}{ }_{\\rho} \\Lambda^{v}{ }_{\\sigma}$.\n- Denote $\\Lambda^{\\mu}{ }_{v} \\approx \\delta^{\\mu}{ }_{v}+\\omega^{\\mu}{ }_{v}$. Check $U^{-1}(\\Lambda) M^{\\mu \\nu} U(\\Lambda) \\approx M^{\\mu v}+\\frac{i}{2} \\omega_{\\rho \\sigma}[M^{\\mu \\nu}, M^{\\rho \\sigma}]$.\n- Identify this expression with $\\wedge^{\\mu}{ }_{\\rho} \\Lambda^{\\nu}{ }_{\\sigma} M^{\\rho \\sigma}$ (when simplifying by $\\omega_{\\rho \\sigma}$, one needs to enforce the antisymmetry of the residual factors).", + "final_answer": [ + "[M^{\\mu \\nu}, M^{\\rho \\sigma}]=\\mathfrak{i}(g^{\\nu \\sigma} M^{\\mu \\rho}+g^{\\mu \\rho} M^{v \\sigma}-g^{\\nu \\rho} M^{\\mu \\sigma}-g^{\\mu \\sigma} M^{v \\rho})" + ], + "answer_type": "Expression", + "answer": "We can write\n\n\\begin{aligned}\n(\\Lambda^{-1} \\Lambda^{\\prime} \\Lambda)_{\\rho \\sigma} & =\\Lambda_{\\rho}^{-1 \\mu} \\Lambda_{\\mu}^{\\prime}{ }^{\\nu} \\Lambda_{v \\sigma}=\\Lambda_{\\rho}^{\\mu}(\\delta_{\\mu}{ }^{\\nu}+\\chi_{\\mu}{ }^{\\nu}+\\mathcal{O}(\\chi^{2})) \\Lambda_{v \\sigma} \\\\\n& =\\delta_{\\rho \\sigma}+\\Lambda_{\\rho}^{\\mu} \\Lambda_{\\sigma}^{v} \\chi_{\\mu \\nu}+\\mathcal{O}(\\chi^{2})\n\\end{aligned}\n\n\nTherefore, we have\n\n\\begin{equation*}\n\\mathrm{U}(\\Lambda^{-1} \\Lambda^{\\prime} \\Lambda)=1+\\frac{i}{2} \\Lambda_{\\rho}^{\\mu} \\Lambda_{\\sigma}^{\\nu} \\chi_{\\mu \\nu} M^{\\rho \\sigma}+\\mathcal{O}(\\chi^{2}) \\tag{*}\n\\end{equation*}\n\n\nOn the other hand, we also have\n\n\\begin{aligned}\n\\mathrm{U}(\\Lambda)^{-1} \\mathcal{M}^{\\mu \\nu} \\mathrm{U}(\\Lambda) & =(1-\\frac{i}{2} \\omega_{\\rho \\sigma} M^{\\rho \\sigma}+\\mathcal{O}(\\omega^{2})) \\mathcal{M}^{\\mu \\nu}(1+\\frac{i}{2} \\omega_{\\rho \\sigma} M^{\\rho \\sigma}+\\mathcal{O}(\\omega^{2})) \\\\\n& =M^{\\mu v}+\\frac{i}{2} \\omega_{\\rho \\sigma}[M^{\\mu \\nu}, M^{\\rho \\sigma}]+\\mathcal{O}(\\omega^{2})\n\\end{aligned}\n\n\nThe second term of the right hand side must coincide at order $\\omega^{1}$ with the coefficient of $\\frac{i}{2} \\chi_{\\mu \\nu}$ in $(*)$ :\n\n\\begin{aligned}\n\\frac{i}{2} \\omega_{\\rho \\sigma}[M^{\\mu v}, M^{\\rho \\sigma}] & =\\Lambda_{\\rho}^{\\mu} \\Lambda_{\\sigma}^{v} M^{\\rho \\sigma}|_{\\text {order } \\omega^{1}} \\\\\n& =(\\delta^{\\mu}{ }_{\\rho} \\omega^{v}{ }_{\\sigma}+\\omega^{\\mu}{ }_{\\rho} \\delta^{v}{ }_{\\sigma}) M^{\\rho \\sigma}=\\omega^{v}{ }_{\\sigma} M^{\\mu \\sigma}+\\omega_{\\rho}^{\\mu} M^{\\rho v} \\\\\n& =\\omega_{\\rho \\sigma}(g^{v \\rho} M^{\\mu \\sigma}+g^{\\mu \\sigma} M^{v \\rho})\n\\end{aligned}\n\n\nAt this stage, we cannot simply \"divide\" by $\\omega_{\\rho \\sigma}$, because $\\omega_{\\rho \\sigma}$ is not linearly independent of $\\omega_{\\sigma \\rho}$. Before we can perform this operation, we should explicitly antisymmetrize the coefficient of $\\omega_{\\rho \\sigma}$ in the right hand side by writing\n\n$$\\frac{i}{2} \\omega_{\\rho \\sigma}[M^{\\mu \\nu}, M^{\\rho \\sigma}]=\\frac{1}{2} \\omega_{\\rho \\sigma}(g^{\\nu \\rho} M^{\\mu \\sigma}+g^{\\mu \\sigma} M^{v \\rho}-g^{\\nu \\sigma} M^{\\mu \\rho}-g^{\\mu \\rho} M^{v \\sigma})$$\n\nThen, we obtain the following expression for the commutator of two generators of the Lorentz algebra\n\n$$[M^{\\mu \\nu}, M^{\\rho \\sigma}]=\\mathfrak{i}(g^{\\nu \\sigma} M^{\\mu \\rho}+g^{\\mu \\rho} M^{v \\sigma}-g^{\\nu \\rho} M^{\\mu \\sigma}-g^{\\mu \\sigma} M^{v \\rho}) $$", + "symbol": { + "$M^{\\mu \\nu}$": "generator of the Lorentz transformation", + "$M^{\\rho \\sigma}$": "generator of the Lorentz transformation", + "$\\chi^{\\mu}{ }_{v}$": "small variation parameter in transformation", + "$\\omega^{\\mu}{ }_{v}$": "infinitesimal parameter for transformation", + "$g^{\\nu \\sigma}$": "Minkowski metric tensor component", + "$g^{\\mu \\rho}$": "Minkowski metric tensor component", + "$\\delta^{\\mu}{ }_{v}$": "Kronecker delta function component", + "$\\Lambda^{\\mu}{ }_{v}$": "Lorentz transformation matrix component", + "$\\wedge^{\\mu}{ }_{\\rho}$": "part of a transformation expression", + "$\\Lambda_{\\rho}^{\\mu}$": "inverse Lorentz transformation matrix component", + "$\\Lambda_{\\sigma}^{v}$": "Lorentz transformation matrix component" + }, + "chapter": "Basics of Quantum Field Theory", + "section": "Basics of Quantum Field Theory" + }, + { + "id": 68, + "topic": "Others", + "question": "Derive the commutators $[\\mathrm{M}^{\\mu \\nu}, \\mathrm{P}^{\\rho}]$ and $[\\mathrm{P}^{\\mu}, \\mathrm{P}^{\\nu}]$. Hints :\n- Note that $\\Lambda^{-1} \\mathrm{a} \\Lambda$ is a translation by $(\\Lambda^{-1} \\mathrm{a})_{\\rho}=\\mathrm{a}_{\\mu} \\wedge^{\\mu}{ }_{\\rho}$.\n- Denote $\\Lambda^{\\mu}{ }_{v} \\approx \\delta^{\\mu}{ }_{v}+\\omega^{\\mu}{ }_{v}$. Check that $U^{-1}(\\Lambda) P^{\\mu} U(\\Lambda) \\approx P^{\\mu}+\\frac{i}{2} \\omega_{\\rho \\sigma}[P^{\\mu}, M^{\\rho \\sigma}]$.\n- Identify this with $\\wedge^{\\mu}{ }_{\\rho} \\mathrm{P}^{\\rho}$.\n- What elementary property of translations implies $[\\mathrm{P}^{\\mu}, \\mathrm{P}^{\\nu}]=0$ ?", + "final_answer": [ + "[\\mathrm{M}^{\\mu \\nu}, \\mathrm{P}^{\\rho}] = \\mathfrak{i}(g^{\\rho \\mu} \\mathrm{P}^{\\nu} - g^{\\rho \\nu} \\mathrm{P}^{\\mu})" + ], + "answer_type": "Expression", + "answer": "The action of $\\Lambda^{-1} \\mathrm{a} \\Lambda$ on a point $x$ can be written explicitly as follows,\n\n\\begin{aligned}\n{[(\\Lambda^{-1} \\mathrm{a} \\Lambda) x]_{\\rho} } & =\\Lambda_{\\rho}^{-1 \\mu}(a_{\\mu}+\\Lambda_{\\mu}{ }^{\\nu} \\chi_{v})=\\Lambda_{\\rho}^{-1 \\mu} a_{\\mu}+\\delta_{\\rho}{ }^{\\nu} \\chi_{v} \\\\\n& =x_{\\rho}+a_{\\mu} \\Lambda_{\\rho}^{\\mu}\n\\end{aligned}\n\n\nTherefore, $\\Lambda^{-1} \\mathrm{a} \\Lambda$ acts as a translation of $x_{\\rho}$ by an amount $a_{\\mu} \\Lambda^{\\mu}{ }_{\\rho}$. Consider now the representation $\\mathrm{U}(\\Lambda^{-1} \\mathrm{a} \\Lambda)$. On the one hand, the previous results tells us that\n\n\\begin{equation*}\n\\mathrm{U}(\\Lambda^{-1} \\mathrm{a} \\Lambda)=1+i \\mathrm{a}_{\\mu} \\Lambda_{\\rho}^{\\mu}{ }_{\\rho}^{\\rho} \\mathrm{P}^{\\rho}+\\mathcal{O}(\\mathrm{a}^{2}) \\tag{*}\n\\end{equation*}\n\n\nOn the other hand, we have\n\n\\begin{aligned}\n\\mathrm{U}(\\Lambda^{-1}) \\mathrm{P}^{\\mu} \\mathrm{U}(\\Lambda) & =(1-\\frac{i}{2} \\omega_{\\rho \\sigma} M^{\\rho \\sigma}+\\mathcal{O}(\\omega^{2})) \\mathrm{P}^{\\mu}(1+\\frac{i}{2} \\omega_{\\rho \\sigma} M^{\\rho \\sigma}+\\mathcal{O}(\\omega^{2})) \\\\\n& =\\mathrm{P}^{\\mu}+\\frac{i}{2} \\omega_{\\rho \\sigma}[\\mathrm{P}^{\\mu}, M^{\\rho \\sigma}]+\\mathcal{O}(\\omega^{2})\n\\end{aligned}\n\n\nThe second term of the right hand side must coincide at order $\\omega^{1}$ with the coefficient of $i a_{\\mu}$ in $(*)$ :\n\n\\begin{aligned}\n\\frac{i}{2} \\omega_{\\rho \\sigma}[\\mathrm{P}^{\\mu}, M^{\\rho \\sigma}] & =\\Lambda_{\\rho}^{\\mu} \\mathrm{P}^{\\rho}|_{\\text {order } \\omega^{1}} \\\\\n& =\\omega_{\\rho}^{\\mu}{ }_{\\rho} \\mathrm{P}^{\\rho}=-\\omega_{\\rho \\sigma} g^{\\mu \\sigma} \\mathrm{P}^{\\rho}=\\frac{1}{2} \\omega_{\\rho \\sigma}(g^{\\mu \\rho} \\mathrm{P}^{\\sigma}-g^{\\mu \\sigma} \\mathrm{P}^{\\rho})\n\\end{aligned}\n\nwhere in the last equality we have performed the antisymmetrization on the indices $\\rho, \\sigma$. Therefore, we conclude that\n\n[P^{\\mu}, M^{\\rho \\sigma}]=\\mathfrak{i}(g^{\\mu \\sigma} P^{\\rho}-g^{\\mu \\rho} P^{\\sigma})\n\n\nThe fact that $P^{\\mu}$ and $P^{v}$ commute simply follows from the fact that the order in which two successive translations are performed does not matter.", + "symbol": { + "$\\mathrm{M}^{\\mu \\nu}$": "Lorentz transformation generator with indices (mu, nu)", + "$\\mathrm{P}^{\\rho}$": "momentum operator with index rho", + "$\\Lambda$": "Lorentz transformation matrix", + "$a_{\\mu}$": "translation vector component with index mu", + "$\\omega^{\\mu}{ }_{v}$": "infinitesimal rotation parameter with indices (mu, v)", + "$x_{\\rho}$": "spatial component with index rho", + "$g^{\\mu \\sigma}$": "metric tensor component with indices (mu, sigma)" + }, + "chapter": "Basics of Quantum Field Theory", + "section": "Basics of Quantum Field Theory" + }, + { + "id": 69, + "topic": "Others", + "question": "Calculate the expression in coordinate space of the retarded propagator given in eq. \\begin{align}\n\\tilde{G}_R^0(\\kappa) = \\frac{i}{(\\kappa_0 + i0^+)^2 - (\\kappa^2 + m^2)}.\n\\end{align}\n\nHint : perform the $\\mathrm{k}_{0}$ integral in the complex plane with the theorem of residues. The remaining integrals are elementary.", + "final_answer": [ + "G_{R}^{0}(x, y) = -\\frac{i}{2 \\pi} \\theta(r^{0}) \\delta(r_{0}^{2}-r^{2})" + ], + "answer_type": "Expression", + "answer": "The free retarded propagator in coordinate space is given by the following Fourier integral:\n$$ G_{R}^{0}(x, y)=i \\int \\frac{d^{4} k}{(2 \\pi)^{4}} \\frac{e^{-i k \\cdot(x-y)}}{(k^{0}+i 0^{+})^{2}-k^{2}}. $$\n\nThe integrand has two poles in the complex plane of the variable $k^{0}$, located at $k^{0}= \\pm|\\mathbf{k}|-\\mathfrak{i} 0^{+}$. The integration over $k^{0}$ can be performed with the theorem of residues by completing the real axis with a semi-circle at infinity. Whether this semi-circle should be in the upper or lower half plane is determined by the request that the exponential factor in the numerator does not diverge when going to infinity in the imaginary direction. To see this, write $k^{0}=k_{r}^{0}+i k_{i}^{0}$. Then, we have\n\ne^{-i k^{0}(x^{0}-y^{0})}=e^{-i k_{r}^{0}(x^{0}-y^{0})} e^{k_{i}^{0}(x^{0}-y^{0})} .\n\n\nThe dangerous factor is the second one, since the argument of the exponential is real. If $x^{0}-y^{0}<0$, this exponential remains bounded if we close the contour in the upper half plane. Since the integrand has no poles on this side, the integral is zero. If on the contrary $x^{0}-y^{0}>0$, we must close the contour in the lower half plane, and the theorem of residues gives non-zero contributions from the two poles. By writing\n\n$$\\frac{\\mathfrak{i}}{(k^{0}+i 0^{+})^{2}-\\mathbf{k}^{2}}=\\frac{\\mathfrak{i}}{2|\\mathbf{k}|}[\\frac{1}{k^{0}+\\mathfrak{i} 0^{+}-|\\mathbf{k}|}-\\frac{1}{k^{0}+\\mathfrak{i} 0^{+}+|\\mathbf{k}|}],$$\n\nwe obtain easily the corresponding residues, and the propagator now reads\n\n$$ G_{R}^{0}(x, y)=\\theta(x^{0}-y^{0}) \\int \\frac{d^{3} \\mathbf{k}}{(2 \\pi)^{3}} \\frac{e^{i \\mathbf{k} \\cdot(x-y)}}{2|\\mathbf{k}|}[e^{-i|\\mathbf{k}|(x^{0}-y^{0})}-e^{i|\\mathbf{k}|(x^{0}-y^{0})}] .$$\n\n\nIn order to make the notations more compact, let us denote $r^{0} \\equiv x^{0}-y^{0}$ and $\\mathbf{r} \\equiv \\boldsymbol{x}-\\mathbf{y}$. It is convenient to perform the integration over $k$ in spherical coordinates with a polar axis in the direction of $r$. This leads to\n\n\\begin{aligned}\nG_{R}^{0}(x, y) & =\\frac{i}{8 \\pi^{2} r} \\theta(r^{0}) \\int_{0}^{\\infty} d k[e^{i k r}-e^{-i k r}][e^{i k r^{0}}-e^{-i k r^{0}}] \\\\\n& =\\frac{i}{8 \\pi^{2} r} \\theta(r^{0}) \\int_{0}^{\\infty} d k[e^{i k(r+r^{0})}+e^{-i k(r+r^{0})}-e^{i k(r-r^{0})}-e^{-i k(r-r^{0})}] \\\\\n& =\\frac{i}{8 \\pi^{2} r} \\theta(r^{0}) \\int_{-\\infty}^{+\\infty} d k[e^{i k(r+r^{0})}-e^{i k(r-r^{0})}] \\\\\n& =\\frac{i}{4 \\pi r} \\theta(r^{0})[\\delta(r+r^{0})-\\delta(r-r^{0})]=-\\frac{i}{2 \\pi} \\theta(r^{0}) \\delta(r_{0}^{2}-r^{2})\n\\end{aligned}\n\n\nThe proportionality to $\\theta(r^{0})$ makes this propagator retarded, while the delta function with support on the light-cone makes it causal. With a mass, the integral over $k$ would be much more complicated (the result is expressible in terms of Bessel functions), with a support restricted to $r_{0}^{2}-r^{2} \\geq 0$ (therefore, it is still causal).", + "symbol": { + "$\\tilde{G}_R^0$": "retarded propagator in momentum space", + "$\\kappa$": "four-momentum", + "$\\kappa_0$": "energy component of four-momentum", + "$m$": "mass", + "$G_{R}^{0}$": "free retarded propagator in coordinate space", + "$x$": "four-position coordinate", + "$y$": "four-position coordinate", + "$k$": "momentum", + "$k^{0}$": "energy component of momentum", + "$\\mathbf{k}$": "spatial components of momentum", + "$x^{0}$": "time component of four-position $x$", + "$y^{0}$": "time component of four-position $y$", + "$r^{0}$": "time component difference between four-positions $x$ and $y$", + "$\\mathbf{r}$": "spatial component difference between four-positions $x$ and $y$", + "$r$": "spatial distance between four-positions $x$ and $y$" + }, + "chapter": "Basics of Quantum Field Theory", + "section": "Basics of Quantum Field Theory" + }, + { + "id": 70, + "topic": "Others", + "question": "Consider a hypothetical quantum field theory with a kinetic term $$\\mathcal{L}_{0} \\equiv-\\frac{1}{2 \\mu^{2}} \\phi(\\square+m^{2})^{2} \\phi,$$ where $\\mu$ is a constant with the dimension of mass. What is the expression for Källen-Lehman spectral function for this theory in this theory?", + "final_answer": [ + "2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})" + ], + "answer_type": "Expression", + "answer": "The free propagator is obtained from the inverse of the operator between the two fields. In the case of the first example, its momentum space expression reads\n\n$$\\mathcal{G}_{\\mathrm{R}}^{0}(p)=-\\frac{\\mathfrak{i} \\mu^{2}}{((p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-m^{2})^{2}} $$\n\nwhere the $\\mathrm{i}^{+}$prescription we have chosen ensures that all the poles are located below the real axis in the complex plane of the variable $p^{0}$. Therefore, this is indeed the retarded propagator. Recall now that the\nfree retarded propagator for the usual scalar kinetic term is\n\n$$\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\frac{\\mathrm{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{m}^{2}}$$\n\n\nThe two propagators are thus related to one another by a derivative with respect to the squared mass. More precisely, one has\n\n$$\\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p)$$\n\n\nThe Källen-Lehman representation of the standard retarded propagator, $G_{R}^{0}(p)$, reads\n\n$$\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\int_{0}^{\\infty} \\frac{\\mathrm{d} M^{2}}{2 \\pi} \\rho(M^{2}) \\frac{\\mathfrak{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{M}^{2}}$$\n\n(Of course, in the case of the free propagator, the spectral function is trivial, namely $\\rho(M^{2})=2 \\pi \\delta(M^{2}-$ $\\mathrm{m}^{2}$ ). In the interacting case, it would be more complicated, but still positive definite). Note that in this representation, the dependence on the mass $m^{2}$ is carried only by the spectral function $\\rho(M^{2})$. Taking a derivative with respect to $\\mathrm{m}^{2}$ of this equation, we obtain\n\n$$\\mathcal{G}_{\\mathrm{R}}^{0}(p)=\\int_{0}^{\\infty} \\frac{\\mathrm{d} M^{2}}{2 \\pi}[-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} \\rho(M^{2})] \\frac{\\mathfrak{i}}{(p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-M^{2}}$$\n\nand the (free) spectral function in the theory with the higher order kinetic term is the factor between the square brackets, i.e.\n\n$$-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} \\rho(M^{2})=2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})$$\n\nThis distribution is not positive definite. Indeed, when applied to a positive definite test function $f(M^{2})$, we obtain by integration by parts\n\n$$ \\int_{0}^{\\infty} \\frac{d M^{2}}{2 \\pi} f(M^{2}) 2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})=-\\mu^{2} f^{\\prime}(m^{2}) $$\n\nthat can have either sign. The non-positivity of this spectral function means that this theory must contain states that contribute negatively to the sum in $$\\begin{align}\n\\rho(M^2) \\equiv 2\\pi \\sum_{\\text{classes } \\alpha} \\delta(M^2 - m_\\alpha^2) \n\\left\\langle 0_{\\text{out}} \\left| \\phi(0) \\right| \\alpha_0 \\right\\rangle\n\\left\\langle \\alpha_0 \\left| \\phi(0) \\right| 0_{\\text{in}} \\right\\rangle.\n\\end{align}$$, and therefore is not unitary in the usual sense.", + "symbol": { + "$\\mu$": "constant with the dimension of mass", + "$\\phi$": "field", + "$m$": "mass", + "$\\mathcal{G}_{\\mathrm{R}}^{0}$": "retarded propagator in the modified theory", + "$G_{\\mathrm{R}}^{0}$": "retarded propagator in the standard theory", + "$\\rho(M^{2})$": "spectral function", + "$M$": "variable related to mass squared", + "$p$": "momentum", + "$p^{0}$": "energy component of momentum", + "$\\mathbf{p}$": "spatial component of momentum" + }, + "chapter": "Basics of Quantum Field Theory", + "section": "Basics of Quantum Field Theory" + }, + { + "id": 71, + "topic": "Others", + "question": "For the theory with the kinetic term $\\mathcal{L}_{0} \\equiv-\\frac{1}{2 \\mu^{2}} \\phi(\\square+m^{2})^{2} \\phi$, what is the relationship between its retarded propagator $\\mathcal{G}_{R}^{0}(p)$ and the free retarded propagator $G_{R}^{0}(p)$ of a standard scalar field? (Express $\\mathcal{G}_{R}^{0}(p)$ in terms of $G_{R}^{0}(p)$)", + "final_answer": [ + "\\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p)" + ], + "answer_type": "Expression", + "answer": "The free propagator is obtained from the inverse of the operator between the two fields. In the case of the first example, its momentum space expression reads\n\n$$ \\mathcal{G}_{\\mathrm{R}}^{0}(p)=-\\frac{\\mathfrak{i} \\mu^{2}}{((p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-m^{2})^{2}} $$\n\nwhere the $\\mathrm{i}^{+}$prescription we have chosen ensures that all the poles are located below the real axis in the complex plane of the variable $p^{0}$. Therefore, this is indeed the retarded propagator. Recall now that the\nfree retarded propagator for the usual scalar kinetic term is\n\n$$\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\frac{\\mathrm{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{m}^{2}}$$\n\n\nThe two propagators are thus related to one another by a derivative with respect to the squared mass. More precisely, one has\n\n$$ \\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p) $$", + "symbol": { + "$\\mathcal{L}_{0}$": "kinetic term", + "$\\mu$": "mass scale factor", + "$\\phi$": "scalar field", + "$\\square$": "d'Alembertian operator", + "$m$": "mass of the field", + "$\\mathcal{G}_{R}^{0}(p)$": "retarded propagator in modified theory", + "$G_{R}^{0}(p)$": "free retarded propagator of a standard scalar field", + "$p$": "momentum", + "$p^{0}$": "energy component of momentum", + "$\\mathbf{p}$": "spatial momentum vector" + }, + "chapter": "Basics of Quantum Field Theory", + "section": "Basics of Quantum Field Theory" + }, + { + "id": 72, + "topic": "Others", + "question": "For the theory with the kinetic term $\\mathcal{L}_{0} \\equiv-\\frac{1}{2 \\mu^{2}} \\phi(\\square+m^{2})^{2} \\phi$, what is its the free retarded propagator?", + "final_answer": [ + "\\frac{\\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{2}^{2}}-\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{1}^{2}}]" + ], + "answer_type": "Expression", + "answer": "The free propagator is obtained from the inverse of the operator between the two fields. \nThe free retarded propagator reads\n\n\\begin{aligned}\n \\mathcal{G}_{\\mathrm{R}}^{0}(\\mathrm{p})&=-\\frac{\\mathfrak{i} \\mu^{2}}{((\\mathrm{p}^{0}+\\mathfrak{i 0 ^ { + }})^{2}-\\mathrm{p}^{2}-\\mathrm{m}_{1}^{2})((\\mathrm{p}^{0}+\\mathfrak{i 0 ^ { + }})^{2}-\\mathrm{p}^{2}-\\mathrm{m}_{2}^{2})} \\\\\n& =\\frac{\\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{2}^{2}}-\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{1}^{2}}] .\n\\end{aligned}", + "symbol": { + "$\\mathcal{L}_{0}$": "kinetic term of the theory", + "$\\mu$": "parameter associated with the kinetic term", + "$\\phi$": "field in the theory", + "$m$": "mass parameter in the kinetic term", + "$\\mathcal{G}_{\\mathrm{R}}^{0}$": "free retarded propagator", + "$\\mathrm{p}^{0}$": "energy component of momentum", + "$\\mathrm{p}$": "momentum", + "$\\mathrm{m}_{1}$": "mass parameter (1)", + "$\\mathrm{m}_{2}$": "mass parameter (2)" + }, + "chapter": "Basics of Quantum Field Theory", + "section": "Basics of Quantum Field Theory" + }, + { + "id": 73, + "topic": "Others", + "question": "For the theory with the kinetic term $-\\frac{1}{2 \\mu^{2}} \\phi(\\square+m_{1}^{2})(\\square+m_{2}^{2}) \\phi$, what is its Källen-Lehman spectral function?", + "final_answer": [ + "\\frac{2 \\pi \\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\delta(M^{2}-m_{2}^{2})-\\delta(M^{2}-m_{1}^{2})]" + ], + "answer_type": "Expression", + "answer": "The free propagator is obtained from the inverse of the operator between the two fields. In the case of the first example, its momentum space expression reads\n\n\\mathcal{G}_{\\mathrm{R}}^{0}(p)=-\\frac{\\mathfrak{i} \\mu^{2}}{((p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-m^{2})^{2}}\n\nwhere the $\\mathrm{i}^{+}$prescription we have chosen ensures that all the poles are located below the real axis in the complex plane of the variable $p^{0}$. Therefore, this is indeed the retarded propagator. Recall now that the\nfree retarded propagator for the usual scalar kinetic term is\n\n\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\frac{\\mathrm{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{m}^{2}}\n\n\nThe two propagators are thus related to one another by a derivative with respect to the squared mass. More precisely, one has\n\n\\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p)\n\n\nThe Källen-Lehman representation of the standard retarded propagator, $G_{R}^{0}(p)$, reads\n\n\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\int_{0}^{\\infty} \\frac{\\mathrm{d} M^{2}}{2 \\pi} \\rho(M^{2}) \\frac{\\mathfrak{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{M}^{2}}\n\n(Of course, in the case of the free propagator, the spectral function is trivial, namely $\\rho(M^{2})=2 \\pi \\delta(M^{2}-$ $\\mathrm{m}^{2}$ ). In the interacting case, it would be more complicated, but still positive definite). Note that in this representation, the dependence on the mass $m^{2}$ is carried only by the spectral function $\\rho(M^{2})$. Taking a derivative with respect to $\\mathrm{m}^{2}$ of this equation, we obtain\n\n\\mathcal{G}_{\\mathrm{R}}^{0}(p)=\\int_{0}^{\\infty} \\frac{\\mathrm{d} M^{2}}{2 \\pi}[-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} \\rho(M^{2})] \\frac{\\mathfrak{i}}{(p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-M^{2}}\n\nand the (free) spectral function in the theory with the higher order kinetic term is the factor between the square brackets, i.e.\n\n-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} \\rho(M^{2})=2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})\n\n\nThis distribution is not positive definite. Indeed, when applied to a positive definite test function $f(M^{2})$, we obtain by integration by parts\n\n\\int_{0}^{\\infty} \\frac{d M^{2}}{2 \\pi} f(M^{2}) 2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})=-\\mu^{2} f^{\\prime}(m^{2})\n\nthat can have either sign. The non-positivity of this spectral function means that this theory must contain states that contribute negatively to the sum in eq. (1.113), and therefore is not unitary in the usual sense.\n\nConsider now the second example, more general, of theory with higher derivatives in the kinetic term. This time, the free retarded propagator reads\n\n\\begin{aligned}\n& \\mathcal{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=-\\frac{\\mathfrak{i} \\mu^{2}}{((\\mathrm{p}^{0}+\\mathfrak{i 0 ^ { + }})^{2}-\\mathrm{p}^{2}-\\mathrm{m}_{1}^{2})((\\mathrm{p}^{0}+\\mathfrak{i 0 ^ { + }})^{2}-\\mathrm{p}^{2}-\\mathrm{m}_{2}^{2})} \\\\\n& =\\frac{\\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{2}^{2}}-\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{1}^{2}}] .\n\\end{aligned}\n\n\nBy the same reasoning as before, we see that this corresponds to the following spectral function,\n\n$$\\frac{2 \\pi \\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\delta(M^{2}-m_{2}^{2})-\\delta(M^{2}-m_{1}^{2})]$$\n\nwhich is also not positive definite (one delta function contributes positively and the other one negatively). The bottom line of this exercise is that kinetic terms with higher derivatives in general lead to serious problems with unitarity. In the case of the second example, the theory could nevertheless have a practical use if $m_{1} \\gg m_{2}$, provided one stays in an energy range much lower than $m_{1}$ in order not to excite the mode that has a negative norm (such a kinetic term is in fact the basis of the Pauli-Villars regularization).", + "symbol": { + "$\\mu$": "parameter in the kinetic term", + "$\\phi$": "scalar field", + "$\\square$": "d'Alembertian operator", + "$m_1$": "first mass parameter in the kinetic term", + "$m_2$": "second mass parameter in the kinetic term", + "$p$": "momentum", + "$p^0$": "energy component of the momentum", + "$\\mathbf{p}$": "spatial momentum", + "$m$": "mass parameter", + "$M$": "mass variable in the Källen-Lehman spectral function", + "$\\rho$": "spectral function" + }, + "chapter": "Basics of Quantum Field Theory", + "section": "Basics of Quantum Field Theory" + }, + { + "id": 74, + "topic": "Others", + "question": "Consider all the 1-loop and 2-loop graphs for the six-point function in a scalar theory with a $\\phi^{4}$ interaction. Write the corresponding Feynman integrals.\nIf the answer exists in an integral, then find the integrand", + "final_answer": [ + "\\frac{(-i \\lambda)^{3} i^{3}}{(\\ell^{2}-m^{2}+i 0^{+})((\\ell+p)^{2}-m^{2}+i 0^{+})((\\ell-q)^{2}-m^{2}+i 0^{+})} \\frac{(-i \\lambda)^{4} i^{2} \\mathfrak{i}^{3}}{(\\ell^{\\prime 2}-m^{2}+i 0^{+})((\\ell+\\ell^{\\prime}+p+r)^{2}-m^{2}+i 0^{+}) (\\ell^{2}-m^{2}+i 0^{+})((\\ell+p)^{2}-m^{2}+i 0^{+})((\\ell-q)^{2}-m^{2}+i 0^{+})} \\frac{(-i \\lambda)^{4} i^{2} \\mathfrak{i}^{3}}{[(\\ell^{2}-m^{2}+i 0^{+})((\\ell+p)^{2}-m^{2}+i 0^{+})((\\ell-q)^{2}-m^{2}+i 0^{+})]^{2}}" + ], + "answer_type": "Expression", + "answer": "$$\\int \\frac{d^{D} \\ell}{(2 \\pi)^{D}} \\frac{(-i \\lambda)^{3} i^{3}}{(\\ell^{2}-m^{2}+i 0^{+})((\\ell+p)^{2}-m^{2}+i 0^{+})((\\ell-q)^{2}-m^{2}+i 0^{+})}$$\n\n\\begin{aligned}\n&\\frac{1}{2} \\int \\frac{d^{D} \\ell d^{D} \\ell^{\\prime}}{(2 \\pi)^{2 D}} \\frac{(-i \\lambda)^{4} i^{2}}{(\\ell^{\\prime 2}-m^{2}+i 0^{+})((\\ell+\\ell^{\\prime}+p+r)^{2}-m^{2}+i 0^{+})} \\\\\n& \\times \\frac{\\mathfrak{i}^{3}}{(\\ell^{2}-m^{2}+i 0^{+})((\\ell+p)^{2}-m^{2}+i 0^{+})((\\ell-q)^{2}-m^{2}+i 0^{+})} . \n\\end{aligned}\n\n\n\\begin{equation}\n\\begin{split}\n&\\frac{1}{2} \\int \\frac{d^{D} \\ell d^{D} \\ell^{\\prime}}{(2 \\pi)^{2 D}}\\frac{(-i \\lambda)^{4} i^{2}}{(\\ell^{2}-m^{2}+i 0^{+})((\\ell+p)^{2}-m^{2}+i 0^{+})((\\ell-q)^{2}-m^{2}+i 0^{+})} . \\\\\n&\\times \\frac{\\mathfrak{i}^{3}}{(\\ell^{2}-m^{2}+i 0^{+})((\\ell+p)^{2}-m^{2}+i 0^{+})((\\ell-q)^{2}-m^{2}+i 0^{+})}\n\\end{split}\n\\end{equation}\n\n\nNote that there are many different ways of attaching six lines labeled 1 to 6 to these basic topologies.", + "symbol": { + "$\\phi$": "scalar field", + "$\\lambda$": "coupling constant", + "$\\ell$": "loop momentum", + "$m$": "mass", + "$p$": "momentum of the particle", + "$q$": "another particle momentum", + "$r$": "yet another particle momentum", + "$D$": "dimension of spacetime", + "$\\ell^{\\prime}$": "additional loop momentum" + }, + "chapter": "Perturbation Theory", + "section": "Perturbation Theory" + }, + { + "id": 75, + "topic": "Others", + "question": "Calculate $\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{v} \\gamma^{\\rho} \\gamma^{\\sigma})$.", + "final_answer": [ + "4(g^{\\rho \\sigma} g^{\\mu \\nu}-g^{\\nu \\sigma} g^{\\mu \\rho}+g^{\\mu \\sigma} g^{\\nu \\rho})" + ], + "answer_type": "Expression", + "answer": "With four Dirac matrices, we can write\n\n\\begin{align}\n\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{v} \\gamma^{\\rho} \\gamma^{\\sigma})= & -\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{v} \\gamma^{\\sigma} \\gamma^{\\rho})+2 g^{\\rho \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu}) \\\\\n= & +\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\sigma} \\gamma^{v} \\gamma^{\\rho})+2 g^{\\rho \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu})-2 g^{v \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\rho}) \\\\\n= & -\\operatorname{tr}(\\gamma^{\\sigma} \\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\rho})+2 g^{\\rho \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu})-2 g^{v \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\rho}) \\\\\n& +2 g^{\\mu \\sigma} \\operatorname{tr}(\\gamma^{\\nu} \\gamma^{\\rho}) .\n\\end{align}\n\nThis gives:\n\n\\begin{align}\n\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\rho} \\gamma^{\\sigma}) & =g^{\\rho \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu})-g^{\\nu \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\rho})+g^{\\mu \\sigma} \\operatorname{tr}(\\gamma^{\\nu} \\gamma^{\\rho}) \\\\\n& =4(g^{\\rho \\sigma} g^{\\mu \\nu}-g^{\\nu \\sigma} g^{\\mu \\rho}+g^{\\mu \\sigma} g^{\\nu \\rho}) .\n\\end{align}", + "symbol": { + "$\\gamma^{\\mu}$": "Dirac gamma matrix with index mu", + "$\\gamma^{v}$": "Dirac gamma matrix with index v", + "$\\gamma^{\\rho}$": "Dirac gamma matrix with index rho", + "$\\gamma^{\\sigma}$": "Dirac gamma matrix with index sigma", + "$g^{\\rho \\sigma}$": "metric tensor component with indices rho and sigma", + "$g^{v \\sigma}$": "metric tensor component with indices v and sigma", + "$g^{\\mu \\nu}$": "metric tensor component with indices mu and nu", + "$g^{\\nu \\sigma}$": "metric tensor component with indices nu and sigma", + "$g^{\\mu \\rho}$": "metric tensor component with indices mu and rho", + "$g^{\\mu \\sigma}$": "metric tensor component with indices mu and sigma", + "$g^{\\nu \\rho}$": "metric tensor component with indices nu and rho" + }, + "chapter": "Quantum Electrodynamics", + "section": "Quantum Electrodynamics" + }, + { + "id": 76, + "topic": "Others", + "question": "Consider two coherent states $|\\chi_{\\text {in }}\\rangle$ and $|\\vartheta_{\\text {in }}\\rangle$. The state $|\\chi_{\\text {in }}\\rangle$ is defined by the function $\\chi(\\mathbf{k}) \\equiv(2 \\pi)^{3} \\chi_{0} \\delta(\\mathbf{k})$, and the state $|\\vartheta_{\\text {in }}\\rangle$ is defined by the function $\\vartheta(\\mathbf{k}) \\equiv(2 \\pi)^{3} \\vartheta_{0} \\delta(\\mathbf{k})$, where $\\chi_0$ and $\\vartheta_0$ are constants, and $\\delta(\\mathbf{k})$ is the 3-dimensional Dirac delta function. Calculate the overlap $\\langle\\vartheta_{\\text {in }} \\mid \\chi_{\\text {in }}\\rangle$.", + "final_answer": [ + "\\exp (-\\frac{V|\\chi_{0}-\\vartheta_{0}|^{2}}{4 m})" + ], + "answer_type": "Expression", + "answer": "Consider now two such coherent states, $|\\chi_{\\text {in }}\\rangle,|\\vartheta_{\\text {in }}\\rangle$, in the special case where their defining functions have only a zero mode: $\\chi(\\mathbf{k}) \\equiv(2 \\pi)^{3} \\chi_{0} \\delta(\\mathbf{k}), \\vartheta(\\mathbf{k}) \\equiv(2 \\pi)^{3} \\vartheta_{0} \\delta(\\mathbf{k})$. The overlap of these two states is given by\n\n\\begin{align}\n\\langle\\vartheta_{\\text {in }} \\mid \\chi_{\\text {in }}\\rangle= & \\exp (-\\frac{1}{2} \\int \\frac{d^{3} \\mathbf{k}}{(2 \\pi)^{3} 2 \\mathrm{E}_{\\mathrm{k}}}[|\\chi(\\mathbf{k})|^{2}+|\\vartheta(\\mathbf{k})|^{2}]) \\\\\n& \\times\\langle 0_{\\text {in }}| \\exp (\\int \\frac{d^{3} \\mathrm{k}}{(2 \\pi)^{3} 2 \\mathrm{E}_{\\mathrm{k}}} \\vartheta^{*}(\\mathbf{k}) \\mathrm{a}_{\\mathbf{k}, \\text { in }}) \\exp (\\int \\frac{\\mathrm{d}^{3} \\mathbf{k}}{(2 \\pi)^{3} 2 \\mathrm{E}_{\\mathrm{k}}} \\chi(\\mathbf{k}) \\mathrm{a}_{\\mathrm{k}, \\text { in }}^{\\dagger})|0_{\\text {in }}\\rangle \\\\\n= & \\exp (-\\frac{1}{2} \\int \\frac{d^{3} \\mathbf{k}}{(2 \\pi)^{3} 2 \\mathrm{E}_{\\mathrm{k}}}|\\chi(\\mathbf{k})-\\vartheta(\\mathbf{k})|^{2}) \\\\\n= & \\exp (-\\frac{V|\\chi_{0}-\\vartheta_{0}|^{2}}{4 m})\n\\end{align}\n\n\nTherefore, spatially homogeneous coherent states (i.e. ground states of quadratic theories shifted by a uniform field), have an exponentially suppressed overlap, and the argument of the exponential is proportional to the volume. Thus, distinct coherent states of this type are orthogonal if the volume is infinite.", + "symbol": { + "$|\\chi_{\\text {in }}\\rangle$": "coherent state defined by chi", + "$|\\vartheta_{\\text {in }}\\rangle$": "coherent state defined by vartheta", + "$\\chi_{0}$": "constant characterizing chi", + "$\\vartheta_{0}$": "constant characterizing vartheta", + "$V$": "volume", + "$m$": "mass" + }, + "chapter": "Spontaneous Symmetry Breaking", + "section": "Spontaneous Symmetry Breaking" + }, + { + "id": 77, + "topic": "Others", + "question": "Using Weyl's prescription for quantization, where a classical quantity $f(q, p)$ is mapped to an operator $F(Q,P)$ by\n\n$$ F(Q, P) \\equiv \\int \\frac{d p d q d \\mu d v}{(2 \\pi)^{2}} f(q, p) e^{i(\\mu(Q-q)+v(P-p))}, $$\n\ncalculate the quantum operator corresponding to $f(q,p) = qp$. \n(You may use the results from previous parts that for $g(q,p) = (\\alpha q + \\beta p)^k$, its Weyl map is $(\\alpha Q + \\beta P)^k$. Specifically, $q^2$ maps to $Q^2$, $p^2$ maps to $P^2$, and $(q+p)^2$ maps to $(Q+P)^2$.)", + "final_answer": [ + "\\frac{1}{2}(QP+PQ)" + ], + "answer_type": "Expression", + "answer": "To obtain the Weyl mapping of $qp$, we use the polarization identity: $q p=\\frac{1}{2}((q+p)^{2}-q^{2}-p^{2})$.\nFrom the previous result $(\\alpha q+\\beta p)^{n} \\rightarrow (\\alpha Q+\\beta P)^{n}$, we have the following specific mappings:\n\nFor $q^2$: set $\\alpha=1, \\beta=0, n=2$. So, $q^{2} \\rightarrow Q^{2}$.\nFor $p^2$: set $\\alpha=0, \\beta=1, n=2$. So, $p^{2} \\rightarrow P^{2}$.\nFor $(q+p)^2$: set $\\alpha=1, \\beta=1, n=2$. So, $(q+p)^{2} \\rightarrow (Q+P)^{2}$.\n\nWe know that $(Q+P)^2 = (Q+P)(Q+P) = Q^2 + QP + PQ + P^2$, because $Q$ and $P$ do not generally commute.\n\nNow, we apply the Weyl mapping to $qp$ using its expression in terms of squares:\n\n\\text{Weyl}[qp] = \\text{Weyl}[\\frac{1}{2}((q+p)^{2}-q^{2}-p^{2})]\n\nDue to the linearity of the Weyl mapping:\n\n\\text{Weyl}[qp] = \\frac{1}{2}(\\text{Weyl}[(q+p)^{2}] - \\text{Weyl}[q^{2}] - \\text{Weyl}[p^{2}])\n\nSubstituting the mapped operators:\n\n\\text{Weyl}[qp] = \\frac{1}{2}((Q+P)^{2} - Q^{2} - P^{2})\n\n\n\\text{Weyl}[qp] = \\frac{1}{2}((Q^{2}+QP+PQ+P^{2}) - Q^{2} - P^{2})\n\n\n\\text{Weyl}[qp] = \\frac{1}{2}(QP+PQ)\n\nTherefore, the Weyl mapping of $qp$ is $\\frac{1}{2}(QP+PQ)$.", + "symbol": { + "$f$": "classical quantity", + "$q$": "position variable", + "$p$": "momentum variable", + "$Q$": "quantum position operator", + "$P$": "quantum momentum operator", + "$\\alpha$": "parameter in linear combination", + "$\\beta$": "parameter in linear combination", + "$k$": "exponent in Weyl map", + "$\\mu$": "integration variable for position", + "$v$": "integration variable for momentum" + }, + "chapter": "Functional Quantization", + "section": "Functional Quantization" + }, + { + "id": 78, + "topic": "Others", + "question": "Consider the fermionic integral,\n\n$$\\langle\\chi_{j_{1}} \\cdots \\chi_{j_{p}} \\bar{\\chi}_{i_{1}} \\cdots \\bar{\\chi}_{i_{q}}\\rangle \\equiv \\operatorname{det}^{-1}(\\boldsymbol{M}) \\int \\prod_{k=1}^{n}[d \\chi_{k} d \\bar{\\chi}_{k}] \\chi_{j_{1}} \\cdots \\chi_{j_{p}} \\bar{\\chi}_{i_{1}} \\cdots \\bar{\\chi}_{i_{q}} \\exp (\\bar{\\chi}^{\\top} \\boldsymbol{M} \\boldsymbol{\\chi}) .$$\n\nAssuming $p=q$, compute the expectation value for $p=q=1$, i.e., find an expression for $\\langle \\chi_{j_1} \\bar{\\chi}_{i_1} \\rangle$.", + "final_answer": [ + "-M_{j_1 i_1}^{-1}" + ], + "answer_type": "Expression", + "answer": "To calculate these integrals, we introduce the generating function:\n\n$$Z[\\overline{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] \\equiv \\operatorname{det}^{-1}(\\boldsymbol{M}) \\int \\prod_{k=1}^{n}[d \\chi_{k} d \\bar{\\chi}_{k}] \\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{\\chi}-\\bar{\\chi}^{\\top} \\boldsymbol{\\eta}) \\exp (\\bar{\\chi}^{\\top} \\mathbf{M} \\boldsymbol{\\chi})$$\n\nThis is a Gaussian Grassmann integral, which evaluates to:\n\n$$Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{M}^{-1} \\boldsymbol{\\eta})$$\n\nExpanding this exponential: \n\n$$Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = 1 + \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} + \\mathcal{O}(\\eta^2 \\bar{\\eta}^2)$$\n\nAlternatively, by expanding the factor $\\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{\\chi}-\\bar{\\chi}^{\\top} \\boldsymbol{\\eta})$ in the definition of $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}]$:\n\n$$\\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{\\chi}-\\bar{\\chi}^{\\top} \\boldsymbol{\\eta}) = (1 + \\sum_a \\bar{\\eta}_a \\chi_a + \\dots ) (1 - \\sum_b \\bar{\\chi}_b \\eta_b + \\dots ) = 1 + \\sum_a \\bar{\\eta}_a \\chi_a - \\sum_b \\bar{\\chi}_b \\eta_b - \\sum_{a,b} \\bar{\\eta}_a \\chi_a \\bar{\\chi}_b \\eta_b + \\dots$$\n\nSo,\n\n$$Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\langle 1 \\rangle + \\sum_a \\bar{\\eta}_a \\langle \\chi_a \\rangle - \\sum_b \\langle \\bar{\\chi}_b \\rangle \\eta_b - \\sum_{a,b} \\bar{\\eta}_a \\langle \\chi_a \\bar{\\chi}_b \\rangle \\eta_b + \\dots$$\n\nSince $\\langle \\chi_a \\rangle = 0$ and $\\langle \\bar{\\chi}_b \\rangle = 0$ (as $p \neq q$ for these terms), we have:\n\n$$Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\langle 1 \\rangle - \\sum_{a,b} \\bar{\\eta}_a \\langle \\chi_a \\bar{\\chi}_b \\rangle \\eta_b + \\dots$$\n\n(Note: $\\langle 1 \\rangle = 1$ by normalization). Comparing the coefficients of $\\bar{\\eta}_{j_1} \\eta_{i_1}$ in both expansions of $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}]$:\nFrom $\\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{M}^{-1} \\boldsymbol{\\eta})$, the coefficient of $\\bar{\\eta}_{j_1} \\eta_{i_1}$ is $M_{j_1 i_1}^{-1}$.\nFrom the series expansion in terms of expectation values, the coefficient of $\\bar{\\eta}_{j_1} \\eta_{i_1}$ is $-\\langle \\chi_{j_1} \\bar{\\chi}_{i_1} \\rangle$.\nTherefore, $M_{j_1 i_1}^{-1} = -\\langle \\chi_{j_1} \\bar{\\chi}_{i_1} \\rangle$, which gives:\n\n$$\\langle \\chi_{j_1} \\bar{\\chi}_{i_1} \\rangle = -M_{j_1 i_1}^{-1}$$", + "symbol": { + "$\\chi_{j_1}$": "fermionic variable with index j_1", + "$\\bar{\\chi}_{i_1}$": "conjugate fermionic variable with index i_1", + "$\\boldsymbol{M}$": "matrix M", + "$M_{j_1 i_1}^{-1}$": "inverse component of matrix M with indices j_1 and i_1", + "$n$": "number of fermionic variables" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 79, + "topic": "Others", + "question": "Consider the fermionic integral,\n$$\n\\langle\\chi_{j_{1}} \\cdots \\chi_{j_{p}} \\bar{\\chi}_{i_{1}} \\cdots \\bar{\\chi}_{i_{q}}\\rangle \\equiv \\operatorname{det}^{-1}(\\boldsymbol{M}) \\int \\prod_{k=1}^{n}[d \\chi_{k} d \\bar{\\chi}_{k}] \\chi_{j_{1}} \\cdots \\chi_{j_{p}} \\bar{\\chi}_{i_{1}} \\cdots \\bar{\\chi}_{i_{q}} \\exp (\\bar{\\chi}^{\\top} \\boldsymbol{M} \\boldsymbol{\\chi}) .\n$$\nAssuming $p=q$, compute the expectation value for $p=q=2$, i.e., find an expression for $\\langle \\chi_{j_1} \\chi_{j_2} \\bar{\\chi}_{i_1} \\bar{\\chi}_{i_2} \\rangle$.", + "final_answer": [ + "M_{j_1 i_2}^{-1} M_{j_2 i_1}^{-1} - M_{j_1 i_1}^{-1} M_{j_2 i_2}^{-1}" + ], + "answer_type": "Expression", + "answer": "We use the generating function $Z[\\overline{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}]$ as defined in the previous problem. The expansion of $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{M}^{-1} \\boldsymbol{\\eta})$ to second order in $\\bar{\\eta}$ and $\\eta$ is: $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = 1 + \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} + \\frac{1}{2!} ( \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} )^2 + \\dots = 1 + \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} + \\frac{1}{2} \\sum_{a,b,c,d} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} \\bar{\\eta}_{c} M_{cd}^{-1} \\eta_{d} + \\dots$ Using anticommutation $\\eta_b \\bar{\\eta}_c = - \\bar{\\eta}_c \\eta_b$: $= 1 + \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} - \\frac{1}{2} \\sum_{a,b,c,d} \\bar{\\eta}_{a} \\bar{\\eta}_{c} M_{ab}^{-1} M_{cd}^{-1} \\eta_{b} \\eta_{d} + \\dots$ As noted, this can be written by antisymmetrizing the $M^{-1}$ terms: $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\dots - \\frac{1}{4} \\sum_{a,c,b,d} \\bar{\\eta}_{a} \\bar{\\eta}_{c} (M_{ab}^{-1} M_{cd}^{-1} - M_{ad}^{-1} M_{cb}^{-1}) \\eta_{b} \\eta_{d} + \\dots$ Expanding $\\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{\\chi}-\\bar{\\chi}^{\\top} \\boldsymbol{\\eta})$ gives terms with two $\\bar{\\eta}$ and two $\\eta$ variables. Comparing coefficients, the result matches $\\langle \\chi_{j_1} \\chi_{j_2} \\bar{\\chi}_{i_1} \\bar{\\chi}_{i_2} \\rangle = M_{j_1 i_2}^{-1} M_{j_2 i_1}^{-1} - M_{j_1 i_1}^{-1} M_{j_2 i_2}^{-1}$.", + "symbol": { + "$\\chi_{j_{1}}$": "fermionic field component with index j1", + "$\\chi_{j_{2}}$": "fermionic field component with index j2", + "$\\bar{\\chi}_{i_{1}}$": "conjugate fermionic field component with index i1", + "$\\bar{\\chi}_{i_{2}}$": "conjugate fermionic field component with index i2", + "$\\bar{\\eta}_{a}$": "component of the conjugate auxiliary field with index a", + "$\\eta_{b}$": "component of the auxiliary field with index b", + "$M_{ab}^{-1}$": "element of the inverse matrix of M at a,b" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 80, + "topic": "Others", + "question": "Consider a general function of a single Grassmann variable $\\chi$, which can be written as $f(\\chi) = a + \\chi b$, where $a$ and $b$ are c-numbers (or objects that commute with Grassmann variables). Introduce a second Grassmann variable $\\eta$ that anticommutes with $\\chi$. Calculate explicitly the integral $\\widetilde{f}(\\eta) \\equiv \\int d \\chi e^{\\chi \\eta} f(\\chi)$. (Recall Grassmann properties: $e^{\\chi \\eta} = 1 + \\chi \\eta$, $\\chi^2 = 0$, and Berezin integration rules $\\int d\\chi = 0$, $\\int d\\chi \\chi = 1$).", + "final_answer": [ + "\\eta a+b" + ], + "answer_type": "Expression", + "answer": "Given $f(\\chi) = a + \\chi b$ and $e^{\\chi \\eta} = 1 + \\chi \\eta$ (since higher powers of $\\chi\\eta$ would involve $\\chi^2$ or $\\eta^2$ which are zero if $\\chi, \\eta$ are single components, or this is the truncated expansion for Grassmann variables).\nThe integral is:\n\\begin{align}\n\\widetilde{f}(\\eta) & =\\int d \\chi e^{\\chi \\eta}(a+\\chi b) \\\\& =\\int d \\chi(1+\\chi \\eta)(a+\\chi b) \\\\\n& =\\int d \\chi(a + \\chi b + \\chi \\eta a + \\chi \\eta \\chi b)\n\\end{align}\nSince $\\chi$ and $\\eta$ are Grassmann variables, they anticommute ($\\chi \\eta = -\\eta \\chi$), and $\\chi^2 = 0$. Thus, the term $\\chi \\eta \\chi b = -\\eta \\chi^2 b = -\\eta (0) b = 0$. \nThe integral simplifies to:$$\\widetilde{f}(\\eta) = \\int d \\chi(a+\\chi(\\eta a+b))$$\nUsing the Berezin integration rules $\\int d\\chi = 0$ and $\\int d\\chi \\chi = 1$: \n$$\\widetilde{f}(\\eta) = a \\int d\\chi (1) + (\\eta a+b) \\int d\\chi \\chi = a \\cdot 0 + (\\eta a+b) \\cdot 1 = \\eta a+b$$", + "symbol": { + "$\\chi$": "Grassmann variable", + "$f$": "general function of a Grassmann variable", + "$a$": "c-number that commutes with Grassmann variables", + "$b$": "c-number that commutes with Grassmann variables", + "$\\eta$": "second Grassmann variable that anticommutes with $\\chi$" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 81, + "topic": "Others", + "question": "Consider two Grassmann variables $\\theta_\\pm$.\n\nFor the operator $\\tau_3 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{-} \\frac{\\partial}{\\partial \\theta_{-}})$, find an eigenfunction corresponding to the eigenvalue $1/2$. Express it using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize.", + "final_answer": [ + "$\\theta_{+}" + ], + "answer_type": "Expression", + "answer": "From the eigenvalue equation $\\tau_3 f = \\lambda f$, setting $\\lambda=1/2$: \n$0 = \\frac{1}{2} a \\implies a=0$ \n$(\\frac{1}{2}-\\frac{1}{2})b=0 \\implies 0 \\cdot b = 0$ (no constraint on $b$) \n$(-\\frac{1}{2}-\\frac{1}{2})c=0 \\implies -c=0 \\implies c=0$ \n$0 = \\frac{1}{2} d \\implies d=0$ \nSo, eigenfunctions for $\\lambda=1/2$ are of the form $b\\theta_+$. Choosing $b=1$ for normalization, an eigenfunction is $\\theta_+$.", + "symbol": { + "$\\theta_{+}$": "Grassmann variable", + "$\\theta_{-}$": "Grassmann variable", + "$\\tau_3$": "operator associated with the problem" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 82, + "topic": "Others", + "question": "Consider two Grassmann variables $\\theta_\\pm$.\n\nFor the operator $\\tau_1 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$, find an eigenfunction corresponding to the eigenvalue $1/2$. Express it using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize.", + "final_answer": [ + "$\\theta_{+} + \\theta_{-}" + ], + "answer_type": "Expression", + "answer": "For $\\lambda=1/2$: $a=0, d=0$. $\\frac{1}{2}c = \\frac{1}{2}b \\implies c=b$. Eigenfunctions are $b\\theta_+ + b\\theta_- = b(\\theta_+ + \\theta_-)$. Normalized: $\\theta_+ + \\theta_-$.", + "symbol": { + "$\\theta_{+}$": "Grassmann variable theta plus", + "$\\theta_{-}$": "Grassmann variable theta minus", + "$\\tau_1$": "operator tau one" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 83, + "topic": "Others", + "question": "Consider two Grassmann variables $\\theta_\\pm$.\n\nFor the operator $\\tau_1 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$, find an eigenfunction corresponding to the eigenvalue $-1/2$. Express it using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize.", + "final_answer": [ + "$\\theta_{+} - \\theta_{-}" + ], + "answer_type": "Expression", + "answer": "For $\\lambda=-1/2$: $a=0, d=0$. $\\frac{1}{2}c = -\\frac{1}{2}b \\implies c=-b$. Eigenfunctions are $b\\theta_+ - b\\theta_- = b(\\theta_+ - \\theta_-)$. Normalized: $\\theta_+ - \\theta_-$.", + "symbol": { + "$\\theta_{+}$": "Grassmann variable (positive component)", + "$\\theta_{-}$": "Grassmann variable (negative component)", + "$\\tau_1$": "operator involving Grassmann variables" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 84, + "topic": "Others", + "question": "Consider two Grassmann variables $\\theta_\\pm$. For the operator $\\tau_2 \\equiv \\frac{i}{2}(\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}})$, find an eigenfunction corresponding to the eigenvalue $-1/2$. Express it using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize.", + "final_answer": [ + "$\\theta_{+} - i\\theta_{-}" + ], + "answer_type": "Expression", + "answer": "For $\\lambda=-1/2$: $a=0, d=0$. $-\\frac{i}{2}c = -\\frac{1}{2}b \\implies ic=b \\implies c=-ib$. Eigenfunctions are $b\\theta_+ - ib\\theta_- = b(\\theta_+ - i\\theta_-)$. Normalized: $\\theta_+ - i\\theta_-$.", + "symbol": { + "$\\tau_2$": "operator", + "$\\theta_{+}$": "variable theta plus", + "$\\theta_{-}$": "variable theta minus", + "$i$": "imaginary unit" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 85, + "topic": "Others", + "question": "Consider two Grassmann variables $\\theta_\\pm$. Given the operators $\\tau_{1 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$ and $\\tau_{2} \\equiv \\frac{i}{2}(\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}})$, calculate the action of the operator $(\\tau_1 - i\\tau_2)$ on the Grassmann variable $\\theta_+$.}", + "final_answer": [ + "\\theta_{-}" + ], + "answer_type": "Expression", + "answer": "$\\tau_1 \\theta_+ = \\frac{1{2}\\theta_-$. $\\tau_2 \\theta_+ = \\frac{i}{2}\\theta_-$. $(\\tau_1 - i\\tau_2)\\theta_+ = \\frac{1}{2}\\theta_- - i(\\frac{i}{2}\\\\theta_-) = \\frac{1}{2}\\theta_- + \\frac{1}{2}\\theta_- = \\theta_-$.}", + "symbol": { + "$\\tau_{1}$": "operator involving Grassmann variables", + "$\\tau_{2}$": "operator involving Grassmann variables", + "$\\theta_{+}$": "Grassmann variable", + "$\\theta_{-}$": "Grassmann variable" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 86, + "topic": "Others", + "question": "Assume that $[X, Y]=i c Y$ where c is a numerical constant. Use the all-orders Baker-CampbellHausdorff formula to calculate $\\ln (e^{i X} e^{i Y})$.", + "final_answer": [ + "i X+\\frac{i c}{e^{c}-1} Y" + ], + "answer_type": "Expression", + "answer": "In this case, we have\n$$\n\\operatorname{ad}_{x}(Y)=-i[X, Y]=c Y, \\quad e^{\\operatorname{ad}_{X}} Y=e^{c} Y\n$$\n\nSince $Y$ commutes with itself, this implies that\n$$\ne^{\\operatorname{tad}_{Y}} e^{\\mathrm{ad}_{x}} \\mathrm{Y}=e^{\\mathrm{c}} \\mathrm{Y}\n$$\n\nUsing this in the general Baker-Campbell-Hausdorff formula gives\n\n$$\\ln (e^{i X} e^{i Y})=i X+i \\int_{0}^{1} d t \\frac{\\ln (e^{c})}{e^{c}-1} Y=i X+\\frac{i c}{e^{c}-1} Y$$\n\n(Obviously, this result goes to $i(X+Y)$ when $\\mathrm{c} \\rightarrow 0$.)", + "symbol": { + "$X$": "operator X", + "$Y$": "operator Y", + "$i$": "imaginary unit", + "$c$": "numerical constant" + }, + "chapter": "Non-Abelian Gauge Symmetry", + "section": "Non-Abelian Gauge Symmetry" + }, + { + "id": 87, + "topic": "Others", + "question": "Calculate the one-loop $\\beta$-function of a scalar field theory with cubic interactions in six spacetime dimensions.", + "final_answer": [ + "\\beta=-\\frac{3 \\lambda^{3}}{4(4 \\pi)^{3}}" + ], + "answer_type": "Expression", + "answer": "Scalar field theory with cubic interactions is renormalizable in six dimensions (the coupling constant of this cubic interaction is dimensionless in six dimensions), and the power counting indicates that the divergences are in the 2-point and 3-point functions. In order to calculate the one-loop $\\\\beta$-function, we need fist to calculate at this order the self-energy and vertex counterterms. Let us start with the self-energy, given by\n$$\n\\Sigma(p) \\equiv \\frac{\\mathfrak{i} \\lambda^{2} \\mu^{2 \\epsilon}}{2} \\int \\frac{d^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{1}{\\ell^{2}(\\ell+p)^{2}}=\\frac{\\mathfrak{i} \\lambda^{2} \\mu^{2 \\epsilon}}{2} \\int_{0}^{1} d x \\int \\frac{d^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{1}{[\\ell^{2}-\\Delta]^{2}}\n$$\nwhere $\\Delta \\equiv-x(1-x) p^{2}$ (note that the loop momentum $\\ell$ has been shifted in the last expression). Performing a Wick's rotation and using standard results for D-dimensional integral gives\n$$\n\\Sigma(p)=-\\frac{\\lambda^{2} \\mu^{2 \\epsilon}}{2(4 \\pi)^{\\mathrm{D} / 2}} \\Gamma(2-\\frac{\\mathrm{D}}{2}) \\int_{0}^{1} \\mathrm{~d} x \\Delta^{\\mathrm{D} / 2-2} .\n$$\n\nThen, define $D \\equiv 6-2 \\epsilon$, and set $\\epsilon$ to zero in all factors, except the $\\Gamma$ function that has a simple pole (with residue -1 ) and the factor $(\\mu^{2} / p^{2})^{\\epsilon}$ that carries the scale dependence. We arrive at\n$$\n\\Sigma(p)=-\\frac{\\lambda^{2} p^{2}}{12(4 \\pi)^{3} \\epsilon}[-\\frac{\\mu^{2}}{p^{2}}]^{\\epsilon}\n$$\n\nAt this order, the field renormalization constant is\n\n\\begin{equation*}\n\\mathrm{Z}=1+\\frac{\\partial \\Sigma}{\\partial \\mathrm{p}^{2}}=1-\\frac{\\lambda^{2}}{12(4 \\pi)^{3} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon} \\tag{*}\n\\end{equation*}\n\nwhere we have chosen the renormalization point $p^{2}=-M^{2}$.\n\nLet us now turn our attention to the 3-point function. Denoting $p, q$ two of the momenta entering in the graph, the one-loop vertex correction reads\n\n\\begin{align}\n\\mu^{\\epsilon} \\Gamma(p, q) & =i \\lambda^{3} \\mu^{3 \\epsilon} \\int \\frac{d^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{1}{\\ell^{2}(\\ell+p)^{2}(\\ell-q)^{2}} \\\\\n& =2 i \\lambda^{3} \\mu^{3 \\epsilon} \\int_{\\substack{0 \\leq x, y \\leq 1 \\\\\nx+y \\leq 1}} d x d y \\int \\frac{d^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{1}{[\\ell^{2}-\\Delta]^{3}},\n\\end{align}\n\nwith $-\\Delta \\equiv x(1-x) p^{2}+y(1-y) q^{2}+2 x y p \\cdot q$. The reason why we denote the left hand side $\\mu^{\\epsilon} \\Gamma$ is that this loop integral is a correction to the 3-point coupling, that has dimension (mass) ${ }^{\\epsilon}$ in $\\mathrm{D}=6-2 \\epsilon$ dimensions. By writing it in this way, $\\Gamma$ is a correction to the dimensionless coupling $\\lambda$. After a Wick's rotation, the pole in this integral reads\n\n\\begin{equation*}\n\\Gamma(p, q)=\\frac{\\lambda^{3}}{2(4 \\pi)^{3} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon} \\tag{**}\n\\end{equation*}\n\nwhere we have chosen the special kinematical configuration $p^{2}=q^{2}=(p+q)^{2}=-M^{2}$ as renormalization point.\n\nFrom the results $(*)$ and $(* *)$, we get the following wavefunction and vertex counterterms,\n$$\n\\delta_{z}=-\\frac{\\lambda^{2}}{12(4 \\pi)^{3} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon}, \\quad \\delta_{\\lambda}=-\\frac{\\lambda^{3}}{2(4 \\pi)^{3} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon}\n$$\nand the $\\beta$-function in six dimensions is then given by\n$$\n\\beta=\\lim _{\\epsilon \\rightarrow 0} M \\frac{\\partial}{\\partial M}[\\frac{3 \\lambda^{3}}{8(4 \\pi)^{3} \\epsilon}(1-2 \\epsilon \\ln (\\frac{M}{\\mu})+\\mathcal{O}(\\epsilon^{2}))]=-\\frac{3 \\lambda^{3}}{4(4 \\pi)^{3}} .\n$$", + "symbol": { + "$\\beta$": "beta function", + "$\\lambda$": "coupling constant", + "$\\mu$": "renormalization scale", + "$\\epsilon$": "dimensional regularization parameter", + "$D$": "spacetime dimensions", + "$p$": "momentum", + "$\\Delta$": "shifted square of momentum", + "$\\Sigma$": "self-energy", + "$\\Gamma$": "vertex correction", + "$Z$": "field renormalization constant", + "$M$": "renormalization point" + }, + "chapter": "Renormalization of Gauge Theories", + "section": "Renormalization of Gauge Theories" + }, + { + "id": 88, + "topic": "Others", + "question": "Calculate the one-loop $\\beta$ function in quantum electrodynamics. How does the electromagnetic coupling strength vary with distance? What is the physical interpretation of this behaviour?", + "final_answer": [ + "\\beta(e)=\\frac{e^{3}}{12 \\pi^{2}}" + ], + "answer_type": "Expression", + "answer": "In order to follow the same procedure as in Exercise 10.1 for calculating the $\\beta$ function in QED, we would need to calculate the one-loop electron self-energy, the one-loop photon self-energy, and the one-loop correction to the electron-photon vertex. An alternative is to use the relation $Z_{1} Z_{2}^{-1} Z_{3}^{-1 / 2} e_{r}=e_{b}$ and the fact that $Z_{1}=Z_{2}$ due to Ward identities. Therefore, the relationship between the bare and renormalized couplings can also be obtained from the photon wavefunction renormalization, i.e. from a unique loop diagram. At one-loop, the photon polarization tensor is given by\n\n$$\\Pi^{\\mu u}(q)=-i e^{2} \\mu^{2 \\epsilon} \\int \\frac{d^{D} \\ell}{(2 \\pi)^{D}} \\frac{\\operatorname{tr}(\\gamma^{\\mu} \\ell \\gamma^{v}(\\ell+ot q))}{\\ell^{2}(\\ell+q)^{2}}$$\nFrom gauge invariance, we expect the tensor structure to be $\\Pi^{\\mu v}(q) \\equiv(g^{\\mu v} q^{2}-q^{\\mu} q^{v}) \\Pi(q^{2})$, and the function $\\Pi(q^{2})$ can be obtained from the trace, since $(D-1) q^{2} \\Pi(q^{2})=\\Pi_{\\mu}^{\\mu}(q)$, that reads\n\n\\begin{align}\n\\Pi_{\\mu}^{\\mu}(\\mathrm{q}) & =4(\\mathrm{D}-2) i e^{2} \\mu^{2 \\epsilon} \\int \\frac{\\mathrm{~d}^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{\\ell \\cdot(\\ell+\\mathrm{q})}{\\ell^{2}(\\ell+\\mathrm{q})^{2}} \\\\\n& =4(\\mathrm{D}-2) i e^{2} \\mu^{2 \\epsilon} \\int_{0}^{1} \\mathrm{dx} \\int \\frac{\\mathrm{~d}^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{\\ell^{2}+\\Delta+\\text { terms odd in } \\ell}{[\\ell^{2}-\\Delta]^{2}}\n\\end{align}\n\nwhere $\\Delta \\equiv-x(1-x) q^{2}$, and the $\\ell$ in the last expression is a shifted integration variable. Performing a Wick's rotation and integrating over $\\ell$ gives\n\n\\begin{align}\n\\Pi(q^{2}) & =e^{2} \\mu^{2 \\epsilon} \\frac{4(\\mathrm{D}-2)}{(\\mathrm{D}-1) \\mathrm{q}^{2}(4 \\pi)^{\\mathrm{D} / 2}}[\\Gamma(1-\\frac{\\mathrm{D}}{2})-2 \\Gamma(2-\\frac{\\mathrm{D}}{2})] \\int_{0}^{1} \\mathrm{~d} x \\Delta^{\\mathrm{D} / 2-1} \\\\\n& =\\frac{4 \\mathrm{e}^{2}}{3(4 \\pi)^{2} \\epsilon}[-\\frac{\\mu^{2}}{\\mathrm{q}^{2}}]^{\\epsilon}\n\\end{align}\n\n\nThe photon wavefunction renormalization factor is\n$$\nZ_{3}=\\frac{1}{1+\\Pi}=1-\\Pi+\\mathcal{O}(e^{4})=1-\\frac{4 e^{2}}{3(4 \\pi)^{2} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon}+\\mathcal{O}(e^{4})\n$$\nthe last equality being at the point $q^{2}=-M^{2}$. From the relation $Z_{3}^{-1 / 2} e_{r}=e_{b}$, the scale dependence of the renormalized coupling constant can be determined from that of $Z_{3}$ (since the bare coupling is by definition scale independent),\n$$\n\\beta(e)=\\frac{e}{2 Z_{3}} M \\frac{\\partial Z_{3}}{\\partial M}=\\frac{e^{3}}{12 \\pi^{2}},\n$$\nwhich is often written as\n$$\n\\beta(\\alpha) \\equiv M \\frac{\\partial \\alpha}{\\partial M}=\\frac{2 \\alpha^{2}}{3 \\pi} .\n$$\nSince the $\\beta$ function of QED is positive, the coupling strength increases with the energy scale, or at shorter distances. Conversely, it decreases at larger distances. This can be understood physically as a screening phenomenon: a test charge polarizes the vacuum around it by arranging virtual electron-antielectron pairs in such a way that they screen the Coulomb potential seen at a distance. This is very similar to Debye screening in plasma physics (but there it is due to the polarization of on-shell charged particles, instead of vacuum fluctuations). As the probe of the electrical field approaches the test charge, the charge it sees is closer and closer to the naked charge of the test particle. For a point-like charge, the latter is infinite since it is not hidden by any screening.", + "symbol": { + "$\\beta$": "beta function", + "$e$": "electron charge", + "$Z_{1}$": "renormalization constant for vertex", + "$Z_{2}$": "renormalization constant for electron field", + "$Z_{3}$": "renormalization constant for photon field", + "$e_{r}$": "renormalized electron charge", + "$e_{b}$": "bare electron charge", + "$\\mu$": "regularization scale", + "$q$": "momentum transfer", + "$\\ell$": "loop momentum", + "$\\Delta$": "shifted integration variable", + "$M$": "momentum scale", + "$\\alpha$": "fine-structure constant" + }, + "chapter": "Renormalization of Gauge Theories", + "section": "Renormalization of Gauge Theories" + }, + { + "id": 89, + "topic": "Others", + "question": "The special conformal transformation is given by $y^{\\mu}=\\frac{x^{\\mu}+b^{\\mu} x^{2}}{1+2 b \\cdot x+b^{2} x^{2}}$. For an infinitesimal 4-vector $b^{\\mu}$, this transformation can be expanded to first order in $b^{\\mu}$ as $y^{\\mu} \\approx x^{\\mu}+\\delta x^{\\mu}$, where $\\delta x^{\\mu} = (x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) b_{\\rho}$ (using $g^{\\mu\\rho}$ to raise the index of $b$). The generator $T_{\\rho}$ (associated with parameter $b^{\\rho}$) is defined such that the transformation on coordinates is $x'^{\\mu} = \\exp (i b^{\\rho} T_{\\rho}) x^{\\mu} \\approx x^{\\mu} + i b^{\\rho} (T_{\\rho} x^{\\mu})$, from which $\\delta x^{\\mu} = i b^{\\rho} (T_{\\rho} x^{\\mu})$. The provided text then gives an expression for an object $T^{\\mu}$ (note the upper index) as $T^{\\mu}=-\\mathfrak{i}(x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) \\partial_{\\rho}$. What is this expression for $T^{\\mu}$?", + "final_answer": [ + "T^{\\mu}=-\\mathfrak{i}(x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) \\partial_{\\rho}" + ], + "answer_type": "Expression", + "answer": "Given the infinitesimal transformation for $y^{\\mu}$:\n$$y^{\\mu}=\\frac{x^{\\mu}+b^{\\mu} x^{2}}{1+2 b \\cdot x+b^{2} x^{2}}$$\nFor infinitesimal $b^{\\mu}$, we expand $(1+2 b \\cdot x+b^{2} x^{2})^{-1} \\approx 1 - (2 b \\cdot x+b^{2} x^{2}) + \\mathcal{O}(b^2) \\approx 1 - 2 b \\cdot x + \\mathcal{O}(b^2)$.\nSo, $y^{\\mu} \\approx (x^{\\mu}+b^{\\mu} x^{2})(1-2b \\cdot x) + \\mathcal{O}(b^2)$\n$y^{\\mu} \\approx x^{\\mu} - 2(b \\cdot x)x^{\\mu} + b^{\\mu}x^2 + \\mathcal{O}(b^2)$.\nLet $b^{\\mu} = g^{\\mu\\sigma}b_{\\sigma}$. Then $b \\cdot x = b_{\\sigma}x^{\\sigma}$.\n$y^{\\mu} \\approx x^{\\mu} + (g^{\\mu\\sigma}x^2 - 2x^{\\mu}x^{\\sigma})b_{\\sigma} + \\mathcal{O}(b^2)$.\nSo, $\\delta x^{\\mu} = (x^2 g^{\\mu \\sigma} - 2x^{\\sigma}x^{\\mu})b_{\\sigma}$. (The problem statement uses $\\rho$ for the summed index of $b$).\n$\\delta x^{\\mu} = (x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) b_{\\rho}$.\nFrom the definition of how generators act on fields $\\Phi(x)$, an infinitesimal transformation $\\delta \\Phi = i \\epsilon^a (G_a \\Phi)$ involves the generator $G_a$. For coordinate transformations $x_\\mu \\rightarrow x_\\mu + \\delta x_\\mu$, where $\\delta x_\\mu = \\epsilon^a \\xi_{a \\mu}(x)$, the generators are often written as $G_a = -i \\xi_a^\nu (x) \\partial_\nu$. \nThe text states that from $\\delta x^{\\mu}$, we can read off the generator $T_{\\rho}$ such that $\\exp (i b^{\\rho} T_{\\rho}) x^{\\mu}}= x^{\\mu}+\\delta x^{\\mu}+\\cdots$.\nIt then provides the expression: $T^{\\mu}=-\\mathfrak{i}(x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) \\partial_{\\rho}$.\nThis expression has a free index $\\mu$ on $T$ and summation over $\\rho$ on the right hand side. This represents a set of four differential operators, one for each value of $\\mu=0,1,2,3$. This $T^{\\mu}$ is the specific quantity requested, as derived/stated in the source material.", + "symbol": { + "$y^{\\mu}$": "transformed coordinate vector", + "$x^{\\mu}$": "original coordinate vector", + "$b^{\\mu}$": "infinitesimal 4-vector parameter", + "$x^{2}$": "square of the coordinate vector", + "$b^{2}$": "square of the vector $b$", + "$b \\cdot x$": "dot product of $b^{\\mu}$ and $x^{\\mu}$", + "$g^{\\mu\\rho}$": "metric tensor (raising the index)", + "$T_{\\rho}$": "generator associated with the parameter $b^{\\rho}$", + "$T^{\\mu}$": "specific generator operator for infinitesimal transformations", + "$\\delta x^{\\mu}$": "change in the coordinate vector $x^{\\mu}$", + "$b_{\\rho}$": "component of the vector $b$ with a lowered index", + "$\\partial_{\\rho}$": "partial derivative with respect to the coordinate $x^{\\rho}$" + }, + "chapter": "Quantum Anomalies", + "section": "Quantum Anomalies" + }, + { + "id": 90, + "topic": "Others", + "question": "In Yang-Mills theory in the temporal gauge $A^{0}=0$ (with coupling $g=1$ for simplicity), what is the expression for the conjugate momentum $\\Pi_a^i$ of the gauge field component $A_a^i$?", + "final_answer": [ + "\\Pi_{a}^{i} = \\partial_{0} A_{\\mathrm{a}}^{i}" + ], + "answer_type": "Expression", + "answer": "By using the temporal gauge $A^{0}=0$, we circumvent the problem that $A_{a}^{0}$ has a vanishing conjugate momentum, because $A^{0}$ is not a dynamical variable in this gauge. Regarding the other components of the gauge potential, the conjugate momentum $\\Pi^{i}$ of $A^{i}$ is given by\n\n$$\\Pi_{a}^{i} \\equiv \\frac{\\partial \\mathcal{L}_{\\mathrm{YM}}}{\\partial \\partial_{0} \\mathcal{A}_{\\mathrm{a}}^{i}}=\\partial_{0} A_{\\mathrm{a}}^{i}$$", + "symbol": { + "$A^{0}$": "gauge field component (0)", + "$g$": "coupling constant", + "$\\Pi_a^i$": "conjugate momentum of gauge field component (a, i)", + "$A_a^i$": "gauge field component (a, i)", + "$\\mathcal{L}_{\\mathrm{YM}}$": "Yang-Mills Lagrangian", + "$\\partial_{0}$": "partial derivative with respect to time" + }, + "chapter": "Localized Field Configurations", + "section": "Localized Field Configurations" + }, + { + "id": 91, + "topic": "Others", + "question": "For Yang-Mills theory in the temporal gauge $A^{0}=0$ (with $g=1$), derive the Hamiltonian density $\\mathcal{H}$. Express it first in terms of the chromo-electric fields $E_a^i = \\Pi_a^i$ and chromo-magnetic fields $B_a^i = \\frac{1}{2} \\epsilon_{ijk} F_a^{jk}$, and then in terms of $\\Pi_a^i$ and $F_a^{ij}$.", + "final_answer": [ + "\\mathcal{H}=\\frac{1}{2}(E_{a}^{i} E_{a}^{i}+B_{a}^{i} B_{a}^{i}) = \\frac{1}{2} \\Pi_{a}^{i} \\Pi_{a}^{i}+\\frac{1}{4} F_{a}^{i j} F_{a}^{i j}" + ], + "answer_type": "Expression", + "answer": "It is convenient to introduce the chromo-electric and chromo-magnetic fields by\n$$\n\\mathrm{E}_{\\mathrm{a}}^{i} \\equiv \\mathrm{~F}_{\\mathrm{a}}^{0 i}=\\Pi_{a}^{i} \\quad(\\text { in } A^{0}=0 \\text { gauge }), \\quad B_{a}^{i} \\equiv \\frac{1}{2} \\epsilon_{i j k} F_{a}^{j k}\n$$\nin terms of which the Lagrangian density can be written as follows,\n\n\\begin{align}\n-\\frac{1}{4} F_{a}^{\\mu \\nu} F_{\\mu v}^{a} & =-\\frac{1}{4}(-F_{a}^{O i} F_{a}^{O i}-F_{a}^{i 0} F_{a}^{i O}+F_{a}^{i j} F_{a}^{i j})=\\frac{1}{4}(2 E_{a}^{i} E_{a}^{i}-\\epsilon_{i j k} \\epsilon_{i j l} B_{a}^{k} B_{a}^{l}) \\\n& =\\frac{1}{4}(2 E_{a}^{i} E_{a}^{i}-(\\delta_{j j} \\delta_{k l}-\\delta_{j k} \\delta_{j l}) B_{a}^{k} B_{a}^{l})=\\frac{1}{2}(E_{a}^{i} E_{a}^{i}-B_{a}^{i} B_{a}^{i}) .\n\\end{align}\n\nThen, the Hamiltonian is given by\n\n$$\\mathcal{H}=\\underbrace{\\Pi_{a}^{i}(\\partial_{0} A_{a}^{i})}_{E_{a}^{i} E_{a}^{i}}-\\mathcal{L}=\\frac{1}{2}(E_{a}^{i} E_{a}^{i}+B_{a}^{i} B_{a}^{i})=\\frac{1}{2} \\Pi_{a}^{i} \\Pi_{a}^{i}+\\frac{1}{4} F_{a}^{i j} F_{a}^{i j} .$$", + "symbol": { + "$A^{0}$": "temporal gauge field component", + "$g$": "coupling constant, set to 1", + "$\\mathcal{H}$": "Hamiltonian density", + "$E_a^i$": "chromo-electric field component", + "$\\Pi_a^i$": "canonical momentum, equivalent to chromo-electric field component", + "$B_a^i$": "chromo-magnetic field component", + "$\\epsilon_{ijk}$": "Levi-Civita symbol", + "$F_a^{jk}$": "field strength tensor (spatial components)", + "$F_a^{ij}$": "field strength tensor", + "$\\partial_{0}$": "time derivative" + }, + "chapter": "Localized Field Configurations", + "section": "Localized Field Configurations" + }, + { + "id": 92, + "topic": "Magnetism", + "question": "A point charge $e$ is located at point $O$ near a system of grounded conductors, inducing charges $e_{a}$ on these conductors. If the charge $e$ is absent and one of the conductors (the $a$-th) has a potential $\\varphi_{a}^{\\prime}$ (with the remaining conductors still grounded), then the potential at point $O$ is $\\varphi_{0}^{\\prime}$. Express the charge $e_{a}$ in terms of $\\varphi_{a}^{\\prime}$ and $\\varphi_{0}^{\\prime}$.", + "final_answer": [ + "e_a = -\\frac{e \\varphi_0'}{\\varphi_a'}" + ], + "answer_type": "Expression", + "answer": "If the charge $e_{a}$ on the conductor gives the conductor a potential $\\varphi_{a}$, while the charge $e_{a}^{\\prime}$ gives the conductor a potential $\\varphi_{a}^{\\prime}$, then we have:\n\n$$\\sum_{a} \\varphi_{a} e_{a}^{\\prime}=\\sum_{a, b} \\varphi_{a} C_{a b} \\varphi_{b}^{\\prime}=\\sum_{a} \\varphi_{a}^{\\prime} e_{a}$$\n\n\nWe apply this relationship to the two states of a system consisting of all conductors and a point charge $e$ (the latter viewed as a limiting case of a small-sized conductor). In one state, there is charge $e$ with charge on the conductors being $e_{a}$ and the potential $\\varphi_{a}=0$. In another state, the charge $e=0$, but one conductor has a potential $\\varphi_{a}^{\\prime} \\neq 0$. Thus we get $e \\varphi_{0}^{\\prime}+e_{a} \\varphi_{a}^{\\prime}=0$, from which follows\n\n$$e_{a}=-\\frac{e \\varphi_{0}^{\\prime}}{\\varphi_{a}^{\\prime}}$$\n\n\nFor example, if the charge $e$ is at a distance $r(r>a)$ from the center of a grounded spherical conductor of radius $a$, then $\\varphi_{0}^{\\prime}=\\varphi_{a}^{\\prime} \\frac{a}{r}$, and the induced charge on the sphere is\n$$\ne_{a}=-\\frac{e a}{r}\n$$\n\nAs another example, consider a charge $e$ between two grounded concentric spheres of radii $a$ and $b$ respectively (the charge is at a distance $r$ from the center, where $a\\Delta$, where $\\Delta$ is a distance such that $a \\ll \\Delta \\ll b$. Hence, when $r<\\Delta$, it can be assumed that this section of the ring is straight, yielding\n$$\n\\int_{\\Delta>r} \\frac{\\mathrm{~d} l}{r}=\\int_{-\\Delta}^{\\Delta} \\frac{\\mathrm{d} l}{\\sqrt{l^{2}+a^{2}}} \\approx 2 \\ln \\frac{2 \\Delta}{a} .\n$$\n\nIn the region $r>\\Delta$, the thickness of the wire can be neglected, which is to say $r$ is assumed to be the distance between two points along the axis of the ring. Thus\n$$\n\\int_{r>\\Delta} \\frac{\\mathrm{d} l}{r}=2 \\int_{\\varphi_{0}}^{\\pi} \\frac{b \\mathrm{~d} \\varphi}{2 b \\sin (\\varphi / 2)}=-2 \\ln \\tan \\frac{\\varphi_{0}}{4}\n$$\n\nIn this expression, $\\varphi$ is the angle subtended by chord $r$ at the center of the ring, and the lower limit of integration is derived from $2 b \\sin (\\frac{\\varphi_{0}}{2})=\\Delta$, yielding $\\varphi_{0} \\approx \\Delta / b$. Adding the two parts of the integral, the quantity $\\Delta$ cancels automatically, and finally, the expression for the capacitance $C$ of the ring is given by\n\n$$C=\\frac{\\pi b}{\\ln (8 b / a)}$$", + "symbol": { + "$C$": "capacitance", + "$b$": "ring radius", + "$a$": "radius of the wire cross-section", + "$\\varphi_{a}$": "potential of the ring", + "$r$": "distance from a point on the ring to its axis", + "$\\Delta$": "intermediate distance scale", + "$l$": "axis element", + "$\\varphi$": "angle subtended at the center of the ring" + }, + "chapter": "Electrostatics of Conductors", + "section": "Electrostatic field energy of conductor" + }, + { + "id": 94, + "topic": "Magnetism", + "question": "An infinitely long cylindrical conductor with radius $R$ is immersed in a uniform transverse electric field with strength $\\mathfrak{C}$. Find the potential distribution $\\varphi(r, \\theta)$ outside the cylinder.", + "final_answer": [ + "\\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}})" + ], + "answer_type": "Expression", + "answer": "Using polar coordinates in the plane perpendicular to the axis of the cylinder. Thus, the solution to the two-dimensional Laplace equation involving a constant vector is\n$$\n\\varphi_{1}=\\text { const } \\cdot \\mathfrak{C} \\cdot \\nabla \\ln r=\\text { const } \\cdot \\frac{\\mathfrak{C} \\cdot \\boldsymbol{r}}{r^{2}} .\n$$\n\nAdding the above with $\\varphi_{0}=-\\boldsymbol{r} \\cdot \\mathfrak{C}$ and letting const $=R^{2}$, we obtain\n$$\n\\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}}) .\n$$", + "symbol": { + "$R$": "radius of the cylindrical conductor", + "$\\mathfrak{C}$": "strength of the uniform transverse electric field", + "$\\varphi$": "potential distribution", + "$r$": "radial distance in polar coordinates", + "$\\theta$": "angular coordinate in polar coordinates" + }, + "chapter": "Electrostatics of Conductors", + "section": "Solutions to electrostatic problems" + }, + { + "id": 95, + "topic": "Magnetism", + "question": "An infinitely long conducting cylinder with a radius $R$ is immersed in a uniform transverse electric field with a strength of $\\mathfrak{C}$. Find the induced surface charge density $\\sigma(\\theta)$ on the surface of the cylinder.", + "final_answer": [ + "\\sigma=\\frac{\\mathfrak{C}}{2 \\pi} \\cos \\theta" + ], + "answer_type": "Expression", + "answer": "The potential is \n\\begin{equation}\n \\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}}).\n\\end{equation}\n\nThus, the surface charge density is\n\n$$\\sigma= -\\frac{1}{4\\pi} \\frac{\\partial \\varphi}{\\partial r} = \\frac{\\mathfrak{C}}{2 \\pi} \\cos \\theta. $$", + "symbol": { + "$R$": "radius of the conducting cylinder", + "$\\mathfrak{C}$": "strength of the uniform transverse electric field", + "$\\sigma(\\theta)$": "induced surface charge density", + "$\\theta$": "polar angle" + }, + "chapter": "Electrostatics of Conductors", + "section": "Solutions to electrostatic problems" + }, + { + "id": 96, + "topic": "Magnetism", + "question": "An infinitely long conducting cylinder with a radius of $R$ is immersed in a uniform transverse electric field with a strength of $\\mathfrak{C}$. Find the induced dipole moment $\\mathscr{P}$ per unit length of the cylinder.", + "final_answer": [ + "\\mathscr{P}=\\mathfrak{C} R^{2} / 2" + ], + "answer_type": "Expression", + "answer": "The potential is \n\\begin{equation*}\n \\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}}).\n\\end{equation*}\n\nThe dipole moment $\\mathscr{P}$ per unit length of the cylinder can be determined by comparing $\\varphi$ with the potential of a two-dimensional dipole field, which is given by\n\n$$2 \\mathscr{P} \\cdot \\nabla \\ln r=\\frac{2 \\mathscr{P} \\cdot \\boldsymbol{r}}{r^{2}},$$\n\n\nTherefore, $\\mathscr{P}=\\mathfrak{C} R^{2} / 2$.", + "symbol": { + "$R$": "radius of the conducting cylinder", + "$\\mathfrak{C}$": "strength of the uniform transverse electric field", + "$\\mathscr{P}$": "induced dipole moment per unit length of the cylinder", + "$r$": "distance from the axis of the cylinder", + "$\\theta$": "angle in polar coordinates" + }, + "chapter": "Electrostatics of Conductors", + "section": "Solutions to electrostatic problems" + }, + { + "id": 97, + "topic": "Magnetism", + "question": "Determine the attraction energy between an electric dipole and a planar conductor surface.", + "final_answer": [ + "\\mathscr{U}=-\\frac{1}{8 x^{3}}(2 \\mathscr{P}_{x}^{2}+\\mathscr{P}_{y}^{2})" + ], + "answer_type": "Expression", + "answer": "Choose the $x$-axis perpendicular to the conductor surface, passing through the point where the dipole is located; let the dipole moment vector $\\mathscr{P}$ lie in the $xy$ plane. The 'image' of the dipole is at the point $-x$ and has a dipole moment $\\mathscr{P}_{x}^{\\prime}=\\mathscr{P}_{x}, \\mathscr{P}_{y}^{\\prime}=-\\mathscr{P}_{y}$. Therefore, the required attraction energy is equivalent to the interaction energy between the dipole and its 'mirror image', and it equals\n\n$$\\mathscr{U}=-\\frac{1}{8 x^{3}}(2 \\mathscr{P}_{x}^{2}+\\mathscr{P}_{y}^{2}) .$$", + "symbol": { + "$x$": "position coordinate perpendicular to the conductor surface", + "$\\mathscr{P}$": "dipole moment vector", + "$\\mathscr{P}_{x}$": "component of the dipole moment vector along the x-axis", + "$\\mathscr{P}_{y}$": "component of the dipole moment vector along the y-axis", + "$\\mathscr{P}_{x}^{\\prime}$": "component of the mirror dipole moment vector along the x-axis", + "$\\mathscr{P}_{y}^{\\prime}$": "component of the mirror dipole moment vector along the y-axis", + "$\\mathscr{U}$": "attraction energy between the dipole and its mirror image" + }, + "chapter": "Electrostatics of Conductors", + "section": "Solutions to electrostatic problems" + }, + { + "id": 98, + "topic": "Magnetism", + "question": "Try to find an expression for the electric dipole moment of a conductive thin cylindrical rod (length $2l$, radius $a$, where $a \\ll l$) placed in an electric field $\\mathfrak{C}$, expressed in terms of the parameter $L=\\ln (2 l / a)-1$. The electric field is parallel to the axis of the rod.", + "final_answer": [ + "\\mathscr{P} = \\mathfrak{C} \\frac{l^{3}}{3 L}[1+\\frac{1}{L}(\\frac{4}{3}-\\ln 2)]" + ], + "answer_type": "Expression", + "answer": "Let $\\tau(z)$ be the charge induced per unit length on the rod's surface; $z$ is the coordinate along the axis of the cylindrical rod, with the origin at the midpoint of the rod axis. The condition of constant potential on the conductor surface takes the form\n$$-\\mathfrak{C} z+\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\int_{-l}^{l} \\frac{\\tau(z^{\\prime}) \\mathrm{d} z^{\\prime} \\mathrm{d} \\varphi}{R}=0, \\quad R=[(z-z)^{2}+4 a^{2} \\sin ^{2} \\frac{\\varphi}{2}]^{1 / 2},$$\n\nwhere $\\varphi$ is the angle between planes through the rod axis and points on its surface separated by distance $R$. In the integral, using the identity $\\tau(z^{\\prime})=\\tau(z)+[\\tau(z^{\\prime})-\\tau(z)]$, the integration is divided into two parts. In the first part of the integration, noting $l \\gg a$, for points not very close to the ends of the rod, we have\n$$\\frac{\\tau(z)}{2 \\pi} \\iint \\frac{\\mathrm{~d} z^{\\prime} \\mathrm{d} \\varphi}{R} \\approx \\frac{\\tau(z)}{2 \\pi} \\int_{0}^{2 \\pi} \\ln \\frac{l^{2}-z^{2}}{a^{2} \\sin ^{2}(\\varphi / 2)} \\mathrm{d} \\varphi=\\tau(z) \\ln \\frac{4(l^{2}-z^{2})}{a^{2}}$$\n\n(using the known integral value $\\int_{0}^{\\pi} \\ln \\sin \\varphi \\mathrm{d} \\varphi=-\\pi \\ln 2$). In the second part of the integral, containing the difference $\\tau(z^{\\prime})-\\tau(z)$, terms containing $a^{2}$ in $R$ can be neglected as they do not cause the integral to diverge. Thus,\n$$\n\\mathfrak{C} z=\\tau(z) \\ln \\frac{4(l^{2}-z^{2})}{a^{2}}+\\int_{-l}^{l} \\frac{\\tau(z^{\\prime})-\\tau(z)}{|z^{\\prime}-z|} \\mathrm{d} z^{\\prime}.\n$$\nThe dependence of $\\tau$ on $z$ is essentially proportional to $z$; in this approximation, the integral in the above equation gives $-2 \\tau(z)$, resulting in\n$$\n\\tau(z)=\\frac{\\mathfrak{C} z}{\\ln [4(l^{2}-z^{2}) / a^{2}]-2}\n$$\n\nThis expression is not applicable near the end points of the rod, but the region of $z$ values for calculating the requested dipole moment is not significant. At the precision adopted here, we have\n\n\\begin{align}\n\\mathscr{P} & =\\int_{-l}^{l} \\tau(z) z \\mathrm{~d} z=\\frac{\\mathfrak{C}}{L} \\int_{0}^{l}[z^{2}-\\frac{z^{2}}{2 L} \\ln (1-\\frac{z^{2}}{l^{2}})] \\mathrm{d} z \\\\\n& =\\mathfrak{C} \\frac{l^{3}}{3 L}[1+\\frac{1}{L}(\\frac{4}{3}-\\ln 2)]\n\\end{align}\n\n(where $L=\\ln (2 l / a)-1$ is a large number).", + "symbol": { + "$l$": "half the length of the cylindrical rod", + "$a$": "radius of the cylindrical rod", + "$\\mathfrak{C}$": "electric field strength", + "$L$": "parameter defined as $\\ln (2 l / a) - 1$", + "$\\tau(z)$": "charge induced per unit length at position $z$", + "$z$": "coordinate along the axis of the cylindrical rod", + "$\\varphi$": "angular coordinate around the rod", + "$\\mathscr{P}$": "electric dipole moment of the rod" + }, + "chapter": "Electrostatics of Conductors", + "section": "Solutions to electrostatic problems" + }, + { + "id": 99, + "topic": "Magnetism", + "question": "Attempt to find another approximate expression for the electric dipole moment of a conductive thin cylindrical rod (length $2l$, radius $a$, and $a \\ll l$) in an electric field $\\mathfrak{C}$, expressed directly in the logarithmic form of $l$ and $a$ $(\\ln (4l/a))$. The field is parallel to the rod's axis.", + "final_answer": [ + "\\mathscr{P}=\\frac{\\mathfrak{C l} l^{3}}{3[\\ln (4 l / a)-7 / 3]}" + ], + "answer_type": "Expression", + "answer": "Let $\\tau(z)$ be the induced charge per unit length on the surface of the rod; $z$ is the coordinate along the axis of the cylinder, with the origin chosen at the midpoint of the rod. The condition of constant potential on the conductor's surface is\n\n$$-\\mathfrak{C} z+\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\int_{-l}^{l} \\frac{\\tau(z^{\\prime}) \\mathrm{d} z^{\\prime} \\mathrm{d} \\varphi}{R}=0, \\quad R=[(z-z)^{2}+4 a^{2} \\sin ^{2} \\frac{\\varphi}{2}]^{1 / 2},\n$$\n\nwhere $\\varphi$ is the angle between the plane through the rod's axis and two points on its surface separated by distance $R$. In the integral, use the identity $\\tau(z^{\\prime})=\\tau(z)+[\\tau(z^{\\prime})-\\tau(z)]$, splitting the integral into two parts. In the first part, noting that $l \\gg a$, for points not very close to the ends of the rod, we have\n$$\n\\frac{\\tau(z)}{2 \\pi} \\iint \\frac{\\mathrm{~d} z^{\\prime} \\mathrm{d} \\varphi}{R} \\approx \\frac{\\tau(z)}{2 \\pi} \\int_{0}^{2 \\pi} \\ln \\frac{l^{2}-z^{2}}{a^{2} \\sin ^{2}(\\varphi / 2)} \\mathrm{d} \\varphi=\\tau(z) \\ln \\frac{4(l^{2}-z^{2})}{a^{2}}\n$$\n(using the known integral result $\\int_{0}^{\\pi} \\ln \\sin \\varphi \\mathrm{d} \\varphi=-\\pi \\ln 2$). In the second part of the integral, containing the difference $\\tau(z^{\\prime})-\\tau(z)$, terms involving $a^{2}$ in $R$ can be omitted as they do not cause divergence of the integral. Thus,\n$$\n\\mathfrak{C} z=\\tau(z) \\ln \\frac{4(l^{2}-z^{2})}{a^{2}}+\\int_{-l}^{l} \\frac{\\tau(z^{\\prime})-\\tau(z)}{|z^{\\prime}-z|} \\mathrm{d} z^{\\prime} .\n$$\nThe dependence of $\\tau$ on $z$ essentially reduces to being proportional to $z$; in this approximation, the integral in the above expression gives $-2 \\tau(z)$, resulting in\n$$\n\\tau(z)=\\frac{\\mathfrak{C} z}{\\ln [4(l^{2}-z^{2}) / a^{2}]-2}\n$$\n\nNear the endpoints of the rod, this expression is not applicable, but for the calculation of the desired dipole moment, this region of $z$-values is not important. To the precision adopted here, we have\n\n\\begin{aligned}\n\\mathscr{P} & =\\int_{-l}^{l} \\tau(z) z \\mathrm{~d} z=\\frac{\\mathfrak{C}}{L} \\int_{0}^{l}[z^{2}-\\frac{z^{2}}{2 L} \\ln (1-\\frac{z^{2}}{l^{2}})] \\mathrm{d} z \\\\\n& =\\mathfrak{C} \\frac{l^{3}}{3 L}[1+\\frac{1}{L}(\\frac{4}{3}-\\ln 2)]\n\\end{aligned}\n\n(where $L=\\ln (2 l / a)-1$ is a large number), or to the same precision\n$$\n\\mathscr{P}=\\frac{\\mathfrak{C l} l^{3}}{3[\\ln (4 l / a)-7 / 3]} .\n$$", + "symbol": { + "$l$": "half-length of the cylindrical rod", + "$a$": "radius of the cylindrical rod", + "$\\mathfrak{C}$": "electric field parallel to the rod's axis", + "$z$": "coordinate along the axis of the cylinder", + "$\\tau(z)$": "induced charge per unit length at position $z$", + "$\\varphi$": "angle on the rod's surface", + "$\\mathscr{P}$": "electric dipole moment of the rod", + "$L$": "a large number, defined as $L=\\ln (2 l / a)-1" + }, + "chapter": "Electrostatics of Conductors", + "section": "Solutions to electrostatic problems" + }, + { + "id": 100, + "topic": "Magnetism", + "question": "Under the influence of a uniform external electric field, consider an uncharged ellipsoid. When the external electric field is only along the $x$ axis of the ellipsoid, find the charge distribution on its surface $\\sigma$.", + "final_answer": [ + "\\sigma=\\mathfrak{C} \\frac{\\nu_{x}}{4 \\pi n^{(x)}}" + ], + "answer_type": "Expression", + "answer": "We first have\n$$\n\\sigma=-\\frac{1}{4 \\pi} \\frac{\\partial \\varphi}{\\partial n}|_{\\xi=0}=-(\\frac{1}{4 \\pi h_{1}}-\\frac{\\partial \\varphi}{\\partial \\xi})_{\\xi=0}\n$$\n(According to equation \n\\begin{align}\nd l^2 &= h_1^2 d \\xi^2 + h_2^2 d \\eta^2 + h_3^2 d \\zeta^2, \\\\\nh_1 &= \\frac{\\sqrt{(\\xi - \\eta)(\\xi - \\zeta)}}{2 R_\\xi}, \\quad \nh_2 = \\frac{\\sqrt{(\\eta - \\zeta)(\\eta - \\xi)}}{2 R_\\eta}, \\quad \nh_3 = \\frac{\\sqrt{(\\zeta - \\xi)(\\xi - \\eta)}}{2 R_\\zeta},\n\\end{align}\nthe length element along the normal direction of the ellipsoid surface is $h_{1} \\mathrm{~d} \\xi$.) With the help of equation \n\\begin{align}\n\\varphi = \\varphi_0 \\left\\{ 1 - \\frac{\\displaystyle \\int_{\\xi}^{\\infty} \\frac{ds}{(s + a^2) R_s}}{\\displaystyle \\int_{0}^{\\infty} \\frac{ds}{(s + a^2) R_s}}, \\right\\}\n\\end{align}\nand considering\n$$\n\\nu_{x}=\\frac{1}{h_{1}} \\frac{\\partial x}{\\partial \\xi}|_{\\xi=0}=\\frac{x}{2 a^{2} h_{1}}|_{\\xi=0},\n$$\n\nwhen the external electric field is along the x-axis direction of the ellipsoid, we obtain\n$$\n\\sigma=\\mathfrak{C} \\frac{\\nu_{x}}{4 \\pi n^{(x)}}\n$$", + "symbol": { + "$x$": "coordinate axis", + "$\\sigma$": "charge distribution on the surface", + "$\\varphi$": "electric potential", + "$h_{1}$": "metric coefficient associated with $\\xi$", + "$\\xi$": "ellipsoidal coordinate", + "$\\eta$": "ellipsoidal coordinate", + "$\\zeta$": "ellipsoidal coordinate", + "$R_{\\xi}$": "radius of curvature associated with $\\xi$", + "$R_{\\eta}$": "radius of curvature associated with $\\eta$", + "$R_{\\zeta}$": "radius of curvature associated with $\\zeta$", + "$\\varphi_0$": "reference electric potential", + "$a$": "characteristic length parameter", + "$s$": "integration variable", + "$\\mathfrak{C}$": "normalization constant", + "$\\nu_{x}$": "component of normal vector along x-axis", + "$n^{(x)}$": "normalization factor for x-direction" + }, + "chapter": "Electrostatics of Conductors", + "section": "conducting ellipsoid" + }, + { + "id": 101, + "topic": "Magnetism", + "question": "Consider an uncharged ellipsoid subjected to a uniform external electric field. When the external electric field is oriented in any direction relative to the ellipsoid's $x, y, z$ axes, find the charge distribution $\\sigma$ on its surface.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\sigma$": "surface charge distribution", + "$x$": "x-axis of ellipsoid", + "$y$": "y-axis of ellipsoid", + "$z$": "z-axis of ellipsoid", + "$\\varphi$": "electric potential", + "$\\xi$": "ellipsoidal coordinate", + "$\\eta$": "ellipsoidal coordinate", + "$\\zeta$": "ellipsoidal coordinate", + "$h_1$": "metric coefficient for $\\xi$", + "$h_2$": "metric coefficient for $\\eta$", + "$h_3$": "metric coefficient for $\\zeta$", + "$R_\\xi$": "radius for ellipsoidal coordinate $\\xi$", + "$R_\\eta$": "radius for ellipsoidal coordinate $\\eta$", + "$R_\\zeta$": "radius for ellipsoidal coordinate $\\zeta$", + "$s$": "integration variable", + "$a$": "semi-axis length of ellipsoid", + "$\\nu_x$": "direction cosine for x-axis", + "$\\mathscr{E}$": "external electric field", + "$n^{(x)}$": "normalization factor for x-component", + "$n_{ik}$": "matrix of normalizing factors", + "$\\nu_i$": "direction cosine in arbitrary direction", + "$\\mathfrak{C}_{k}$": "component of external electric field in direction $k$", + "$\\mathfrak{C}_{x}$": "x-component of external electric field", + "$\\mathfrak{C}_{y}$": "y-component of external electric field", + "$\\mathfrak{C}_{z}$": "z-component of external electric field" + }, + "chapter": "Electrostatics of Conductors", + "section": "conducting ellipsoid" + }, + { + "id": 102, + "topic": "Magnetism", + "question": "For a prolate rotational ellipsoid conductor, find the external potential $\\varphi$ when its symmetry axis is perpendicular to the external field (specifically referring to the scenario described in the solution where the field is in the $z$ axis direction).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varphi$": "external potential", + "$\\varphi_0$": "initial potential", + "$z$": "coordinate axis direction (z-axis)", + "$a$": "semi-major axis of the ellipsoid", + "$b$": "semi-minor axis of the ellipsoid", + "$\\xi$": "local coordinate parameter", + "$R_s$": "radius related function", + "$\\mathfrak{C}$": "proportionality constant" + }, + "chapter": "Electrostatics of Conductors", + "section": "conducting ellipsoid" + }, + { + "id": 103, + "topic": "Magnetism", + "question": "For an oblate rotating ellipsoidal conductor, when its symmetry axis is perpendicular to the external field (specifically referring to the case described in the solution where the field is in the $x$ axis direction), find the potential $\\varphi$ outside the conductor.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varphi$": "potential outside the conductor", + "$\\varphi_0$": "reference potential", + "$a$": "semi-major axis of the oblate ellipsoid", + "$c$": "semi-minor axis of the oblate ellipsoid", + "$x$": "x axis direction", + "$\\xi$": "integration variable", + "$\\mathfrak{c}$": "constant related to the system" + }, + "chapter": "Electrostatics of Conductors", + "section": "conducting ellipsoid" + }, + { + "id": 104, + "topic": "Magnetism", + "question": "Consider a uniform electric field of magnitude \\( \\mathfrak{C} \\) existing along the positive z-axis in the half-space \\( z<0 \\) (i.e., at \\( z \\to -\\infty \\), the electric field is \\( \\vec{E} = \\mathfrak{C}\\hat{k} \\), corresponding to the potential \\( \\varphi = -\\mathfrak{C}z \\)). This electric field is constrained by a grounded conductive plane with a circular hole of radius \\( a \\) centered at the origin, \\( z=0 \\). Determine the expression for the potential \\( \\varphi \\) throughout the entire space (which can be expressed in either oblate spheroidal coordinates \\( \\xi, \\eta \\) or Cartesian coordinates \\( z \\)).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathfrak{C}$": "magnitude of the electric field", + "$z$": "position coordinate along the z-axis", + "$\\varphi$": "electric potential", + "$a$": "radius of the circular hole", + "$\\pi$": "mathematical constant pi", + "$\\xi$": "oblate spheroidal coordinate", + "$\\eta$": "oblate spheroidal coordinate" + }, + "chapter": "Electrostatics of Conductors", + "section": "conducting ellipsoid" + }, + { + "id": 105, + "topic": "Magnetism", + "question": "In the same physical scenario as the previous sub-question (that is, in the half-space $z<0$, there exists a uniform electric field $\\mathfrak{C}$ along the positive $z$-axis, constrained by a grounded conductive plane at $z=0$ with a circular hole of radius $a$), the expression for the electric potential $\\varphi$ is known to be $\\varphi=-\\mathfrak{C} \\frac{z}{\\pi}[\\arctan \\frac{a}{\\sqrt{\\xi}}-\\frac{a}{\\sqrt{\\xi}}]$. Try to find the expression for the surface charge density $\\sigma$ on the lower side of the conductive plane ($z=0^-, \\rho > a$) in terms of the cylindrical radial distance $\\rho$ and hole radius $a$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$z$": "coordinate along the z-axis", + "$\\mathfrak{C}$": "uniform electric field along the positive z-axis", + "$a$": "radius of the circular hole", + "$\\varphi$": "electric potential", + "$\\sigma$": "surface charge density", + "$\\rho$": "cylindrical radial distance", + "$\\xi$": "variable related to the radial distance" + }, + "chapter": "Electrostatics of Conductors", + "section": "conducting ellipsoid" + }, + { + "id": 106, + "topic": "Magnetism", + "question": "Assume a charged spherical conductor is cut in half, try to determine the mutual repulsive force between the two hemispheres.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$F$": "force", + "$E$": "electric field", + "$\\theta$": "angle", + "$e$": "total charge", + "$a$": "radius of the sphere" + }, + "chapter": "Electrostatics of Conductors", + "section": "Force on a Conductor" + }, + { + "id": 107, + "topic": "Magnetism", + "question": "A spherical conductor is cut into two halves, determine the mutual repulsive force between the two hemispheres. The conductor sphere is uncharged and is in a uniform external electric field $\\mathfrak{C}$ perpendicular to the interface.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathfrak{C}$": "external electric field", + "$E$": "electric field on the sphere surface", + "$\\theta$": "angle with respect to the perpendicular interface", + "$F$": "mutual repulsive force", + "$a$": "radius of the sphere" + }, + "chapter": "Electrostatics of Conductors", + "section": "Force on a Conductor" + }, + { + "id": 108, + "topic": "Magnetism", + "question": "For waves propagating on the charged surface of a liquid conductor in a gravitational field, determine the stability conditions of this surface.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\omega$": "frequency", + "$k$": "wave number", + "$\\rho$": "density of the liquid", + "$g$": "gravitational acceleration", + "$\\sigma_{0}$": "surface charge density", + "$\\alpha$": "surface tension" + }, + "chapter": "Electrostatics of Conductors", + "section": "Force on a Conductor" + }, + { + "id": 109, + "topic": "Magnetism", + "question": "Find the stability condition for a charged spherical droplet with respect to small deformations.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$x$": "horizontal position along the x-axis", + "$z$": "vertical position along the z-axis", + "$\\zeta$": "vertical displacement of the liquid surface", + "$a$": "amplitude of the vertical displacement", + "$k$": "wave number", + "$\\omega$": "angular frequency of the wave", + "$t$": "time", + "$E_z$": "electric field in the z-direction", + "$E$": "electric field magnitude", + "$\\sigma_0$": "surface charge density", + "$\\varphi$": "electric potential", + "$\\varphi_1$": "small correction to the electric potential", + "$F_s$": "additional negative pressure acting on the surface", + "$\\rho_0$": "initial or reference liquid density", + "$\\rho$": "liquid density", + "$g$": "acceleration due to gravity", + "$\\alpha$": "surface tension coefficient", + "$\\Phi$": "velocity potential of the liquid", + "$A$": "amplitude related to the velocity potential" + }, + "chapter": "Electrostatics of Conductors", + "section": "Force on a Conductor" + }, + { + "id": 110, + "topic": "Magnetism", + "question": "Find the stability condition (Rayleigh, 1882) of a charged spherical droplet relative to splitting into two identical smaller droplets (large deformation). Assume the original droplet has a charge of $e$ and a radius of $a$, while each smaller droplet after splitting has a charge of $e/2$ and a radius of $a/2^{1/3}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$e$": "charge of the original droplet", + "$a$": "radius of the original droplet", + "$\\sigma_0$": "surface charge density", + "$k$": "wave number", + "$\\omega$": "angular frequency of the wave", + "$\\rho$": "liquid density", + "$\\alpha$": "surface tension coefficient", + "$g$": "acceleration due to gravity", + "$E_z$": "electric field above the surface", + "$E$": "electric field", + "$\\varphi$": "electric potential", + "$\\zeta$": "vertical displacement of the liquid surface", + "$\\Phi$": "velocity potential of the liquid", + "$A$": "amplitude of the velocity potential" + }, + "chapter": "Electrostatics of Conductors", + "section": "Force on a Conductor" + }, + { + "id": 111, + "topic": "Magnetism", + "question": "An infinitely long straight charged wire (with charge per unit length of $e$) is parallel to the interface between two media with different dielectric constants ($\\varepsilon_1$ and $\\varepsilon_2$ respectively), and the distance from the interface is $h$. Determine the potential $\\varphi_1$ in medium 1 ($\\varepsilon_1$).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$e$": "charge per unit length of the wire", + "$\\varepsilon_1$": "dielectric constant of medium 1", + "$\\varepsilon_2$": "dielectric constant of medium 2", + "$h$": "distance from the interface", + "$\\varphi_1$": "electric potential in medium 1", + "$e'$": "image charge per unit length", + "$r$": "distance from observation point to the original wire", + "$r'$": "distance from observation point to the image wire" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric constant" + }, + { + "id": 112, + "topic": "Magnetism", + "question": "An infinitely long straight conductor (with a linear charge of $e$) is parallel to the interface between two media with different dielectric constants ($\\varepsilon_1$ and $\\varepsilon_2$ respectively) and at a distance $h$ from the interface. Determine the potential $\\varphi_2$ within medium 2 ($\\varepsilon_2$).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$e$": "linear charge density of the conductor", + "$\\varepsilon_1$": "dielectric constant of medium 1", + "$\\varepsilon_2$": "dielectric constant of medium 2", + "$h$": "distance from the conductor to the interface", + "$\\varphi_2$": "potential within medium 2", + "$r$": "distance from the observation point to the original wire" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric constant" + }, + { + "id": 113, + "topic": "Magnetism", + "question": "Find the torque $K$ acting on a rotational ellipsoid in a uniform electric field $\\mathfrak{C}$, with a dielectric constant of $\\varepsilon$. The volume of the ellipsoid is $V$, $\\alpha$ is the angle between the direction of $\\mathfrak{C}$ and the symmetry axis of the ellipsoid, and $n$ is the depolarization coefficient along this axis.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$K$": "torque", + "$\\mathfrak{C}$": "uniform electric field", + "$\\varepsilon$": "dielectric constant", + "$V$": "volume of the ellipsoid", + "$\\alpha$": "angle between the electric field and symmetry axis", + "$n$": "depolarization coefficient along the symmetry axis", + "$\\mathscr{P}$": "dipole moment of the ellipsoid" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric Ellipsoid" + }, + { + "id": 114, + "topic": "Magnetism", + "question": "For a conductive rotating ellipsoid ($\\varepsilon \\rightarrow \\infty$), in a uniform electric field $\\mathfrak{C}$, find the torque $K$ acting on it. The volume of the ellipsoid is $V$, $\\alpha$ is the angle between the direction of $\\mathfrak{C}$ and the symmetry axis of the ellipsoid, and $n$ is the depolarization factor along this axis.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon$": "permittivity", + "$\\mathfrak{C}$": "uniform electric field", + "$K$": "torque", + "$V$": "volume of the ellipsoid", + "$\\alpha$": "angle between the electric field direction and the ellipsoid's symmetry axis", + "$n$": "depolarization factor along the symmetry axis", + "$\\mathscr{P}$": "dipole moment of the ellipsoid" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric Ellipsoid" + }, + { + "id": 115, + "topic": "Magnetism", + "question": "A hollow dielectric sphere (dielectric constant $\\varepsilon$, with inner and outer radii $b$ and $a$, respectively) is placed in a uniform external electric field $\\mathfrak{E}$. Determine the field inside the cavity of the sphere.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon$": "dielectric constant", + "$b$": "inner radius of the sphere", + "$a$": "outer radius of the sphere", + "$\\mathfrak{E}$": "external electric field", + "$A$": "constant related to potential in region 1", + "$B$": "constant related to potential in region 3", + "$C$": "constant related to potential in region 2", + "$D$": "constant related to potential in region 2", + "$\\boldsymbol{E}_{3}$": "electric field inside the cavity" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric Ellipsoid" + }, + { + "id": 116, + "topic": "Magnetism", + "question": "Determine the height $h$ by which the liquid surface inside a vertical parallel-plate capacitor rises.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$h$": "height of the liquid surface", + "$\\widetilde{F}$": "symbol representing a force or energy term", + "$\\rho$": "density of the liquid", + "$g$": "acceleration due to gravity", + "$\\varepsilon$": "relative permittivity of the liquid", + "$E$": "electric field strength" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Thermodynamic Relations for Dielectrics in an Electric Field" + }, + { + "id": 117, + "topic": "Magnetism", + "question": "If the object is not in a vacuum, but in a medium with a dielectric constant of $\\varepsilon^{(e)}$, find the formula of $\\mathscr{F}-\\mathscr{F}_{0}$.\nIf the answer exists in an integral, then find the integrand", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon^{(e)}$": "dielectric constant of the medium", + "$\\mathscr{F}$": "current force in medium", + "$\\mathscr{F}_{0}$": "force in vacuum", + "$\\mathfrak{C}$": "electric displacement field source term", + "$\\boldsymbol{D}$": "electric displacement vector", + "$\\boldsymbol{E}$": "electric field vector" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Total free energy of dielectric" + }, + { + "id": 118, + "topic": "Magnetism", + "question": "Consider a capacitor composed of two conducting surfaces separated by a distance $h$, with $h$ being smaller than the dimensions of the capacitor plates. The space between the capacitor plates is filled with a material of dielectric constant $\\varepsilon_{1}$. A small sphere with radius $a \\ll h$ and dielectric constant $\\varepsilon_{2}$ is placed inside the capacitor. Determine the change in the capacitance of the capacitor.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$h$": "distance between the capacitor plates", + "$\\varepsilon_{1}$": "dielectric constant of the material between the capacitor plates", + "$a$": "radius of the small sphere", + "$\\varepsilon_{2}$": "dielectric constant of the small sphere", + "$\\varphi$": "potential difference between the capacitor plates", + "$\\widetilde{\\mathscr{F}}$": "free energy when the potential is constant", + "$C_{0}$": "original capacitance of the capacitor", + "$\\mathfrak{C}$": "electric field strength", + "$\\mathscr{F}$": "free energy", + "$\\mathscr{F}_{0}$": "initial free energy without the sphere", + "$\\boldsymbol{D}$": "electric displacement vector", + "$\\varepsilon^{(e)}$": "effective dielectric constant", + "$\\boldsymbol{E}$": "electric field vector", + "$\\mathbf{E}^{(i)}$": "electric field inside the sphere", + "$C$": "capacitance with the sphere" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Electrostriction of Isotropic Dielectrics" + }, + { + "id": 119, + "topic": "Magnetism", + "question": "Try to determine the potential $\\varphi$ produced by a point charge $e$ inside an anisotropic homogeneous medium (with the point charge located at the origin and the principal axes of the dielectric tensor $\\varepsilon_{ik}$ aligned along the $x, y, z$ axes), and express it using the principal dielectric constants $\\varepsilon^{(x)}, \\varepsilon^{(y)}, \\varepsilon^{(z)}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varphi$": "potential", + "$e$": "point charge", + "$\\varepsilon_{ik}$": "dielectric tensor component", + "$\\varepsilon^{(x)}$": "principal dielectric constant in the x direction", + "$\\varepsilon^{(y)}$": "principal dielectric constant in the y direction", + "$\\varepsilon^{(z)}$": "principal dielectric constant in the z direction", + "$x$": "Cartesian coordinate along the x-axis", + "$y$": "Cartesian coordinate along the y-axis", + "$z$": "Cartesian coordinate along the z-axis" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric properties of crystals" + }, + { + "id": 120, + "topic": "Magnetism", + "question": "Determine the potential $\\varphi$ generated by a point charge $e$ in an anisotropic homogeneous medium, using tensor notation that does not depend on the choice of coordinate system.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varphi$": "potential", + "$e$": "point charge", + "$\\varepsilon^{(x)}$": "permittivity along the x-axis", + "$\\varepsilon^{(y)}$": "permittivity along the y-axis", + "$\\varepsilon^{(z)}$": "permittivity along the z-axis", + "$|\\varepsilon|$": "determinant of permittivity tensor", + "$\\varepsilon_{i k}$": "permittivity tensor", + "$\\varepsilon_{i k}^{-1}$": "inverse permittivity tensor", + "$x$": "x-coordinate", + "$y$": "y-coordinate", + "$z$": "z-coordinate", + "$x_{i}$": "coordinate component i", + "$x_{k}$": "coordinate component k" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric properties of crystals" + }, + { + "id": 121, + "topic": "Magnetism", + "question": "An anisotropic dielectric sphere with a radius of $a$ (principal values of the dielectric tensor are $\\varepsilon^{(x)}, \\varepsilon^{(y)}, \\varepsilon^{(z)}$, with principal axes along the $x, y, z$ axes, respectively) is in a uniform external electric field $\\boldsymbol{\\mathfrak{C}} = (\\mathfrak{C}_x, \\mathfrak{C}_y, \\mathfrak{C}_z)$ (in vacuum). Determine the $x$ component of the torque $K_x$ acting on the sphere.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the sphere", + "$\\varepsilon^{(x)}$": "dielectric tensor principal value along the x-axis", + "$\\varepsilon^{(y)}$": "dielectric tensor principal value along the y-axis", + "$\\varepsilon^{(z)}$": "dielectric tensor principal value along the z-axis", + "$\\mathfrak{C}_x$": "x component of the external electric field", + "$\\mathfrak{C}_y$": "y component of the external electric field", + "$\\mathfrak{C}_z$": "z component of the external electric field", + "$K_x$": "x component of the torque acting on the sphere", + "$\\mathcal{P}_y$": "y component of the total dipole moment", + "$\\mathcal{P}_z$": "z component of the total dipole moment" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric properties of crystals" + }, + { + "id": 122, + "topic": "Magnetism", + "question": "An anisotropic dielectric sphere with a radius of $a$ (the principal values of the dielectric tensor are $\\varepsilon^{(x)}, \\varepsilon^{(y)}, \\varepsilon^{(z)}$, with principal axes along the $x, y, z$ directions, respectively) is placed in a uniform external electric field $\\boldsymbol{\\mathfrak{C}} = (\\mathfrak{C}_x, \\mathfrak{C}_y, \\mathfrak{C}_z)$ (in vacuum). Determine the $y$ component of the torque $K_y$ acting on the sphere.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the sphere", + "$\\varepsilon^{(x)}$": "dielectric tensor principal value along x-axis", + "$\\varepsilon^{(y)}$": "dielectric tensor principal value along y-axis", + "$\\varepsilon^{(z)}$": "dielectric tensor principal value along z-axis", + "$\\mathfrak{C}_x$": "x-component of the external electric field", + "$\\mathfrak{C}_y$": "y-component of the external electric field", + "$\\mathfrak{C}_z$": "z-component of the external electric field", + "$E_i$": "electric field intensity component inside the sphere", + "$P_i$": "polarization intensity component of the sphere", + "$\\mathcal{P}_i$": "total dipole moment component of the sphere", + "$K_y$": "y-component of the torque acting on the sphere" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric properties of crystals" + }, + { + "id": 123, + "topic": "Magnetism", + "question": "An anisotropic dielectric sphere with a radius of $a$ (principal values of the dielectric tensor are $\\varepsilon^{(x)}, \\varepsilon^{(y)}, \\varepsilon^{(z)}$, with principal axes along the $x, y, z$ axes respectively) is placed in a uniform external electric field $\\boldsymbol{\\mathfrak{C}} = (\\mathfrak{C}_x, \\mathfrak{C}_y, \\mathfrak{C}_z)$ in vacuum. Determine the $z$ component $K_z$ of the torque acting on the sphere.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the sphere", + "$\\varepsilon^{(x)}$": "principal value of the dielectric tensor along the x-axis", + "$\\varepsilon^{(y)}$": "principal value of the dielectric tensor along the y-axis", + "$\\varepsilon^{(z)}$": "principal value of the dielectric tensor along the z-axis", + "$x$": "x-axis", + "$y$": "y-axis", + "$z$": "z-axis", + "$\\mathfrak{C}$": "external electric field vector", + "$\\mathfrak{C}_x$": "x component of the external electric field", + "$\\mathfrak{C}_y$": "y component of the external electric field", + "$\\mathfrak{C}_z$": "z component of the external electric field", + "$E_i$": "component of the electric field intensity inside the sphere", + "$i$": "index representing x, y, or z axis", + "$P_i$": "component of the polarization intensity", + "$\\mathcal{P}_i$": "component of the total dipole moment", + "$\\boldsymbol{K}$": "torque vector acting on the sphere", + "$K_z$": "z component of the torque on the sphere" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric properties of crystals" + }, + { + "id": 124, + "topic": "Magnetism", + "question": "Consider a dielectric sphere (with radius $a$) placed in a uniform external electric field $\\mathfrak{C}$, sliced into two halves by a plane perpendicular to the field direction. Find the attraction force between the two hemispheres.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the dielectric sphere", + "$\\mathfrak{C}$": "external electric field", + "$\\varepsilon$": "dielectric constant of the sphere", + "$\\mathbf{E}$": "electric field strength near the sphere surface", + "$\\boldsymbol{E}^{(i)}$": "electric field strength inside the sphere", + "$\\boldsymbol{D}^{(i)}$": "electric displacement within the slit", + "$E_{r}$": "radial component of the electric field on the outer surface", + "$E_{\\theta}$": "tangential component of the electric field on the outer surface", + "$\\theta$": "angle between the radial vector and the electric field direction", + "$F$": "attraction force between the two hemispheres" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Electricity in solids" + }, + { + "id": 125, + "topic": "Magnetism", + "question": "Try to determine the shape change of a dielectric sphere in a uniform external electric field.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathscr{P}$": "polarization vector", + "$\\mathfrak{C}$": "external electric field strength", + "$V$": "volume of the sphere", + "$\\varepsilon^{(x)}$": "dielectric constant along the x axis", + "$n$": "effective medium parameter", + "$\\varepsilon_{ik}$": "permittivity tensor", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\delta_{ik}$": "Kronecker delta", + "$a_{1}$": "elastic constant related to strain", + "$u_{ik}$": "strain tensor component", + "$a_{2}$": "elastic constant related to strain", + "$u_{ll}$": "trace of strain tensor", + "$u_{x x}$": "strain component in the x direction", + "$u_{y y}$": "strain component in the y direction", + "$a$": "semi-major axis of the ellipsoid", + "$b$": "semi-minor axis of the ellipsoid", + "$R$": "radius of the original sphere", + "$\\mu$": "shear modulus" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Electricity in solids" + }, + { + "id": 126, + "topic": "Magnetism", + "question": "Determine the Young's modulus of a non-pyroelectric piezoelectric material parallel plate thin slab under the following conditions (the ratio of tensile stress to relative tensile strain): The slab is under tensile stress between the plates of a short-circuited capacitor.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\boldsymbol{E}$": "electric field", + "$\\sigma_{ik}$": "stress tensor component", + "$\\sigma_{zz}$": "tensile stress (z-axis)", + "$u_{zz}$": "relative tensile strain (z-axis)", + "$\\mu_{zzzz}$": "material coefficient related to Young's modulus (zzzz component)", + "$E$": "Young's modulus", + "$E_{x}$": "electric field component along x-axis", + "$E_{y}$": "electric field component along y-axis", + "$E_{z}$": "electric field component along z-axis", + "$D_{z}$": "electric displacement field component along z-axis", + "$\\varepsilon_{zz}$": "permittivity component (zz)", + "$\\gamma_{z,zz}$": "piezoelectric coupling coefficient (z, zz component)", + "$\\sigma_{xx}$": "tensile stress (x-axis)", + "$u_{xx}$": "relative tensile strain (x-axis)", + "$\\mu_{xxxx}$": "material coefficient related to Young's modulus (xxxx component)", + "$\\gamma_{z,xx}$": "piezoelectric coupling coefficient (z, xx component)" + }, + "chapter": "Electrostatics of Conductors", + "section": "Piezoelectric" + }, + { + "id": 127, + "topic": "Magnetism", + "question": "Try to derive the expression or equation for the velocity of sound within a piezoelectric medium.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$u_{i k}$": "displacement component", + "$\\sigma_{i k}$": "stress tensor component", + "$\\widetilde{F}$": "modified free energy", + "$F_{0}$": "initial free energy", + "$\\lambda_{i k l m}$": "elasticity tensor component", + "$\\varepsilon_{i k}$": "dielectric tensor component", + "$E_{i}$": "electric field component", + "$D_{i 0}$": "electric displacement component", + "$\\beta_{i, k l}$": "piezoelectric tensor component", + "$\\rho$": "density of the medium", + "$\\boldsymbol{u}$": "displacement vector", + "$\\nu_{i k}$": "strain component", + "$\\varphi$": "electric potential", + "$\\omega$": "angular frequency", + "$k_{k}$": "wave vector component", + "$\\boldsymbol{k}$": "wave vector", + "$\\delta_{i k}$": "Kronecker delta", + "$l$": "subscript index indicating a specific component", + "$m$": "another subscript index indicating a specific component", + "$p$": "another subscript index indicating a specific component", + "$r$": "another subscript index indicating a specific component", + "$s$": "another subscript index indicating a specific component", + "$q$": "another subscript index indicating a specific component" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Piezoelectric" + }, + { + "id": 128, + "topic": "Magnetism", + "question": "The piezoelectric crystals belonging to the $C_{6 v}$ crystal class are constrained by the surface plane ( $xz$ plane) of the symmetry axis ($z$ axis). Try to determine the velocity of surface waves propagating perpendicular to the symmetry axis (along the $x$ axis) which undergo displacement $u_{z}$ and potential $\\varphi$ oscillations", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$C_{6 v}$": "crystal class", + "$x$": "coordinate axis perpendicular to the symmetry axis", + "$z$": "symmetry axis", + "$u_{z}$": "displacement oscillation", + "$\\varphi$": "potential oscillation", + "$\\rho$": "density", + "$\\sigma_{i k}$": "stress tensor component", + "$\\lambda_{i k l m}$": "elastic stiffness tensor component", + "$\\beta_{l, i k}$": "piezoelectric tensor component", + "$E_{l}$": "electric field component", + "$\\varepsilon_{i k}$": "permittivity tensor component", + "$\\sigma_{zx}$": "stress component in the zx plane", + "$\\sigma_{zy}$": "stress component in the zy plane", + "$D_{x}$": "electric displacement in the x direction", + "$D_{y}$": "electric displacement in the y direction", + "$E_{x}$": "electric field component in the x direction", + "$E_{y}$": "electric field component in the y direction", + "$u_{zx}$": "strain component in the zx plane", + "$u_{zy}$": "strain component in the zy plane", + "$\\beta$": "piezoelectric coupling constant", + "$\\lambda$": "effective stiffness constant", + "$\\varepsilon$": "permittivity", + "$D_{z}$": "pyroelectric displacement", + "$D_{0}$": "constant pyroelectric displacement", + "$t$": "time", + "$\\Delta$": "Laplacian operator", + "$\\bar{\\lambda}$": "modified stiffness constant", + "$\\psi$": "auxiliary function", + "$\\varphi^{(i)}$": "internal potential", + "$\\varphi^{(e)}$": "external potential", + "$\\varkappa$": "decay constant", + "$k$": "wave number", + "$\\omega$": "angular frequency", + "$\\Lambda$": "dimensionless constant" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Piezoelectric" + }, + { + "id": 129, + "topic": "Magnetism", + "question": "Given the second-order tensor $\\sigma_{ik}$, with its symmetric part $s_{ik}$ and its antisymmetric part formed by the axial vector $\\boldsymbol{a}$ (specific definitions are found in the symbols table), express the determinant $|\\sigma|$ of the tensor $\\sigma_{ik}$ in terms of the components of $s_{ik}$ and $\\boldsymbol{a}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\sigma_{ik}$": "second-order tensor", + "$s_{ik}$": "symmetric part of the tensor", + "$\\boldsymbol{a}$": "axial vector", + "$s_{xx}$": "component of the symmetric tensor along x", + "$s_{yy}$": "component of the symmetric tensor along y", + "$s_{zz}$": "component of the symmetric tensor along z", + "$a_x$": "component of the axial vector along x", + "$a_y$": "component of the axial vector along y", + "$a_z$": "component of the axial vector along z", + "$s_{i k}$": "general component of the symmetric part of the tensor", + "$a_{i}$": "component of the axial vector", + "$a_{k}$": "component of the axial vector" + }, + "chapter": "Constant current", + "section": "Hall Effect" + }, + { + "id": 130, + "topic": "Magnetism", + "question": "Given a second-order tensor $\\sigma_{ik}$, its symmetric part is $s_{ik}$, and the antisymmetric part is formed by the axial vector $\\boldsymbol{a}$. Given its determinant as $|\\sigma|=|s|+s_{i k} a_{i} a_{k}$. Try to express the axial vector $b_i$ of the antisymmetric part of its inverse tensor $\\sigma_{ik}^{-1}$ using the components of $s_{i k}$ and $\\boldsymbol{a}$ (i.e., $\\sigma_{ik}^{-1} = \\rho_{ik} + \\epsilon_{ikl}b_l$, where $\\epsilon_{ikl}b_l$ is the antisymmetric part).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\sigma_{ik}$": "second-order tensor", + "$s_{ik}$": "symmetric part of the tensor", + "$\\boldsymbol{a}$": "axial vector forming the antisymmetric part", + "$|\\sigma|$": "determinant of the tensor", + "$b_i$": "axial vector of the antisymmetric part of the inverse tensor", + "$\\rho_{ik}$": "symmetric part of inverse tensor", + "$\\epsilon_{ikl}$": "Levi-Civita symbol", + "$a_i$": "component of the axial vector", + "$a_k$": "component of the axial vector", + "$b_l$": "component of the axial vector forming the antisymmetric part", + "$\\sigma_{ik}^{-1}$": "inverse tensor" + }, + "chapter": "Constant current", + "section": "Hall Effect" + }, + { + "id": 131, + "topic": "Magnetism", + "question": "Assuming two parallel planar plates (made of the same metal $A$) are immersed in an electrolyte solution $AX$. In the case of a very small current $j$, derive the expression for the effective resistivity of the solution $\\frac{\\mathscr{E}}{l j}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$A$": "metal of the plates", + "$j$": "current density", + "$l$": "distance between the plates", + "$c_1$": "surface concentration at one plate", + "$c_2$": "surface concentration at the other plate", + "$\\rho$": "resistivity", + "$D$": "diffusion coefficient", + "$\\beta$": "constant related to the system", + "$m$": "mass of the ion", + "$e$": "elementary charge", + "$\\mathscr{E}$": "potential difference between the two plates", + "$\\sigma$": "conductivity", + "$\\zeta$": "electric potential related to concentration" + }, + "chapter": "Constant current", + "section": "Diffusive electricity phenomenon" + }, + { + "id": 132, + "topic": "Magnetism", + "question": "Consider a circular line current with a radius $a$. Try to find the radial component $B_r$ of the magnetic field in cylindrical coordinates.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the circular line current", + "$B_r$": "radial component of the magnetic field", + "$r$": "radial distance in cylindrical coordinates", + "$\\varphi$": "angular coordinate in cylindrical coordinates", + "$z$": "vertical coordinate in cylindrical coordinates", + "$A_{\\varphi}$": "azimuthal component of the vector potential", + "$J$": "current density", + "$c$": "speed of light", + "$R$": "distance from line element to observation point", + "$\\theta$": "angle variable in elliptic integral transformation", + "$k$": "parameter in elliptic integrals", + "$K$": "complete elliptic integral of the first kind", + "$E$": "complete elliptic integral of the second kind" + }, + "chapter": "Static Magnetic Field", + "section": "Magnetic field of constant current" + }, + { + "id": 133, + "topic": "Magnetism", + "question": "Consider a circular line current with a radius of $a$. Try to find the axial component $B_z$ of the magnetic field it produces in cylindrical coordinates.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the circular line current", + "$B_z$": "axial component of the magnetic induction", + "$r$": "radial distance in the cylindrical coordinate system", + "$\\varphi$": "angular coordinate in the cylindrical coordinate system", + "$z$": "axial coordinate in the cylindrical coordinate system", + "$A_{\\varphi}$": "azimuthal component of the vector potential", + "$J$": "current density", + "$c$": "speed of light in vacuum", + "$R$": "distance from element of current loop to point of interest", + "$\\theta$": "variable introduced to simplify integral", + "$k$": "parameter for elliptic integrals", + "$K$": "complete elliptic integral of the first kind", + "$E$": "complete elliptic integral of the second kind" + }, + "chapter": "Static Magnetic Field", + "section": "Magnetic field of constant current" + }, + { + "id": 134, + "topic": "Magnetism", + "question": "Try to find the 'internal' part of the self-inductance $L_i$ of a closed thin wire with a circular cross-section.\n\n\\footnotetext{\n(1) The assertion in the main text that the self-inductance does not depend on the current distribution actually applies not only to the approximation (34.1), but also to subsequent approximations that do not contain large logarithmic terms (this corresponds to considering the coefficient in front of $l / a$ in the argument of the logarithm); see the exercises in this section.\n(2) In exercises $1-6$, it is assumed that the susceptibility of the medium is $\\mu_{e}=1$.\n}", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$L_i$": "internal part of the self-inductance", + "$r$": "distance to the wire axis", + "$a$": "radius of the wire", + "$H$": "magnetic field", + "$J$": "current density", + "$c$": "speed of light in vacuum (used in calculations with units involving electromagnetism)", + "$l$": "length of the closed wire", + "$\\mu_i$": "permeability inside the wire", + "$\\mu_{e}$": "susceptibility of the medium" + }, + "chapter": "Static Magnetic Field", + "section": "Self-inductance of a wire conductor" + }, + { + "id": 135, + "topic": "Magnetism", + "question": "Try to determine the self-inductance of a thin circular ring (radius $b$) made from a wire with a circular cross-section (radius $a$).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$b$": "radius of the thin circular ring", + "$a$": "radius of the wire's circular cross-section", + "$L_e$": "self-inductance associated with external magnetic field", + "$L_{i}$": "self-inductance associated with internal magnetic field", + "$L$": "total self-inductance", + "$\\varphi$": "central angle subtended by the chord on the ring", + "$\\varphi_{0}$": "initial central angle for integration", + "$\\mu_{i}$": "relative permeability of the material" + }, + "chapter": "Static Magnetic Field", + "section": "Self-inductance of a wire conductor" + }, + { + "id": 136, + "topic": "Magnetism", + "question": "A current flows through a wire loop $(\\mu_{i}=1)$; try to determine the elongation of the loop under the magnetic field generated by this current.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mu_{i}$": "permeability (set to 1 in this context)", + "$J$": "current density", + "$L$": "inductance", + "$q$": "generalized coordinate", + "$a$": "radius of the wire", + "$b$": "radius parameter for the loop", + "$\\sigma_{\\|}$": "stress parallel to the wire axis", + "$\\sigma_{\\perp}$": "stress perpendicular to the wire axis", + "$c$": "speed of light", + "$E$": "Young's modulus", + "$\\sigma$": "Poisson's ratio" + }, + "chapter": "Static Magnetic Field", + "section": "Self-inductance of a wire conductor" + }, + { + "id": 137, + "topic": "Magnetism", + "question": "Seek the first-order correction value of the cylindrical helical tube's self-inductance, due to the field distortion near both ends of the cylindrical helical tube when $l / h$ (with $\\mu_{e}=1$).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$l$": "characteristic length related to the cylindrical helical tube", + "$h$": "length of the cylindrical helical tube", + "$\\mu_{e}$": "magnetic permeability, given as 1 in this problem", + "$L$": "self-inductance of the cylindrical helical tube", + "$J$": "surface current density", + "$\\boldsymbol{g}_{1}$": "tangential vector on the helical surface", + "$\\boldsymbol{g}_{2}$": "tangential vector on the helical surface", + "$R$": "distance function in the integral", + "$g$": "surface current density", + "$n$": "number of turns per unit length of the helix", + "$b$": "radius of the cylindrical helical tube", + "$\\varphi$": "angle between diametrical planes", + "$z_{1}$": "position variable along the helical tube", + "$z_{2}$": "position variable along the helical tube", + "$\\zeta$": "difference $z_{2} - z_{1}$" + }, + "chapter": "Static Magnetic Field", + "section": "Self-inductance of a wire conductor" + }, + { + "id": 138, + "topic": "Magnetism", + "question": "If a planar circuit is placed on the surface of a semi-infinite medium with a permeability of $\\mu_{e}$, find the factor by which the self-inductance of the planar circuit changes. We neglect the internal part of the conductor's self-inductance.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mu_{e}$": "permeability of the medium", + "$\\boldsymbol{H}_{0}$": "magnetic field due to the current in absence of medium", + "$\\boldsymbol{H}$": "magnetic field in the vacuum half-space", + "$\\boldsymbol{B}$": "magnetic field in the medium", + "$B_{n}$": "normal component of the magnetic field", + "$\\boldsymbol{H}_{t}$": "tangential component of the magnetic field" + }, + "chapter": "Static Magnetic Field", + "section": "Self-inductance of a wire conductor" + }, + { + "id": 139, + "topic": "Magnetism", + "question": "Given a straight wire carrying a current $J$ parallel to an infinitely long cylindrical conductor with a radius $a$ (permeability $\\mu$) at a distance $l$ from the axis of the cylinder, determine the force on the straight wire.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$J$": "current", + "$a$": "radius of the cylindrical conductor", + "$\\mu$": "permeability of the cylindrical conductor", + "$l$": "distance from the axis of the cylinder", + "$J^{\\prime}$": "symbolic current related to $J$ and $\\mu$", + "$J^{\\prime \\prime}$": "another symbolic current related to $J$ and $\\mu$", + "$b$": "another distance parameter relevant to the configuration" + }, + "chapter": "Static Magnetic Field", + "section": "Forces in a magnetic field" + }, + { + "id": 140, + "topic": "Magnetism", + "question": "Try to determine the average magnetization intensity of a polycrystal in a strong magnetic field $(H \\gg 4 \\pi M)$, where the microcrystals have uniaxial symmetry.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$H$": "magnetic field intensity", + "$M$": "magnetization intensity", + "$\\theta$": "angle between easy magnetization direction and magnetization vector", + "$\\psi$": "angle between easy magnetization direction and magnetic field vector", + "$\\vartheta$": "small angle close to difference between $\\theta$ and $\\psi$", + "$\\beta$": "parameter related to the material's magnetic properties", + "$\\overline{M}$": "average magnetization intensity" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Ferromagnetism and Antiferromagnetism" + }, + { + "id": 141, + "topic": "Magnetism", + "question": "For cubic symmetric microcrystals, try to determine the average magnetization intensity of a polycrystal in a strong magnetic field $(H \\gg 4 \\pi M)$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$M_x$": "x-component of magnetization", + "$M_y$": "y-component of magnetization", + "$M_z$": "z-component of magnetization", + "$H_x$": "x-component of magnetic field", + "$H_y$": "y-component of magnetic field", + "$H_z$": "z-component of magnetic field", + "$\\lambda$": "Lagrange multiplier", + "$M$": "magnitude of magnetization", + "$H$": "magnitude of magnetic field", + "$\\beta$": "coefficients related to magnetization properties", + "$\\vartheta$": "angle between magnetization and magnetic field", + "$\\bar{M}$": "average magnetization" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Ferromagnetism and Antiferromagnetism" + }, + { + "id": 142, + "topic": "Magnetism", + "question": "Try to find out the relative elongation of a ferromagnetic cubic crystal depending on the magnetization direction $\\boldsymbol{m}$ and the measurement direction $\\boldsymbol{n}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\boldsymbol{m}$": "magnetization direction", + "$\\boldsymbol{n}$": "measurement direction", + "$u_{i k}$": "strain tensor component", + "$a_{1}$": "coefficient of magnetization direction (x-component)", + "$a_{2}$": "coefficient of magnetization direction (mixed components)", + "$m_x$": "x-component of magnetization direction", + "$m_y$": "y-component of magnetization direction", + "$m_z$": "z-component of magnetization direction", + "$n_x$": "x-component of measurement direction", + "$n_y$": "y-component of measurement direction", + "$n_z$": "z-component of measurement direction", + "$\\nu_{xx}$": "normal stress component (x-axis)", + "$\\nu_{xy}$": "shear stress component in xy-plane" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Magnetostriction of ferromagnets" + }, + { + "id": 143, + "topic": "Magnetism", + "question": "The easy magnetization axes of a cubic ferromagnet align along the three edges of the cube (specifically the $x, y, z$ axes). The magnetic domains are magnetized parallel or antiparallel to the $z$ axis, and the domain walls are distributed parallel to the (100) plane. Determine the surface tension of the domain wall in this case.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$x$": "x-axis coordinate", + "$y$": "y-axis coordinate", + "$z$": "z-axis coordinate", + "$M$": "magnetization vector", + "$\\alpha$": "parameter related to inhomogeneous anisotropy", + "$K$": "anisotropy constant", + "$\\beta$": "parameter related to anisotropy energy", + "$m_{x}$": "component of magnetization in the x-direction", + "$m_{y}$": "component of magnetization in the y-direction", + "$m_{z}$": "component of magnetization in the z-direction", + "$\\theta$": "angle between magnetization vector and the z-axis", + "$\\Delta_{(100)}$": "surface tension of the domain wall parallel to the (100) plane" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Surface tension of domain walls" + }, + { + "id": 144, + "topic": "Magnetism", + "question": "The easy magnetization axes of a cubic ferromagnet are along the three edges of the cube (namely the $x, y, z$ axes). The magnetic domains are magnetized parallel or antiparallel to the $z$-axis, and the domain walls are distributed parallel to the (110) plane. Determine the surface tension of the domain walls in this scenario.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$x$": "x-axis of the cubic ferromagnet", + "$y$": "y-axis of the cubic ferromagnet", + "$z$": "z-axis of the cubic ferromagnet", + "$\\boldsymbol{M}$": "magnetization vector", + "$M$": "magnitude of the magnetization vector", + "$\\theta$": "angle between magnetization vector and z-axis", + "$\\nu_{\\mathrm{non}-\\mathrm{u}}$": "non-uniform anisotropy energy", + "$A$": "constant involving magnetic parameters", + "$B$": "constant related to magnetic anisotropy", + "$\\beta$": "anisotropy constant", + "$\\alpha$": "constant related to surface tension", + "$\\xi$": "coordinate perpendicular to the domain wall plane", + "$\\Delta$": "surface tension of the domain wall", + "$\\Delta_{(110)}$": "surface tension of the domain wall in the (110) plane" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Surface tension of domain walls" + }, + { + "id": 145, + "topic": "Magnetism", + "question": "If the transition between magnetic domains is not achieved through the rotation of $M$ but by changing the magnitude of $M$ (i.e., when $M$ changes sign after passing through zero), determine the surface tension of the domain wall in a uniaxial crystal. The free energy's dependence on $M$ (at $\\boldsymbol{H}=0$) takes the expanded form corresponding to the situation near the Curie point as given by Equation \\begin{align*}\n\\tilde{\\Phi} = \\Phi_0 + AM^2 + BM^4 - MH - \\frac{H^2}{8\\pi},\n\\end{align*}.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$M$": "magnetization", + "$H$": "magnetic field", + "$\\tilde{\\Phi}$": "free energy density", + "$\\Phi_0$": "initial free energy density", + "$A$": "coefficient of quadratic term in free energy expansion", + "$B$": "coefficient of quartic term in free energy expansion", + "$\\alpha_{1}$": "coefficient related to inhomogeneity in magnetization", + "$M_{z}$": "component of magnetization along the z-axis", + "$M_{0}$": "equilibrium value of magnetization within the domain", + "$m$": "normalized magnetization vector", + "$T_c$": "Curie temperature", + "$x$": "axis perpendicular to the domain wall plane", + "$\\Delta$": "surface tension of the domain wall", + "$T$": "temperature", + "$\\beta$": "dimensionless parameter related to the energy" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Surface tension of domain walls" + }, + { + "id": 146, + "topic": "Magnetism", + "question": "The parallel plane magnetic domains extend perpendicularly to the surface of the ferromagnetic material without changing the direction of magnetization . Try to derive and present the exact mathematical expression for the magnetic field energy per unit surface area near the surface of the ferromagnet (expressed in terms of $\\zeta(3)$).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\zeta(3)$": "Apéry's constant, which is the value of the Riemann zeta function at 3", + "$\\sigma$": "surface charge density", + "$M$": "magnetization or magnetic polarization", + "$z$": "coordinate perpendicular to the surface plane", + "$x$": "coordinate parallel to the surface plane", + "$a$": "width of the magnetic domain", + "$c_{n}$": "Fourier series coefficient", + "$b_{n}$": "coefficient for series solution of potential field" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Magnetic domain structure of ferromagnets" + }, + { + "id": 147, + "topic": "Superconductivity", + "question": "Find the magnetic moment ${ }^{(1)}$ of a superconducting disk perpendicular to the external magnetic field.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$c$": "radius of the disk approaching zero", + "$a$": "radius of the spheroid", + "$\\varepsilon^{(i)}$": "permittivity of the dielectric (internal)", + "$n^{(x)}$": "demagnetizing factor along the x-axis", + "$\\mathscr{M}$": "magnetic moment", + "$\\mathfrak{H}$": "external magnetic field" + }, + "chapter": "Superconductivity", + "section": "Superconducting current" + }, + { + "id": 148, + "topic": "Superconductivity", + "question": "Seek the heat capacity of a superconducting ellipsoid in the intermediate state.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\tilde{\\phi}_t$": "thermodynamic potential", + "$V$": "volume", + "$H_{\\text{cr}}$": "critical magnetic field", + "$n$": "demagnetization factor", + "$\\mathfrak{H}$": "external magnetic field", + "$T$": "temperature", + "$\\mathscr{C}_{t}$": "total heat capacity", + "$\\mathscr{C}_{s}$": "heat capacity in superconducting state", + "$H_{\\mathrm{cr}}^{\\prime}$": "derivative of critical magnetic field with respect to temperature", + "$H_{\\mathrm{cr}}^{\\prime \\prime}$": "second derivative of critical magnetic field with respect to temperature" + }, + "chapter": "Superconductivity", + "section": "Intermediate structure" + }, + { + "id": 149, + "topic": "Magnetism", + "question": "An isotropic conducting sphere with radius $a$ is in a uniform periodic external magnetic field. Determine the expression for its magnetic polarizability $\\alpha$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the sphere", + "$\\alpha$": "magnetic polarizability", + "$k$": "wave number", + "$\\mathfrak{H}$": "external magnetic field strength", + "$\\beta$": "constant related to axial vector", + "$V$": "volume of the sphere, expressed as $\\frac{4 \\pi a^3}{3}$", + "$r$": "radial distance from the center of the sphere", + "$\\boldsymbol{H}^{(i)}$": "magnetic field inside the sphere", + "$\\boldsymbol{H}^{(e)}$": "external magnetic field", + "$\\boldsymbol{n}$": "unit vector in the radial direction", + "$\\boldsymbol{A}$": "vector potential", + "$f$": "spherically symmetric solution function", + "$\\varphi$": "scalar potential function" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Depth of magnetic field penetration into a conductor" + }, + { + "id": 150, + "topic": "Magnetism", + "question": "An isotropic conducting sphere with a radius of $a$ is in a uniform periodic external magnetic field, with its magnetic susceptibility given by $\\alpha = \\alpha^{\\prime} + \\mathrm{i} \\alpha^{\\prime \\prime}$. Determine the expression for the real part of its magnetic susceptibility $\\alpha^{\\prime}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the sphere", + "$\\alpha$": "magnetic susceptibility", + "$\\alpha^{\\prime}$": "real part of the magnetic susceptibility", + "$\\alpha^{\\prime \\prime}$": "imaginary part of the magnetic susceptibility", + "$k$": "wave number related to the sphere", + "$\\delta$": "skin depth" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Depth of magnetic field penetration into a conductor" + }, + { + "id": 151, + "topic": "Magnetism", + "question": "An isotropic conductive sphere with a radius of $a$ is in a uniform periodic external magnetic field. Its magnetic susceptibility is $\\alpha = \\alpha^{\\prime} + \\mathrm{i} \\alpha^{\\prime \\prime}$. Determine the expression for the imaginary part of its magnetic susceptibility $\\alpha^{\\prime \\prime}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the sphere", + "$\\alpha$": "magnetic susceptibility", + "$\\alpha^{\\prime}$": "real part of the magnetic susceptibility", + "$\\alpha^{\\prime \\prime}$": "imaginary part of the magnetic susceptibility", + "$k$": "wave number related to skin depth", + "$\\delta$": "skin depth" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Depth of magnetic field penetration into a conductor" + }, + { + "id": 152, + "topic": "Magnetism", + "question": "Find the smallest value of the attenuation coefficient for the magnetic field inside a conducting sphere.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$H$": "magnetic field magnitude", + "$H_{r}$": "radial component of the magnetic field", + "$\\gamma$": "attenuation coefficient", + "$k$": "real scalar constant relating to wave number", + "$r$": "radial distance from center of the sphere", + "$a$": "radius of the sphere", + "$\\sigma$": "conductivity of the sphere", + "$c$": "speed of light" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Depth of magnetic field penetration into a conductor" + }, + { + "id": 153, + "topic": "Magnetism", + "question": "Two inductively coupled circuits respectively contain self-inductances $L_{1}$ and $L_{2}$ and capacitances $C_{1}$ and $C_{2}$. Determine the intrinsic frequencies of electric oscillations within these coupled circuits (we neglect resistances $R_{1}$ and $R_{2}$).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$L_{1}$": "self-inductance of the first circuit", + "$L_{2}$": "self-inductance of the second circuit", + "$L_{12}$": "mutual inductance between the two circuits", + "$C_{1}$": "capacitance of the first circuit", + "$C_{2}$": "capacitance of the second circuit", + "$R_{1}$": "resistance of the first circuit", + "$R_{2}$": "resistance of the second circuit", + "$\\omega_{1}$": "intrinsic frequency of the first circuit", + "$\\omega_{2}$": "intrinsic frequency of the second circuit", + "$c$": "speed of light (used in calculation context)" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Capacitors in a quasi-constant current loop" + }, + { + "id": 154, + "topic": "Magnetism", + "question": "A uniformly magnetized sphere rotates uniformly around an axis parallel to the magnetization direction. Determine the unipolar induced electromotive force between one pole of the uniformly magnetized sphere and the equator.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathbf{B}$": "magnetic field vector", + "$\\mathbf{v}$": "velocity vector", + "$\\mathbf{H}$": "magnetic field strength vector", + "$\\sigma$": "electrical conductivity", + "$c$": "speed of light", + "$\\mathcal{E}$": "electromotive force (EMF) along a path", + "$\\mathbf{r}$": "position vector", + "$\\boldsymbol{\\Omega}$": "angular velocity vector", + "$d\\mathbf{l}$": "infinitesimal line element", + "$B_{0}$": "magnetic induction inside the sphere", + "$a$": "radius of the sphere", + "$H$": "magnetic field strength", + "$M$": "magnetization", + "$\\mathscr{M}$": "total magnetic moment of the sphere", + "$\\mathscr{E}$": "induced electromotive force" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Movement of a conductor in a magnetic field" + }, + { + "id": 155, + "topic": "Superconductivity", + "question": "Determine the current generated inside the superconducting ring when its uniform rotation stops.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$J$": "current density", + "$m$": "mass", + "$c$": "speed of light", + "$e$": "elementary charge", + "$L$": "self-inductance", + "$\\Phi_e$": "external magnetic flux", + "$\\Phi_0$": "constant reference magnetic flux", + "$\\Omega$": "angular velocity", + "$b$": "radius of the superconducting ring", + "$a$": "radius of the wire cross-section" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "The excitation of current by acceleration" + }, + { + "id": 156, + "topic": "Magnetism", + "question": "Try to determine the absorption coefficient of Alfven waves in an incompressible fluid (assuming this coefficient is very small).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\gamma$": "absorption coefficient", + "$\\bar{Q}$": "average energy dissipated per unit volume per unit time", + "$\\bar{q}$": "average energy flux density of the wave", + "$\\rho$": "density", + "$T$": "temperature", + "$s$": "entropy per unit mass", + "$\\mathbf{v}$": "fluid velocity vector", + "$\\sigma'$": "stress tensor", + "$\\kappa$": "thermal conductivity", + "$c$": "speed of light", + "$\\sigma$": "conductivity", + "$\\mathbf{H}$": "magnetic field vector", + "$Q$": "dissipation", + "$\\eta$": "dynamic viscosity", + "$\\boldsymbol{v}$": "wave-induced velocity", + "$\\boldsymbol{h}$": "wave-induced magnetic field", + "$q_{x}$": "energy flux density in x-direction", + "$H_{x}$": "magnetic field component in x-direction", + "$\\nu$": "wave frequency", + "$u_{A}$": "Alfven speed", + "$\\omega$": "angular frequency of the wave" + }, + "chapter": "Magnetohydrodynamics", + "section": "Magnetohydrodynamics waves" + }, + { + "id": 157, + "topic": "Magnetism", + "question": "Try to find the law of rotational discontinuity expanding with time.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$x$": "spatial coordinate", + "$v_{x}$": "velocity component in the x-direction", + "$H_{x}$": "magnetic field component in the x-direction", + "$\\mathbf{v}_1$": "velocity vector on one side of the discontinuity", + "$\\mathbf{H}_1$": "magnetic field vector on one side of the discontinuity", + "$\\mathbf{v}_2$": "velocity vector on the other side of the discontinuity", + "$\\mathbf{H}_2$": "magnetic field vector on the other side of the discontinuity", + "$u_{x}$": "velocity component u in the x-direction", + "$\\boldsymbol{u}$": "some velocity field", + "$\\mathbf{u}'$": "perturbation of the velocity field u", + "$\\mathbf{v}'$": "perturbation of the velocity field v", + "$\\rho$": "density", + "$P$": "pressure", + "$P'$": "perturbation of the pressure", + "$\\mathbf{H}$": "magnetic field vector", + "$c$": "speed of light", + "$\\sigma$": "conductivity", + "$\\nu$": "kinematic viscosity", + "$\\boldsymbol{u}_{t}$": "transverse component of the velocity field u", + "$\\boldsymbol{v}_{t}$": "transverse component of the velocity field v", + "$t$": "time", + "$\\delta$": "width of the discontinuity" + }, + "chapter": "Magnetohydrodynamics", + "section": "Tangential and rotational discontinuities" + }, + { + "id": 158, + "topic": "Magnetism", + "question": "A dielectric sphere in vacuum rotates in a constant magnetic field $\\mathfrak{H}$, determine the electric field produced around the sphere.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathfrak{H}$": "constant magnetic field", + "$\\boldsymbol{H}^{(i)}$": "magnetic field inside the sphere", + "$\\varepsilon$": "permittivity of the sphere", + "$\\mu$": "permeability of the sphere", + "$c$": "speed of light in vacuum", + "$\\boldsymbol{\\Omega}$": "angular velocity of rotation", + "$a$": "radius of the sphere", + "$\\boldsymbol{n}$": "unit vector in the direction of position vector", + "$D_{i k}$": "electric quadrupole moment tensor of the sphere", + "$\\mathfrak{H}_{i}$": "component of constant magnetic field", + "$\\Omega_{k}$": "component of angular velocity" + }, + "chapter": "Electromagnetic wave equation", + "section": "Electrodynamics of Moving Dielectrics" + }, + { + "id": 159, + "topic": "Magnetism", + "question": "A magnetized dielectric sphere (with dielectric constant $\\varepsilon$) rotates uniformly in a vacuum around its own axis parallel to the magnetization direction (the $z$-axis) with angular velocity $\\Omega$. This rotation generates an electric field around the sphere. To describe this electric field, it is necessary to calculate the electric quadrupole moment. Determine the $D_{zz}$ component of the electric quadrupole tensor generated by this rotating sphere. The sphere has a radius $a$, total magnetic moment $\\mathscr{M}$, and the speed of light in vacuum $c$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon$": "dielectric constant", + "$z$": "axis parallel to the magnetization direction", + "$\\Omega$": "angular velocity", + "$a$": "radius of the sphere", + "$\\mathscr{M}$": "total magnetic moment of the sphere", + "$c$": "speed of light in vacuum", + "$B^{(i)}$": "magnetic field inside the sphere", + "$H^{(i)}$": "magnetic field intensity inside the sphere", + "$M$": "constant magnetization", + "$\\boldsymbol{D}$": "electric displacement field", + "$\\boldsymbol{E}$": "electric field", + "$\\boldsymbol{v}$": "velocity vector", + "$\\boldsymbol{B}$": "magnetic field vector", + "$\\boldsymbol{H}$": "magnetic field intensity", + "$\\varphi^{(e)}$": "electric potential outside the sphere", + "$\\varphi^{(i)}$": "electric potential inside the sphere", + "$D_{zz}$": "electric quadrupole moment component", + "$\\theta$": "angle between the normal and the z-axis" + }, + "chapter": "Electromagnetic wave equation", + "section": "Electrodynamics of Moving Dielectrics" + }, + { + "id": 160, + "topic": "Magnetism", + "question": "A magnetized metallic sphere (considered as the case of dielectric constant $\\varepsilon \\rightarrow \\infty$) rotates uniformly in vacuum around its own axis parallel to the magnetization direction (the $z$-axis) with an angular velocity $\\Omega$. This rotation will generate an electric field around the sphere. Determine the $D_{zz}$ component of the electric quadrupole moment tensor produced by the metallic sphere to describe its external electric field. The sphere radius is $a$, the total magnetic moment is $\\mathscr{M}$, and the speed of light in vacuum is $c$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon$": "dielectric constant", + "$z$": "axis parallel to magnetization direction", + "$\\Omega$": "angular velocity", + "$a$": "sphere radius", + "$\\mathscr{M}$": "total magnetic moment", + "$c$": "speed of light in vacuum", + "$\\mathbf{M}$": "magnetization", + "$B^{(i)}$": "magnetic field inside the sphere", + "$H^{(i)}$": "magnetic intensity inside the sphere", + "$\\mathbf{D}$": "electric displacement vector", + "$\\mathbf{E}$": "electric field", + "$\\mathbf{v}$": "velocity in the rotating frame", + "$D_{zz}$": "component of the electric quadrupole moment tensor", + "$\\varphi^{(e)}$": "electric field potential outside the sphere", + "$\\varphi^{(i)}$": "electric field potential inside the sphere", + "$\\theta$": "angle between normal and direction of $z$-axis", + "$D_{xx}$": "component of the electric quadrupole moment tensor", + "$D_{yy}$": "component of the electric quadrupole moment tensor" + }, + "chapter": "Electromagnetic wave equation", + "section": "Electrodynamics of Moving Dielectrics" + }, + { + "id": 161, + "topic": "Magnetism", + "question": "Try to find the dispersion relation of magnetostatic oscillations in an unbounded medium.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mu_{ik}$": "permeability tensor component", + "$\\omega$": "angular frequency", + "$\\psi$": "scalar field function related to magnetostatic oscillations", + "$x_i$": "Cartesian coordinate (i-component)", + "$x_k$": "Cartesian coordinate (k-component)", + "$\\mu_{xx}$": "permeability tensor component (xx)", + "$\\mu_{yy}$": "permeability tensor component (yy)", + "$\\mu_{zz}$": "permeability tensor component (zz)", + "$\\mu_{xy}$": "permeability tensor component (xy)", + "$\\mu_{yx}$": "permeability tensor component (yx)", + "$\\omega_M$": "characteristic angular frequency related to magnetization", + "$\\omega_H$": "angular frequency related to external magnetic field", + "$\\beta$": "parameter related to propagation in the medium", + "$\\mu(\\omega)$": "frequency-dependent permeability", + "$\\theta$": "angle between wave vector and easy magnetization axis", + "$\\boldsymbol{k}$": "wave vector", + "$\\boldsymbol{r}$": "position vector", + "$\\gamma$": "gyromagnetic ratio", + "$M$": "magnetization" + }, + "chapter": "Electromagnetic wave equation", + "section": "Dispersion of magnetic permeability" + }, + { + "id": 162, + "topic": "Magnetism", + "question": "The surface of an infinite parallel plate is perpendicular to the easy magnetization axis, and an external magnetic field $\\mathfrak{H}$ is applied along this axis direction. Determine the non-uniform resonance frequency within this plate.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathfrak{H}$": "external magnetic field", + "$\\psi^{(i)}$": "internal potential field", + "$\\psi^{(e)}$": "external potential field", + "$\\varphi^{(i)}$": "internal potential", + "$\\varphi^{(e)}$": "external potential", + "$z$": "coordinate axis perpendicular to the plate", + "$L$": "half the thickness of the plate", + "$A$": "amplitude constant for the internal potential", + "$B$": "amplitude constant for the external potential", + "$k_{z}$": "wave vector component in the z direction", + "$k_{x}$": "wave vector component in the x direction", + "$\\mu$": "relative permeability", + "$n^{(z)}$": "demagnetizing coefficient in the z direction", + "$M$": "magnetization", + "$\\beta$": "scaling factor for the magnetization", + "$\\omega_M$": "magnetization-related angular frequency", + "$\\omega_H$": "field-related angular frequency", + "$\\mu_{xx}$": "permeability component xx", + "$\\mu_{yy}$": "permeability component yy", + "$\\mu_{zz}$": "permeability component zz", + "$\\mu_{xy}$": "permeability component xy", + "$\\mu_{yx}$": "permeability component yx", + "$\\omega$": "vibration frequency", + "$\\gamma$": "gyromagnetic ratio", + "$\\theta$": "angle between wave vector and z axis", + "$\\boldsymbol{k}$": "wave vector" + }, + "chapter": "Electromagnetic wave equation", + "section": "Dispersion of magnetic permeability" + }, + { + "id": 163, + "topic": "Magnetism", + "question": "Calculate the reflection coefficient when light is almost grazing the surface of a material with $\\varepsilon$ close to 1 from a vacuum.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$R_{\\perp}$": "perpendicular reflection coefficient", + "$R_{\\parallel}$": "parallel reflection coefficient", + "$\\theta_0$": "angle of incidence", + "$\\theta_2$": "angle of refraction", + "$\\varphi_{0}$": "incidence angle offset from grazing", + "$\\varepsilon$": "relative permittivity" + }, + "chapter": "Propagation of electromagnetic waves", + "section": "Reflection and refraction of waves" + }, + { + "id": 164, + "topic": "Magnetism", + "question": "Find the reflection coefficient $R_{\\perp}$ when a wave is incident from vacuum onto the surface of a medium where both $\\varepsilon$ and $\\mu$ are different from 1, with the electric field vector perpendicular to the plane of incidence.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$R_{\\perp}$": "reflection coefficient for electric field perpendicular to the plane of incidence", + "$\\varepsilon$": "permittivity of the medium", + "$\\mu$": "permeability of the medium", + "$\\theta_{0}$": "angle of incidence" + }, + "chapter": "Propagation of electromagnetic waves", + "section": "Reflection and refraction of waves" + }, + { + "id": 165, + "topic": "Magnetism", + "question": "Find the reflection coefficient $R_{\\|}$ when a wave is incident on the surface of a medium with both $\\varepsilon$ and $\\mu$ different from 1, and when the electric field vector is parallel to the plane of incidence.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$R_{\\|}$": "reflection coefficient for parallel polarization", + "$\\varepsilon$": "permittivity", + "$\\mu$": "permeability", + "$\\theta_{0}$": "angle of incidence" + }, + "chapter": "Propagation of electromagnetic waves", + "section": "Reflection and refraction of waves" + }, + { + "id": 166, + "topic": "Magnetism", + "question": "For metals with impedance determined by formula \\begin{align*}\n\\zeta = (1 - i)\\sqrt{\\frac{\\omega\\mu}{8\\pi\\sigma}}\n\\end{align*} (a special case with a flat surface having low impedance), and assuming its permeability $\\mu=1$, try to determine the ratio of its thermal radiation intensity to the absolute blackbody surface radiation intensity ($I/I_0$). The ratio should be expressed in terms of the angular frequency of radiation $\\omega$ and the conductivity of the metal $\\sigma$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$I$": "thermal radiation intensity from the flat surface", + "$I_{0}$": "absolute blackbody surface radiation intensity", + "$\\zeta$": "impedance", + "$\\mu$": "permeability", + "$\\omega$": "angular frequency of radiation", + "$\\sigma$": "conductivity of the metal" + }, + "chapter": "Propagation of electromagnetic waves", + "section": "Surface impedance of metal" + }, + { + "id": 167, + "topic": "Magnetism", + "question": "Determine the dependence of the radiation intensity of a dipole emitter immersed in a homogeneous isotropic medium on the medium's permittivity $\\varepsilon$ and magnetic permeability $\\mu$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon$": "permittivity", + "$\\mu$": "magnetic permeability", + "$I$": "radiation intensity", + "$I_{0}$": "radiation intensity in vacuum", + "$\\boldsymbol{E}$": "electric field", + "$\\boldsymbol{H}$": "magnetic field", + "$\\omega$": "angular frequency", + "$c$": "speed of light in vacuum", + "$\\mathbf{B}$": "magnetic flux density", + "$\\mathbf{D}$": "electric displacement field", + "$\\mathbf{j}_{\\text{ex}}$": "external current density", + "$\\boldsymbol{A}^{\\prime}$": "vector potential in medium", + "$R_{0}$": "distance from the source", + "$\\boldsymbol{k}^{\\prime}$": "wave vector in medium", + "$\\boldsymbol{H}_{0}$": "magnetic field in vacuum", + "$\\boldsymbol{E}_{0}$": "electric field in vacuum" + }, + "chapter": "Propagation of electromagnetic waves", + "section": "Reciprocity principle" + }, + { + "id": 168, + "topic": "Magnetism", + "question": "For the E wave in a circular waveguide with a radius of $a$, provide the expression for its attenuation coefficient $\\alpha$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the circular waveguide", + "$\\alpha$": "attenuation coefficient", + "$\\omega$": "angular frequency", + "$\\zeta^{\\prime}$": "characteristic impedance of the medium", + "$c$": "speed of light in vacuum", + "$k_z$": "propagation constant along the z-axis", + "$\\kappa$": "wavenumber in the medium", + "$E_z$": "electric field component in the z direction", + "$H_z$": "magnetic field component in the z direction" + }, + "chapter": "Propagation of electromagnetic waves", + "section": "Propagation of electromagnetic waves in waveguides" + }, + { + "id": 169, + "topic": "Magnetism", + "question": "Provide the expression for the attenuation coefficient $\\alpha$ of H modes in a circular (radius $a$) cross-section waveguide.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\alpha$": "attenuation coefficient", + "$a$": "radius of the circular cross-section waveguide", + "$\\omega$": "angular frequency", + "$\\zeta'$": "some characteristic impedance factor", + "$\\kappa$": "propagation constant transversal to the direction of energy flow", + "$k_z$": "propagation constant in the z-direction", + "$c$": "speed of light in vacuum", + "$E_z$": "z-component of the electric field", + "$H_z$": "z-component of the magnetic field", + "$n$": "mode number", + "$\\varkappa$": "another propagation constant or related factor" + }, + "chapter": "Propagation of electromagnetic waves", + "section": "Propagation of electromagnetic waves in waveguides" + }, + { + "id": 170, + "topic": "Magnetism", + "question": "Linearly polarized light is scattered by small particles with random orientations, where the particles' polarizability tensor has three distinct principal values. It is known that the scalar constants describing the average of the electric dipole moment tensor are $A = \\frac{1}{15}(2 \\alpha_{i i} \\alpha_{k k}^{*}-\\alpha_{i k} \\alpha_{i k}^{*})$ and $B = \\frac{1}{30}(3 \\alpha_{i k} \\alpha_{i k}^{*}-\\alpha_{i i} \\alpha_{k k}^{*})$. Determine the expression for the depolarization ratio $\\frac{I_y}{I_x}$ of the scattered light, where $\\theta$ is the angle between the incident light electric field $\\boldsymbol{E}$ and the scattering direction $\\boldsymbol{n}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$A$": "scalar constant related to the electric dipole moment tensor", + "$B$": "scalar constant related to the electric dipole moment tensor", + "$\\alpha_{i i}$": "principal value of polarizability tensor", + "$\\alpha_{k k}^{*}$": "complex conjugate of principal value of polarizability tensor", + "$\\alpha_{i k}$": "element of polarizability tensor", + "$\\alpha_{i k}^{*}$": "complex conjugate of element of polarizability tensor", + "$I_y$": "intensity of scattered light in the y-direction", + "$I_x$": "intensity of scattered light in the x-direction", + "$\\theta$": "angle between the incident light electric field and the scattering direction", + "$\\boldsymbol{E}$": "incident light electric field", + "$\\boldsymbol{n}$": "scattering direction", + "$\\omega$": "angular frequency of light", + "$c$": "speed of light", + "$R$": "distance from scattering particle", + "$\\mathscr{P}$": "polarization of the scattered light", + "$\\delta_{i l}$": "Kronecker delta function" + }, + "chapter": "Propagation of electromagnetic waves", + "section": "Scattering of electromagnetic waves by small particles" + }, + { + "id": 171, + "topic": "Magnetism", + "question": "Try to express the components of the ray vector $s$ in terms of the components of $\\boldsymbol{n}$ within the principal dielectric axes.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$s$": "components of the ray vector", + "$\\boldsymbol{n}$": "unit vector in principal dielectric axes", + "$n_x$": "x-component of unit vector", + "$n_y$": "y-component of unit vector", + "$n_z$": "z-component of unit vector", + "$s_x$": "x-component of the ray vector", + "$s_y$": "y-component of the ray vector", + "$s_z$": "z-component of the ray vector", + "$\\varepsilon^{(x)}$": "dielectric permittivity along the x-axis", + "$\\varepsilon^{(y)}$": "dielectric permittivity along the y-axis", + "$\\varepsilon^{(z)}$": "dielectric permittivity along the z-axis" + }, + "chapter": "Electromagnetic waves in anisotropic media", + "section": "Optical properties of uniaxial crystals" + }, + { + "id": 172, + "topic": "Magnetism", + "question": "Find the polarization of the reflected light when linearly polarized light is perpendicularly incident from vacuum onto the surface of an anisotropic object induced by a magnetic field.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\boldsymbol{H}$": "magnetic field vector", + "$E$": "electric field", + "$E_{z}$": "z-component of the electric field", + "$H_{x}$": "x-component of the magnetic field", + "$E_{y}$": "y-component of the electric field", + "$H_{y}$": "y-component of the magnetic field", + "$E_{x}$": "x-component of the electric field", + "$n$": "refractive index", + "$\\boldsymbol{E}_{1}$": "electric field vector of the reflected wave", + "$\\boldsymbol{E}_{0}$": "electric field vector of the incident wave", + "$\\boldsymbol{E}_{0}^{+}$": "component of the incident electric field vector with positive circular polarization", + "$\\boldsymbol{E}_{0}^{-}$": "component of the incident electric field vector with negative circular polarization", + "$E_{0 x}^{+}$": "x-component of the positively polarized incident electric field", + "$E_{0 y}^{+}$": "y-component of the positively polarized incident electric field", + "$E_{0 x}^{-}$": "x-component of the negatively polarized incident electric field", + "$E_{0 y}^{-}$": "y-component of the negatively polarized incident electric field", + "$n_{+}$": "refractive index for positive polarization", + "$n_{-}$": "refractive index for negative polarization", + "$n_{0}$": "initial refractive index", + "$G_z$": "anisotropic parameter affecting the refractive indices", + "$g_z$": "component of the anisotropic parameter", + "$E_{1 x}$": "x-component of the reflected electric field", + "$E_{1 y}$": "y-component of the reflected electric field", + "$g$": "anisotropic parameter vector", + "$\\theta$": "angle between the incident direction and the vector $\\boldsymbol{g}$" + }, + "chapter": "Electromagnetic waves in anisotropic media", + "section": "Magneto-optical effect" + }, + { + "id": 173, + "topic": "Superconductivity", + "question": "Attempt to find the limiting law of the dependence of the surface tension coefficient $\\alpha$ of liquid nitrogen near absolute zero on temperature", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\alpha$": "surface tension coefficient", + "$\\alpha_{0}$": "surface tension coefficient at zero temperature", + "$T$": "temperature", + "$\\omega_{\\alpha}$": "frequency related to surface vibrations", + "$\\rho$": "liquid density", + "$k$": "wave vector of vibrations", + "$\\hbar$": "reduced Planck's constant", + "$\\omega$": "vibration frequency" + }, + "chapter": "Superfluidity", + "section": "Superfluidity" + }, + { + "id": 174, + "topic": "Superconductivity", + "question": "Try to find the dispersion relation of impurity particles in a moving superfluid $\\varepsilon_{\\text {imp}}(p)$, given that in a stationary fluid the dispersion relation is $\\varepsilon_{\\text {imp}}^{(0)}(p)$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon_{\\text{imp}}(p)$": "dispersion relation of impurity particles in a moving superfluid", + "$\\varepsilon_{\\text{imp}}^{(0)}(p)$": "dispersion relation of impurity particles in a stationary fluid", + "$p$": "momentum", + "$p_{0}$": "initial momentum of the impurity atom", + "$m$": "mass of the impurity atom", + "$v$": "velocity of the moving superfluid", + "$M$": "mass associated with the fluid's motion", + "$m_{\\text{eff}}^{*}$": "effective mass of impurity particles in the moving fluid" + }, + "chapter": "Superfluidity", + "section": "Superfluidity" + }, + { + "id": 175, + "topic": "Superconductivity", + "question": "Try to find the dispersion relation of small oscillations of a rectilinear vortex line (W. Thomson, 1880).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$z$": "coordinate axis along vortex line", + "$r$": "displacement vector of points on vortex line", + "$t$": "time", + "$k$": "wave number", + "$\\omega$": "angular frequency", + "$\\kappa$": "circulation strength", + "$R_{0}$": "radius of curvature", + "$\\lambda$": "wavelength", + "$a$": "core radius of vortex line", + "$b$": "binormal vector magnitude", + "$n_{z}$": "unit vector along z axis", + "$n_{\\varepsilon}$": "perturbation direction unit vector" + }, + "chapter": "Superfluidity", + "section": "Quantum vortex" + }, + { + "id": 176, + "topic": "Superconductivity", + "question": "A neutron with an initial velocity $v$ scatters within a liquid. Determine the conditions under which an excitation with momentum $p$ and energy $\\varepsilon(p)$ can be produced.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$v$": "initial velocity of the neutron", + "$p$": "momentum of the excitation", + "$\\varepsilon(p)$": "energy of the excitation with momentum p", + "$m$": "mass of the neutron", + "$P$": "initial momentum of the neutron", + "$V$": "velocity component relevant to the condition being determined", + "$\\theta$": "angle between the initial momentum P and the excitation momentum p", + "$p_{\\mathrm{c}}$": "critical momentum value", + "$q$": "momentum within the interval related to phonon radiation" + }, + "chapter": "Superfluidity", + "section": "Fission of quasiparticles" + }, + { + "id": 177, + "topic": "Superconductivity", + "question": "Try to find the magnetic moment of a superconducting sphere of radius $R \\ll \\delta$ in a magnetic field under the London situation.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$R$": "radius of the superconducting sphere", + "$\\delta$": "London penetration depth", + "$\\mathfrak{G}$": "external magnetic field", + "$n_{s}$": "density of superconducting electrons", + "$e$": "elementary charge", + "$m$": "electron mass", + "$c$": "speed of light", + "$M$": "magnetic moment" + }, + "chapter": "Superconductivity", + "section": "Superconductivity current" + }, + { + "id": 178, + "topic": "Superconductivity", + "question": "For superconductors with parameter $\\kappa \\ll 1$, find the first-order correction to the penetration depth in a weak magnetic field.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\kappa$": "parameter for superconductors", + "$\\psi$": "superconducting wavefunction", + "$A$": "magnetic vector potential", + "$B$": "magnetic field", + "$\\mathfrak{S}$": "boundary value of magnetic field derivative", + "$H_{\\mathrm{e}}$": "external magnetic field", + "$\\delta_{\\text{eff}}$": "effective penetration depth", + "$\\delta$": "penetration depth at zero correction", + "$\\psi_{1}$": "first-order correction to superconducting wavefunction", + "$A_{1}$": "first-order correction to magnetic vector potential" + }, + "chapter": "Superconductivity", + "section": "Surface tension at the boundary between the superconducting phase and the normal phase" + }, + { + "id": 179, + "topic": "Superconductivity", + "question": "Attempt to find the critical field of a superconducting small sphere with radius $R \\ll \\delta$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$R$": "radius of the small sphere", + "$\\delta$": "characteristic penetration depth", + "$n$": "normal phase", + "$s$": "superconducting phase", + "$e$": "elementary charge", + "$A$": "vector potential", + "$c$": "speed of light", + "$m$": "mass of the electron", + "$\\hbar$": "reduced Planck's constant", + "$\\boldsymbol{A}$": "vector potential in bold", + "$\\mathfrak{S}$": "magnitude of vector potential related to the field", + "$E_{0}$": "average energy of perturbation", + "$\\mathfrak{G}$": "magnitude related to the uniform field", + "$a$": "parameter related to critical field condition", + "$H_{\\mathrm{c}}^{\\text {(sphere) }}$": "critical field for a superconducting sphere", + "$H_{\\mathrm{o}}$": "reference field strength" + }, + "chapter": "Superconductivity", + "section": "Two types of superconductors" + }, + { + "id": 180, + "topic": "Superconductivity", + "question": "Calculate the interaction energy of two vortices separated by $d \\gg \\xi$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$d$": "distance between two vortices", + "$\\xi$": "coherence length", + "$F_{\\text{vertex}}$": "free energy of the vortex system", + "$\\mathbf{B}$": "magnetic field", + "$\\delta$": "penetration depth", + "$\\phi_0$": "magnetic flux quantum", + "$\\varepsilon_{12}$": "interaction energy between two vortices", + "$B_{1}$": "magnetic field of first vortex", + "$B_{2}$": "magnetic field of second vortex", + "$r_{0}$": "radius of cylindrical surface around the vortex", + "$f_{1}$": "surface surrounding the first vortex", + "$f_{2}$": "surface surrounding the second vortex" + }, + "chapter": "Superconductivity", + "section": "Mixed structure" + }, + { + "id": 181, + "topic": "Superconductivity", + "question": "A thin film (thickness $d \\ll \\xi(T)$) is placed in a weak magnetic field perpendicular to its plane. Find the magnetic moment of the film when the temperature $T$ $>T_{\\mathrm{c}}, T-T_{\\mathrm{c}} \\ll T_{\\mathrm{c}}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$d$": "thickness of the film", + "$\\xi(T)$": "temperature-dependent coherence length", + "$T$": "temperature", + "$T_{\\mathrm{c}}$": "critical temperature", + "$p_{z}$": "quantized momentum component perpendicular to the film", + "$E$": "energy", + "$e$": "elementary charge", + "$\\hbar$": "reduced Planck's constant", + "$\\mathfrak{H}$": "magnetic field strength", + "$m$": "mass of charge carriers", + "$c$": "speed of light", + "$a$": "energy related constant", + "$5 S$": "multiplicative factor involving the area", + "$S$": "area of the film", + "$\\Delta F$": "change in free energy", + "$M$": "magnetic moment of the film", + "$\\mathscr{Q}$": "generalized coordinate for the free energy", + "$\\mathscr{G}$": "given function related to the magnetic moment expression", + "$\\alpha$": "positive constant related to the temperature dependence" + }, + "chapter": "Superconductivity", + "section": "Diamagnetic susceptibility above the phase transition point" + }, + { + "id": 182, + "topic": "Superconductivity", + "question": "Under the conditions of the previous question, determine the magnetic moment of a small sphere with radius $R \\ll \\xi(T)$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$R$": "radius of the small sphere", + "$\\xi(T)$": "temperature-dependent parameter", + "$m$": "mass", + "$e$": "elementary charge", + "$\\hbar$": "reduced Planck's constant", + "$c$": "speed of light in vacuum", + "$\\mathbf{A}$": "vector potential", + "$E$": "energy eigenvalue", + "$E_{0}$": "minimum energy eigenvalue", + "$T$": "temperature", + "$T_{\\mathrm{c}}$": "critical temperature", + "$a$": "constant parameter", + "$\\mathfrak{S}$": "function related to magnetic moment calculations", + "$\\mathscr{S}_{g}$": "scalar function specific to the material or geometry", + "$\\alpha$": "coefficient function of temperature" + }, + "chapter": "Superconductivity", + "section": "Diamagnetic susceptibility above the phase transition point" + }, + { + "id": 183, + "topic": "Magnetism", + "question": "Find the energy spectrum of spin wave quanta in an uniaxial ferromagnet of the 'easy magnetization plane' type $(K<0)$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$M_{0}$": "equilibrium magnetization", + "$M$": "magnetization", + "$\\varepsilon$": "energy of spin wave quanta", + "$\\beta$": "constant (beta)", + "$\\alpha$": "constant (alpha)", + "$k$": "wave number", + "$K$": "anisotropy constant", + "$n_{x}$": "unit vector along x-axis", + "$n_{y}$": "unit vector along y-axis", + "$m$": "vector in yz plane", + "$\\theta$": "polar angle", + "$\\varphi$": "azimuthal angle" + }, + "chapter": "magnetic", + "section": "Spin wave quantum spectroscopy in ferromagnets" + }, + { + "id": 184, + "topic": "Magnetism", + "question": "Calculate the spin wave quantum part of thermodynamic quantities (energy) at temperature $T \\ll \\varepsilon(0)$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$T$": "temperature", + "$\\varepsilon(0)$": "energy at zero wave vector", + "$k$": "quasi-momentum", + "$\\theta$": "angle related to propagation direction", + "$\\varepsilon(k)$": "energy at wave vector k", + "$\\beta$": "constant related to the material", + "$K$": "anisotropy constant", + "$M$": "magnetization", + "$A$": "coefficient related to crystal type", + "$\\alpha$": "parameter related to cubic crystals", + "$\\alpha_{2}$": "parameter related to uniaxial crystals", + "$V$": "volume", + "$E_{\\operatorname{mag}}$": "magnetic energy", + "$M_{\\operatorname{mag}}$": "magnetic magnetization", + "$C_{\\operatorname{mag}}$": "magnetic specific heat" + }, + "chapter": "magnetic", + "section": "Spin wave quantum thermodynamic quantities in ferromagnets" + }, + { + "id": 185, + "topic": "Magnetism", + "question": "In the exchange approximation, determine the spatial correlation function of magnetization fluctuations at distances $r \\gg a$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\hat{m}_{x}$": "magnetization operator in the x-direction", + "$\\hat{m}_{y}$": "magnetization operator in the y-direction", + "$\\beta$": "inverse temperature or thermodynamic beta", + "$M$": "magnetization", + "$\\mathbf{r}$": "position vector", + "$k$": "index representing x or y direction", + "$n_{k}$": "occupation number of spin-wave quantum states", + "$\\varphi_{i k}$": "spatial correlation function of magnetization fluctuations", + "$T$": "temperature", + "$\\alpha$": "constant related to cubic ferromagnet", + "$i$": "index representing x or y direction" + }, + "chapter": "magnetic", + "section": "Spin wave quantum thermodynamic quantities in ferromagnets" + }, + { + "id": 186, + "topic": "Magnetism", + "question": "Given $S \\gg 1$, try to find the correction terms related to the interaction of heat capacity of a cubic lattice. In this lattice, only the exchange integral between a pair of neighboring atoms (along the cubic axes) is non-zero.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$S$": "total spin of the system", + "$J_{mn}$": "exchange integral between atoms at positions m and n", + "$k$": "wave vector in reciprocal space", + "$\\mathbf{r}_m$": "position vector of atom m", + "$\\mathbf{r}_n$": "position vector of atom n", + "$J_{0}$": "exchange integral between a pair of nearest neighbor atoms", + "$a$": "lattice constant", + "$V$": "volume of the system", + "$k_{1x}$": "x-component of wave vector k1", + "$k_{2x}$": "x-component of wave vector k2", + "$k_{1y}$": "y-component of wave vector k1", + "$k_{2y}$": "y-component of wave vector k2", + "$k_{1z}$": "z-component of wave vector k1", + "$k_{2z}$": "z-component of wave vector k2", + "$U(k_{1}, k_{2} ; k_{1}, k_{2})$": "energy correction term involving wave vectors k1 and k2", + "$\\varepsilon(k)$": "energy of magnon quanta with wave vector k", + "$\\beta$": "inverse temperature (1/kT)", + "$\\mathfrak{H}$": "external magnetic field", + "$M_{\\text {int }}$": "interaction part of magnetization", + "$M$": "magnetization", + "$C_{\\text{int}}$": "interaction part of heat capacity", + "$N$": "number of lattice sites or atoms", + "$T$": "temperature" + }, + "chapter": "magnetic", + "section": "Interaction of spin wave quanta" + }, + { + "id": 187, + "topic": "Magnetism", + "question": "Try to determine the interaction law between an atom and a metal wall at 'large' distances.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon_{10}$": "permittivity of medium 1", + "$\\varepsilon_{20}$": "permittivity of medium 2", + "$\\hbar$": "reduced Planck's constant", + "$c$": "speed of light", + "$l$": "separation distance between atom and wall", + "$L$": "distance between the atom and the wall", + "$a$": "interaction constant", + "$n$": "number density of atoms", + "$\\alpha_{2}$": "polarizability of medium 2" + }, + "chapter": "Electromagnetic Fluctuations", + "section": "Molecular forces and limiting cases of solid-solid interactions" + }, + { + "id": 188, + "topic": "Strongly Correlated Systems", + "question": "Determine the correlation function $\\nu(r)$ of a Bose liquid at temperature $T \\ll T_{\\lambda}$, at distances $r \\gtrsim \\hbar u / T$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\nu(r)$": "correlation function of the Bose liquid at distance r", + "$r$": "distance", + "$T$": "temperature", + "$T_{\\lambda}$": "lambda transition temperature for Bose liquid", + "$\\hbar$": "reduced Planck's constant", + "$u$": "speed of sound in the medium", + "$k$": "wave number", + "$m$": "mass of particles in the liquid", + "$a$": "interatomic distance", + "$\\omega$": "angular frequency" + }, + "chapter": "Fluid Dynamics Fluctuations", + "section": "Summation Rule for Shape Factors" + }, + { + "id": 189, + "topic": "Strongly Correlated Systems", + "question": "In a Bose superfluid, the condensate wave function exhibits fluctuations. Try to find the asymptotic form of this fluctuation correlation function at large distances.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\Phi$": "condensate wave function", + "$\\Omega$": "total thermodynamic potential", + "$V$": "volume", + "$T$": "temperature", + "$\\mu$": "chemical potential", + "$\\rho_{s}$": "superfluid density", + "$v_{s}$": "superfluid velocity", + "$m$": "mass", + "$\\hbar$": "reduced Planck's constant", + "$k$": "wave vector", + "$\\delta \\Phi_{k}$": "fluctuation amplitude in Fourier space", + "$G(r)$": "fluctuation correlation function", + "$n_{0}$": "condensate density", + "$r$": "distance", + "$r_{\\mathrm{c}}$": "correlation radius", + "$\\sigma$": "critical exponent", + "$\\zeta$": "appropriate critical exponent", + "$\\beta$": "critical exponent related to order parameter", + "$\\nu$": "critical exponent related to correlation length", + "$\\xi$": "exponent in scaling relation", + "$T_{\\lambda}$": "lambda point temperature" + }, + "chapter": "Fluid Dynamics Fluctuations", + "section": "Summation Rule for Shape Factors" + }, + { + "id": 190, + "topic": "Semiconductors", + "question": "If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is\n\\begin{align*}\nE(\\mathbf{k}) = E_0 - A - \\delta J(\\cos\\pi a k_x \\cos\\pi a k_y \\cos\\pi a k_z)，\n\\end{align*}\nwhere $J$ is the overlap integral. Calculate the bandwidth of the body-centered cubic lattice;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$k_{x}$": "wave vector component in x-direction", + "$k_{y}$": "wave vector component in y-direction", + "$k_{z}$": "wave vector component in z-direction", + "$E_{\\min }$": "minimum energy in the band", + "$E_{0}$": "reference energy level", + "$A$": "energy offset", + "$J$": "exchange interaction energy", + "$E_{\\max }$": "maximum energy in the band", + "$\\Delta E$": "bandwidth" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 191, + "topic": "Semiconductors", + "question": "If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is\n\\begin{align*}\nE(\\mathbf{k}) = E_0 - A - \\delta J(\\cos\\pi a k_x \\cos\\pi a k_y \\cos\\pi a k_z)，\n\\end{align*}\nwhere $J$ is the overlap integral. Calculate the effective mass of electrons at the bottom of the band;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$k$": "wave vector magnitude", + "$E(\\boldsymbol{k})$": "energy as a function of wave vector", + "$E_{0}$": "baseline energy level", + "$A$": "some constant", + "$J$": "exchange energy", + "$a$": "lattice constant", + "$k_{x}$": "x-component of wave vector", + "$k_{y}$": "y-component of wave vector", + "$k_{z}$": "z-component of wave vector", + "$E_{\\min}$": "minimum energy", + "$h$": "Planck's constant", + "$m_{\\mathrm{b}}^{*}$": "effective mass of electrons at the bottom of the band", + "$E_{\\max}$": "maximum energy", + "$\\delta k_{x}$": "deviation in x-component of wave vector", + "$\\delta k_{y}$": "deviation in y-component of wave vector", + "$\\delta k_{z}$": "deviation in z-component of wave vector", + "$m_{\\mathrm{t}}^{*}$": "effective mass of electrons at the top of the band" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 192, + "topic": "Semiconductors", + "question": "If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is\n\\begin{align*}\nE(\\mathbf{k}) = E_0 - A - \\delta J(\\cos\\pi a k_x \\cos\\pi a k_y \\cos\\pi a k_z)，\n\\end{align*}\nwhere $J$ is the overlap integral. Calculate the effective mass of electrons at the top of the band;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$k$": "wave vector", + "$E$": "energy", + "$E_{0}$": "initial energy offset", + "$A$": "constant energy term", + "$J$": "exchange interaction term", + "$a$": "lattice constant", + "$k_{x}$": "x-component of wave vector", + "$k_{y}$": "y-component of wave vector", + "$k_{z}$": "z-component of wave vector", + "$E_{\\min}$": "minimum energy level", + "$h$": "Planck's constant", + "$m_{\\mathrm{b}}^{*}$": "effective mass at the bottom of the band", + "$E_{\\max}$": "maximum energy level", + "$m_{\\mathrm{t}}^{*}$": "effective mass at the top of the band", + "$\\delta k_{x}$": "small deviation in x-component of wave vector", + "$\\delta k_{y}$": "small deviation in y-component of wave vector", + "$\\delta k_{z}$": "small deviation in z-component of wave vector" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 193, + "topic": "Semiconductors", + "question": "By considering only the nearest-neighbor interactions and using the tight-binding method, the energy band of s-state electrons in a simple cubic crystal is derived as\n$$\nE(\\boldsymbol{k})=E_{0}-A-2 J(\\cos 2 \\pi a k_{x}+\\cos 2 \\pi a k_{y}+\\cos 2 \\pi a k_{z})\n$$ Find the bandwidth $(\\Delta E)$ ;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\Delta E$": "bandwidth", + "$E$": "energy", + "$\\boldsymbol{k}$": "wave vector", + "$E_{0}$": "energy level constant", + "$A$": "atomic energy correction", + "$\\boldsymbol{R}_{\\mathrm{s}}$": "position vector of the reference atom", + "$\\boldsymbol{R}_{\\mathrm{n}}$": "position vector of the nearest neighbors", + "$J_{\\mathrm{sn}}$": "overlap integral between s-state electrons", + "$a$": "lattice constant", + "$E_{\\min}$": "minimum energy", + "$E_{\\max}$": "maximum energy", + "$J$": "interaction energy constant", + "$k_{x}$": "x-component of wave vector", + "$k_{y}$": "y-component of wave vector", + "$k_{z}$": "z-component of wave vector" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 194, + "topic": "Semiconductors", + "question": "The energy $E$ near the top of the valence band of a semiconductor crystal can be expressed as: $E(k)=E_{\\text {max }}-10^{26} k^{2}(\\mathrm{erg})$. Now, remove an electron with the wave vector $k=10^{7} \\mathrm{i} / \\mathrm{cm}$, and find the speed of the hole left by this electron.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E$": "energy", + "$E_{\\text{max}}$": "maximum energy near top of valence band", + "$k$": "wave vector", + "$k_{x}$": "wave vector component in x-direction", + "$k_{y}$": "wave vector component in y-direction", + "$k_{z}$": "wave vector component in z-direction", + "$m_{\\mathrm{n}}^{*}$": "effective mass of electron", + "$m_{\\mathrm{p}}^{*}$": "effective mass of hole", + "$v_{x}$": "velocity component in x-direction", + "$v_{y}$": "velocity component in y-direction", + "$v_{z}$": "velocity component in z-direction", + "$h$": "Planck's constant" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 195, + "topic": "Semiconductors", + "question": "In a one-dimensional periodic potential, the wavefunctions of electrons take the following form:\n$\\psi_{k}(x)=\\sin \\frac{\\pi}{a} x$: \nTry to use Bloch's theorem to point out the wave vector $\\mathbf{k}$ values within the reduced Brillouin zone.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\psi_{k}$": "wavefunction of electrons with wave vector k", + "$k$": "wave vector", + "$a$": "lattice constant", + "$x$": "position" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 196, + "topic": "Semiconductors", + "question": "In a one-dimensional periodic potential, the wavefunctions of electrons take the following form:\n$\\psi_{k}(x)=i \\cos \\frac{3 \\pi}{a} x$ ；\nTry to use Bloch's theorem to point out the wave vector $\\mathbf{k}$ values within the reduced Brillouin zone.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\psi_{k}$": "wavefunction of electrons with wave vector $\\mathbf{k}$", + "$x$": "position variable", + "$a$": "lattice constant", + "$\\mathbf{k}$": "wave vector" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 197, + "topic": "Semiconductors", + "question": "The electron wave function moving in a one-dimensional periodic potential field has the following form: $\\psi_{k}(x)=\\sum_{l=-\\infty}^{\\infty}(-1)^{l} f(x-l a)$ . Here $a$ is the lattice constant of the one-dimensional lattice, $f(x)$ is a certain function, try using Bloch's theorem to indicate the values of the wave vector $\\boldsymbol{k}$ within the reduced Brillouin zone.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\psi_{k}$": "electron wave function", + "$x$": "position", + "$a$": "lattice constant", + "$f(x)$": "certain function", + "$\\boldsymbol{k}$": "wave vector", + "$k$": "wave vector" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 198, + "topic": "Semiconductors", + "question": "It is known that the electronic energy band of a one-dimensional crystal can be expressed as\n$$\nE(k)=\\frac{h^{2}}{m_{0} a^{2}}(\\frac{7}{8}-\\cos 2 \\pi k a+\\frac{1}{8} \\cos 6 \\pi k a)\n$$\n\nWhere $a$ is the lattice constant. Try to find: the width of the energy band;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E$": "energy", + "$k$": "wave number", + "$h$": "Planck's constant", + "$m_{0}$": "rest mass", + "$a$": "lattice constant", + "$\\Delta E$": "energy band width" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 199, + "topic": "Semiconductors", + "question": "It is known that the electron energy band of a one-dimensional crystal can be expressed as\n$$\nE(k)=\\frac{h^{2}}{m_{0} a^{2}}(\\frac{7}{8}-\\cos 2 \\pi k a+\\frac{1}{8} \\cos 6 \\pi k a)\n$$\n\nwhere $a$ is the lattice constant. Try to find the velocity of the electron at wave vector $k$;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$k$": "wave vector", + "$E$": "energy", + "$v$": "velocity", + "$h$": "Planck's constant", + "$m_{0}$": "rest mass of the electron", + "$a$": "lattice constant" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 200, + "topic": "Semiconductors", + "question": "It is known that the electronic energy band of a one-dimensional crystal can be expressed as\n$$\nE(k)=\\frac{h^{2}}{m_{0} a^{2}}(\\frac{7}{8}-\\cos 2 \\pi k a+\\frac{1}{8} \\cos 6 \\pi k a)\n$$\n\nwhere $a$ is the lattice constant. Find: Find the effective mass of electrons at the bottom of the band.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", + "$h$": "Planck's constant", + "$E$": "energy", + "$k$": "wave vector", + "$m$": "mass", + "$m_{0}$": "free electron rest mass" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 201, + "topic": "Semiconductors", + "question": "It is known that the electronic band of a one-dimensional crystal can be written as\n$$\nE(k)=\\frac{h^{2}}{m_{0} a^{2}}(\\frac{7}{8}-\\cos 2 \\pi k a+\\frac{1}{8} \\cos 6 \\pi k a)\n$$\n\nIn this equation, $a$ is the lattice constant. Try to find the effective mass of electrons at the top of the band.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", + "$h$": "Planck's constant", + "$E$": "energy", + "$k$": "wave vector", + "$m$": "mass", + "$m_{0}$": "electron rest mass" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 202, + "topic": "Semiconductors", + "question": "Given the lattice constant of a two-dimensional square lattice is $a$, if the electron energy can be expressed as\n$$\nE(k)=\\frac{h^{2}(k_{x}^{2}+k_{y}^{2})}{2 m_{\\mathrm{n}}^{*}}\n$$\n\nTry to find the density of states.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "lattice constant", + "$E$": "energy", + "$k_{x}$": "wave vector component in x-direction", + "$k_{y}$": "wave vector component in y-direction", + "$m_{\\mathrm{n}}^{*}$": "effective mass of electron", + "$h$": "Planck's constant", + "$S$": "area of the crystal" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 203, + "topic": "Semiconductors", + "question": "Calculate the number of quantum states per unit volume between the energy $E=E_{\\mathrm{c}}$ and $E=E_{\\mathrm{c}}+100(\\frac{h^{2}}{8 m_{\\mathrm{n}}^{*} L^{2}})$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E$": "energy", + "$E_{\\mathrm{c}}$": "conduction band minimum energy", + "$h$": "Planck's constant", + "$m_{\\mathrm{n}}^{*}$": "effective mass of an electron", + "$L$": "length or characteristic dimension", + "$g_{\\mathrm{c}}$": "density of states", + "$m_{\\mathrm{dn}}$": "some effective mass (context-dependent)" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 204, + "topic": "Semiconductors", + "question": "For two pieces of n-type silicon material, at a certain temperature $T$, the ratio of the electron densities of the first piece to the second piece is $n_{1} / n_{2}=\\mathrm{e}$ (e is the base of the natural logarithm). If the Fermi level of the first piece of material is $3 k_{0} T$ below the conduction band edge, find the position of the Fermi level in the second piece of material.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E_{\\mathrm{F} 1}$": "Fermi level of the first piece of material", + "$E_{\\mathrm{F} 2}$": "Fermi level of the second piece of material", + "$k_{0}$": "Boltzmann constant or a specific constant related to temperature", + "$T$": "temperature", + "$E_{\\mathrm{c}}$": "conduction band edge", + "$n_{1}$": "carrier concentration in the first piece of material", + "$n_{2}$": "carrier concentration in the second piece of material" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 205, + "topic": "Semiconductors", + "question": "For a p-type semiconductor, in the ionization region of impurities, the known relation is $\\frac{p_{0}(p_{0}+N_{\\mathrm{D}})}{N_{\\mathrm{A}}-N_{\\mathrm{D}}-p_{0}}=\\frac{N_{\\mathrm{v}}}{g} \\exp (-\\frac{E_{\\mathrm{A}}-E_{\\mathrm{v}}}{k_{0} T})$. When the condition $p_{0} \\ll N_{\\mathrm{D}}$ is satisfied, find the expression for hole density $p_{0}$. In the formula, $g$ is the spin degeneracy of the acceptor level.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$p_{0}$": "hole density", + "$N_{\\mathrm{D}}$": "donor density", + "$N_{\\mathrm{A}}$": "acceptor density", + "$N_{\\mathrm{v}}$": "effective density of states in the valence band", + "$g$": "spin degeneracy of the acceptor level", + "$E_{\\mathrm{A}}$": "energy level of the acceptor", + "$E_{\\mathrm{v}}$": "energy level at the valence band edge", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$E_{\\mathrm{F}}$": "Fermi energy" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 206, + "topic": "Semiconductors", + "question": "There is an n-type semiconductor, in addition to the donor concentration $N_{\\mathrm{D}}$, it also contains a small amount of acceptors, with a concentration of $N_{\\mathrm{A}}$. Find the expression for the electron concentration under weak ionization conditions.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$N_{\\mathrm{D}}$": "donor concentration", + "$N_{\\mathrm{A}}$": "acceptor concentration", + "$E_{\\mathrm{A}}$": "energy level of the acceptor", + "$n_{0}$": "electron concentration in the conduction band", + "$n_{\\mathrm{D}}^{+}$": "ionized donor concentration", + "$E_{\\mathrm{D}}$": "energy level of the donor", + "$E_{\\mathrm{F}}$": "Fermi level", + "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", + "$E_{\\mathrm{c}}$": "energy of the conduction band edge", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$\\Delta E_{\\mathrm{D}}$": "ionization energy of the donor" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 207, + "topic": "Semiconductors", + "question": "Please explain why at room temperature, for a certain semiconductor, the electron concentration $n=n_{i} \\sqrt{\\mu_{\\mathrm{p}} / \\mu_{\\mathrm{n}}}$ results in the minimum electrical conductivity $\\sigma$. In this equation, $n_{\\mathrm{i}}$ is the intrinsic carrier concentration, and $\\mu_{\\mathrm{p}}, ~ \\mu_{\\mathrm{n}}$ are the mobilities of holes and electrons respectively. Find the hole concentration under the above condition.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$n$": "electron concentration", + "$n_{i}$": "intrinsic carrier concentration", + "$\\mu_{\\mathrm{p}}$": "mobility of holes", + "$\\mu_{\\mathrm{n}}$": "mobility of electrons", + "$\\sigma$": "electrical conductivity", + "$n_{0}$": "electron concentration when conductivity is minimized", + "$p_{0}$": "hole concentration", + "$q$": "elementary charge" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 208, + "topic": "Semiconductors", + "question": "Suppose a semiconductor crystal is subjected to an electric field $\\boldsymbol{E}$ and a magnetic field $\\boldsymbol{B}$, with $\\boldsymbol{E}$ in the $x-y$ plane and $\\boldsymbol{B}$ along the $z$ direction, try to derive the distribution function of semiconductor electrons in the electromagnetic field considering the multiple interactions of the magnetic field with electrons.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\boldsymbol{E}$": "electric field vector", + "$\\boldsymbol{B}$": "magnetic field vector", + "$E_{x}$": "electric field component along x-axis", + "$E_{y}$": "electric field component along y-axis", + "$B_{z}$": "magnetic field component along z-axis", + "$a_{x}$": "acceleration component along x-axis", + "$a_{y}$": "acceleration component along y-axis", + "$a_{z}$": "acceleration component along z-axis", + "$v_{x}$": "velocity component along x-axis", + "$v_{y}$": "velocity component along y-axis", + "$v_{z}$": "velocity component along z-axis", + "$m$": "mass", + "$q$": "charge", + "$\\mathscr{E}_{x}$": "electric field perturbation along x-axis", + "$\\mathscr{E}_{y}$": "electric field perturbation along y-axis", + "$\\varphi_{1}$": "perturbation function related to velocity component x", + "$\\varphi_{2}$": "perturbation function related to velocity component y", + "$f_{0}$": "equilibrium distribution function", + "$\\tau$": "relaxation time", + "$l$": "mean free path", + "$f_{1}$": "perturbation function related to the component of distribution function along x", + "$f_{2}$": "perturbation function related to the component of distribution function along y", + "$k$": "dimensionless parameter involving charge, mean free path, mass, and magnetic field", + "$\\boldsymbol{v}$": "velocity vector" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 209, + "topic": "Semiconductors", + "question": "Assuming $\\tau_{\\mathrm{n}}=\\tau_{\\mathrm{p}}=\\tau_{0}$ is a constant that does not change with the doping density in the sample, find the value of conductivity when the small-signal lifetime of the sample reaches its maximum.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\tau_{\\mathrm{n}}$": "electron lifetime", + "$\\tau_{\\mathrm{p}}$": "hole lifetime", + "$\\tau_{0}$": "constant lifetime", + "$N_{\\mathrm{t}}$": "trap density", + "$r_{\\mathrm{p}}$": "recombination rate for holes", + "$r_{\\mathrm{n}}$": "recombination rate for electrons", + "$n_{0}$": "initial electron density", + "$p_{0}$": "initial hole density", + "$n_{1}$": "perturbation electron density", + "$p_{1}$": "perturbation hole density", + "$n_{\\mathrm{i}}$": "intrinsic carrier density", + "$e_{\\mathrm{ni}}$": "elementary charge (in intrinsic condition)", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$\\tau_{\\max}$": "maximum lifetime", + "$E_{\\mathrm{t}}$": "trap energy level", + "$E_{\\mathrm{i}}$": "intrinsic energy level", + "$k$": "Boltzmann constant", + "$T$": "temperature" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 210, + "topic": "Semiconductors", + "question": "Assume $\\tau_{\\mathrm{p}}=\\tau_{\\mathrm{n}}=\\tau_{0}$, and based on the small signal lifetime formula\n\n\\tau=\\tau_{\\mathrm{p}} \\frac{n_{0}+n_{1}}{n_{0}+p_{0}}+\\tau_{\\mathrm{n}} \\frac{p_{0}+p_{1}}{n_{0}+p_{0}}\n\n\ndiscuss the relationship between the lifetime $\\tau$ and the position of the recombination center level $E_{\\mathrm{t}}$ in the band gap, and briefly explain its physical significance.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\tau_{\\mathrm{p}}$": "hole lifetime", + "$\\tau_{\\mathrm{n}}$": "electron lifetime", + "$\\tau_{0}$": "reference lifetime", + "$\\tau$": "total carrier lifetime", + "$n_{0}$": "equilibrium electron concentration", + "$n_{1}$": "excess electron concentration", + "$p_{0}$": "equilibrium hole concentration", + "$p_{1}$": "excess hole concentration", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$E_{\\mathrm{t}}$": "recombination center energy level", + "$E_{\\mathrm{i}}$": "intrinsic Fermi level", + "$R_{0}$": "universal gas constant", + "$T$": "temperature", + "$k$": "Boltzmann constant", + "$E_{\\mathrm{e}}$": "conduction band energy", + "$E_{\\mathrm{v}}$": "valence band energy", + "$E_{\\mathrm{c}}$": "energy at conduction band minimum" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 211, + "topic": "Semiconductors", + "question": "Let $f_{\\mathrm{t}}$ be the probability that the composite center energy level $E_{\\mathrm{t}}$ is occupied by an electron, $N_{\\mathrm{c}}$ and $N_{\\mathrm{v}}$ are the effective density of states of the conduction band and the valence band, respectively. Let $f_{\\mathrm{t}}$ be the probability that the composite center energy level $E_{\\mathrm{t}}$ is occupied by an electron, and $N_{\\mathrm{c}}$ be the effective density of states of the conduction band. Consider the rate equation for electrons:\n \\frac{\\mathrm{d} n}{\\mathrm{~d} t}=-\\frac{n(1-f_{\\mathrm{t}})}{\\tau_{\\mathrm{n}}}+\\frac{N_{\\mathrm{c}} f_{\\mathrm{t}}}{\\tau_{\\mathrm{n}}^{\\prime}} \nwhere $\\tau_{\\mathrm{n}}$ and $\\tau_{\\mathrm{n}}^{\\prime}$ are the characteristic time constants related to electron capture and emission, respectively. Under thermal equilibrium conditions, $\\tau_{\\mathrm{n}}^{\\prime}$ can be expressed in terms of $\\tau_{\\mathrm{n}}$, $N_{\\mathrm{c}}$, and parameter $n_1$ (where $n_1 = N_{\\mathrm{c}} \\exp (-\\frac{E_{\\mathrm{c}}-E_{\\mathrm{t}}}{k_{0} T})$, $E_c$ is the conduction band edge energy, $k_0$ is the Boltzmann constant, and $T$ is the temperature). Find the expression for $\\tau_{\\mathrm{n}}^{\\prime}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$f_{\\mathrm{t}}$": "probability that the composite center energy level is occupied by an electron", + "$E_{\\mathrm{t}}$": "composite center energy level", + "$N_{\\mathrm{c}}$": "effective density of states of the conduction band", + "$n$": "electron density in the conduction band", + "$\\tau_{\\mathrm{n}}$": "characteristic time constant for electron capture", + "$\\tau_{\\mathrm{n}}^{\\prime}$": "characteristic time constant for electron emission", + "$n_1$": "factor related to the effective density of states and energy difference", + "$E_{\\mathrm{c}}$": "conduction band edge energy", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$f_{10}$": "probability that the composite center energy level is occupied by electrons in thermal equilibrium", + "$E_{\\mathrm{f}}$": "Fermi level energy", + "$\\tau_{\\mathrm{p}}^{\\prime}$": "characteristic time constant for hole emission", + "$\\tau_{\\mathrm{p}}$": "characteristic time constant for hole capture", + "$p_{1}$": "factor for hole-related processes", + "$N_{\\mathrm{v}}$": "effective density of states of the valence band" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 212, + "topic": "Semiconductors", + "question": "Let $f_{\\mathrm{t}}$ be the probability that the composite center energy level $E_{\\mathrm{t}}$ is occupied by electrons, and $N_{\\mathrm{c}}$ and $N_{\\mathrm{v}}$ be the effective density of states for the conduction band and the valence band, respectively. Derive the small-signal lifetime formula.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$n$": "electron density", + "$p$": "hole density", + "$\\Delta p$": "change in hole density", + "$\\tau$": "carrier lifetime", + "$n_{\\mathrm{i}}$": "intrinsic carrier density", + "$\\tau_{\\mathrm{p}}$": "hole lifetime", + "$n_{1}$": "electron density under recombination center condition", + "$\\tau_{\\mathrm{n}}$": "electron lifetime", + "$p_{1}$": "hole density under recombination center condition", + "$n_{0}$": "equilibrium electron density", + "$p_{0}$": "equilibrium hole density", + "$\\Delta n$": "change in electron density" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 213, + "topic": "Semiconductors", + "question": "A square pulse with an appropriate frequency is irradiated onto an n-type semiconductor sample and is uniformly absorbed inside the sample to generate nonequilibrium carriers at a generation rate of $g_{\\mathrm{p}}$. The lifetime of nonequilibrium holes is $\\tau_{\\mathrm{p}}$, and the pulse width is $\\Delta t=3 \\tau_{\\mathrm{p}}$. Assume the moment the pulse light starts irradiating is $t=0$, find the expression for the concentration of nonequilibrium holes $\\Delta p(\\Delta t)$ at the moment the light pulse ends ($t=\\Delta t$).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$t$": "time", + "$\\Delta t$": "duration of the light pulse", + "$\\Delta p$": "concentration of nonequilibrium holes", + "$g_{\\mathrm{p}}$": "generation rate of holes", + "$\\tau_{\\mathrm{p}}$": "lifetime of holes" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 214, + "topic": "Semiconductors", + "question": "Using the bipolar diffusion theory, consider the case of intrinsic semiconductors (i.e., the electron concentration $n$ is equal to the hole concentration $p$). It is known that the general expression for the bipolar diffusion coefficient is $D^{*}=\\frac{(n+p) D_{\\mathrm{n}} D_{\\mathrm{p}}}{n D_{\\mathrm{n}}+p D_{\\mathrm{p}}}$. Try to derive the specific expression for the bipolar diffusion coefficient $D^{*}$ in intrinsic semiconductors.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$n$": "electron concentration", + "$p$": "hole concentration", + "$D^{*}$": "bipolar diffusion coefficient", + "$D_{\\mathrm{n}}$": "electron diffusion coefficient", + "$D_{\\mathrm{p}}$": "hole diffusion coefficient", + "$\\mu^{*}$": "bipolar mobility", + "$\\Delta n$": "change in electron concentration", + "$\\Delta p$": "change in hole concentration" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 215, + "topic": "Semiconductors", + "question": "For a silicon pn junction, the doping concentrations of the p and n regions are $N_{\\mathrm{A}}=9 \\times 10^{15} \\mathrm{~cm}^{-3}$ and $N_{\\mathrm{D}}=2 \\times 10^{16} \\mathrm{~cm}^{-3}$, respectively; the hole and electron mobilities in the p region are $350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, respectively, while in the n region, they are $300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. Assume the lifetime of the non-equilibrium carriers in both regions is $1 \\mu \\mathrm{~s}$, and the cross-sectional area of the pn junction is $10^{-2} \\mathrm{~cm}^{2} ; \\frac{q}{k_{0} T}=38.7(\\frac{1}{V})$. When a forward voltage $V_{\\mathrm{F}}=0.65 \\mathrm{~V}$ is applied, calculate: Determine the expression for the hole diffusion current variation with $x$ in the n region.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$I_{\\mathrm{pD}}$": "hole diffusion current", + "$x$": "position in the n region", + "$A$": "cross-sectional area", + "$q$": "elementary charge", + "$D_{\\mathrm{p}}$": "hole diffusion coefficient", + "$p$": "hole concentration", + "$I_{\\mathrm{nD}}$": "electron diffusion current", + "$D_{\\mathrm{n}}^{\\prime}$": "modified electron diffusion coefficient", + "$n$": "electron concentration", + "$k_{0}$": "Boltzmann constant relation factor", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$I_{\\mathrm{nt}}$": "electron drift current", + "$I$": "total current in the n region", + "$E$": "electric field due to external forward voltage", + "$E^{\\prime}$": "electric field due to diffusion imbalance" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 216, + "topic": "Semiconductors", + "question": "There is a silicon pn junction, with doping concentrations in the p-region and n-region of $N_{\\mathrm{A}}=9 \\times 10^{15} \\mathrm{~cm}^{-3}$ and $N_{\\mathrm{D}}=2 \\times 10^{16} \\mathrm{~cm}^{-3}$ respectively; the hole and electron mobilities in the p-region are $350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ respectively, and in the n-region, the hole and electron mobilities are $300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ respectively; assume the lifetime of non-equilibrium carriers in both regions is $1 \\mu \\mathrm{~s}, \\mathrm{pn}$ junction cross-sectional area is $10^{-2} \\mathrm{~cm}^{2} ; \\frac{q}{k_{0} T}=38.7(\\frac{1}{V})$ . When the applied forward voltage $V_{\\mathrm{F}}=0.65 \\mathrm{~V}$, try to find: Determine the expression for the electron diffusion current variation with $x$ in the n-region.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$x$": "position in the n-region", + "$I_{\\mathrm{pD}}$": "hole diffusion current", + "$I_{\\mathrm{nD}}$": "electron diffusion current", + "$D_{\\mathrm{p}}$": "hole diffusion coefficient", + "$D_{\\mathrm{n}}^{\\prime}$": "modified electron diffusion coefficient", + "$k_{0}$": "constant related to diffusion", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$I_{\\mathrm{nt}}$": "electron drift current", + "$I$": "total current", + "$E$": "electric field in the n-region from voltage drop", + "$E^{\\prime}$": "electric field from faster electron diffusion" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 217, + "topic": "Semiconductors", + "question": "There is a silicon pn junction, with doping concentrations in the p-region and n-region being $N_{\\mathrm{A}}=9 \\times 10^{15} \\mathrm{~cm}^{-3}$ and $N_{\\mathrm{D}}=2 \\times 10^{16} \\mathrm{~cm}^{-3}$, respectively. The hole and electron mobilities in the p-region are $350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, respectively, and in the n-region, the hole and electron mobilities are $300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, respectively. Assume the lifetime of non-equilibrium carriers in both regions is $1 \\mu \\mathrm{~s}$, the area of the pn junction is $10^{-2} \\mathrm{~cm}^{2}$; $\\frac{q}{k_{0} T}=38.7(\\frac{1}{V})$. When a forward bias voltage of $V_{\\mathrm{F}}=0.65 \\mathrm{~V}$ is applied, determine: Determine the expression for the variation of electron drift current with $x$ in the n-region.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$I_{\\mathrm{pD}}$": "hole diffusion current", + "$I_{\\mathrm{nD}}$": "electron diffusion current", + "$I_{\\mathrm{nt}}$": "electron drift current", + "$I$": "total current", + "$A$": "cross-sectional area", + "$q$": "elementary charge", + "$D_{\\mathrm{p}}$": "hole diffusion coefficient", + "$D_{\\mathrm{n}}^{\\prime}$": "effective electron diffusion coefficient", + "$k_{0}$": "Boltzmann constant", + "$T$": "absolute temperature", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$x$": "position in the n-region", + "$E$": "electric field due to voltage drop", + "$E^{\\prime}$": "electric field due to faster electron diffusion" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 218, + "topic": "Semiconductors", + "question": "There is a silicon pn junction, with the doping concentrations of the p-region and n-region being $N_{\\mathrm{A}}=9 \\times 10^{15} \\mathrm{~cm}^{-3}$ and $N_{\\mathrm{D}}=2 \\times 10^{16} \\mathrm{~cm}^{-3}$ respectively; the hole and electron mobilities in the p-region are $350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ respectively, while in the n-region the hole and electron mobilities are $300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ respectively; assuming the minority carrier lifetime in both regions is $1 \\mu \\mathrm{~s}, \\mathrm{pn}$ junction cross-sectional area is $10^{-2} \\mathrm{~cm}^{2} ; \\frac{q}{k_{0} T}=38.7(\\frac{1}{V})$. When a forward bias voltage $V_{\\mathrm{F}}=0.65 \\mathrm{~V}$ is applied, find: Determine the expression for the total electron current in the n-region as a function of $x$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$x$": "position in the n-region", + "$I_{\\mathrm{pD}}$": "hole diffusion current", + "$I_{\\mathrm{nD}}$": "electron diffusion current", + "$D_{\\mathrm{n}}^{\\prime}$": "effective electron diffusion coefficient", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$q$": "elementary charge", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$I_{\\mathrm{nt}}$": "electron drift current", + "$I$": "total current in the n-region", + "$E$": "electric field due to voltage drop", + "$E^{\\prime}$": "electric field due to electron diffusion", + "$I_{\\mathrm{n}}$": "total electron current" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 219, + "topic": "Semiconductors", + "question": "A metal contacts uniformly doped $n-Si$ material to form a Schottky barrier diode. Given the barrier height on the semiconductor side $q V_{\\mathrm{D}}=0.6 \\mathrm{eV}, N_{\\mathrm{D}}=5 \\times 10^{16} \\mathrm{~cm}^{-3}$, find the relationship curve of $1 / C^{2}$ versus $(V_{\\mathrm{D}}-V)$ under a 5V reverse bias voltage.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$q$": "elementary charge", + "$V_{\\mathrm{D}}$": "barrier potential difference on the semiconductor side", + "$N_{\\mathrm{D}}$": "semiconductor dopant concentration", + "$C$": "capacitance", + "$d$": "space charge region width", + "$E_{\\mathrm{M}}$": "maximum electric field at the semiconductor interface", + "$\\varepsilon_0$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", + "$N$": "dopant concentration in the semiconductor", + "$V$": "applied voltage" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 220, + "topic": "Semiconductors", + "question": "A metal plate is 0.4 $\\mu \\mathrm{~m}$ away from n-type silicon, forming a parallel plate capacitor, with dry air in between having a relative permittivity $\\varepsilon_{\\mathrm{ra}}=1$. When a negative voltage is applied to the metal side, the semiconductor is in a depletion state. Find the expression for depletion layer width $X_{\\mathrm{d}}$ when $V_{\\mathrm{s}}=0.4 \\mathrm{~V}$;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$X_{\\mathrm{d}}$": "depletion layer width", + "$V_{\\mathrm{s}}$": "surface potential", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", + "$\\varepsilon_{0}$": "vacuum permittivity", + "$q$": "elementary charge", + "$N_{\\mathrm{D}}$": "donor concentration", + "$V_{\\mathrm{sm}}$": "surface potential at maximum depletion layer width", + "$V_{\\mathrm{B}}$": "built-in potential", + "$k_{0}$": "Boltzmann constant in appropriate units", + "$T$": "temperature", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$X_{\\mathrm{dm}}$": "maximum depletion layer width" + }, + "chapter": "Semiconductor Surface and MIS Structure", + "section": "Semiconductor Surface and MIS Structure" + }, + { + "id": 221, + "topic": "Semiconductors", + "question": "A metal plate is separated from n-type silicon by a distance of $0.4 \\mu \\mathrm{~m}$, forming a parallel plate capacitor. The relative permittivity of the dry air in between is $\\varepsilon_{\\mathrm{ra}}=1$. When a negative voltage is applied to the metal end, the semiconductor is in a depletion state. Find the expression for the maximum depletion layer width $X_{\\mathrm{dm}}$ when $V_{\\mathrm{s}}=0.4 \\mathrm{~V}$;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$X_{\\mathrm{dm}}$": "maximum depletion layer width", + "$V_{\\mathrm{s}}$": "surface potential", + "$X_{\\mathrm{d}}$": "depletion layer width", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", + "$\\varepsilon_{0}$": "vacuum permittivity", + "$q$": "elementary charge", + "$N_{\\mathrm{D}}$": "donor concentration", + "$V_{\\mathrm{sm}}$": "surface potential at maximum depletion layer width", + "$V_{\\mathrm{B}}$": "built-in potential", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration" + }, + "chapter": "Semiconductor Surface and MIS Structure", + "section": "Semiconductor Surface and MIS Structure" + }, + { + "id": 222, + "topic": "Semiconductors", + "question": "For n-type GaAs with a thickness of 0.08 cm, a current of 50 mA is applied in the $x$ direction, and a magnetic field of 0.5 T is applied in the $z$ direction, resulting in a Hall voltage of -0.4 mV, find: Given the resistivity of the material is $1.5 \\times 10^{-3} \\Omega \\cdot \\mathrm{~cm}$, find the carrier mobility.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mu_{\\mathrm{H}}$": "Hall mobility", + "$R_{\\mathrm{H}}$": "Hall coefficient", + "$\\sigma_{0}$": "conductivity", + "$\\rho_{0}$": "resistivity" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 223, + "topic": "Semiconductors", + "question": "Assume the relaxation time $\\tau$ is constant, and try to calculate the Hall coefficient of an n-type semiconductor.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\tau$": "relaxation time", + "$\\mathscr{E}$": "external electric field", + "$\\boldsymbol{B}$": "magnetic field", + "$e$": "elementary charge", + "$m_{\\mathrm{n}}$": "effective mass of the electron", + "$\\omega$": "cyclotron frequency", + "$v_{x}$": "velocity component in x direction", + "$v_{y}$": "velocity component in y direction", + "$v_{x 0}$": "initial velocity component in x direction", + "$v_{y 0}$": "initial velocity component in y direction", + "$n$": "electron density", + "$j_{x}$": "current density in x direction", + "$j_{y}$": "current density in y direction", + "$\\mathscr{E}_{x}$": "electric field component in x direction", + "$\\mathscr{E}_{y}$": "electric field component in y direction", + "$\\mathscr{C}_{y}$": "unknown field component in y direction", + "$R$": "Hall coefficient" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 224, + "topic": "Semiconductors", + "question": "In the experiment of measuring the Hall coefficient of a semiconductor, the current $J_{x}$ induced by the external electric field $\\varepsilon_{x}$ is called the original current. The transverse current generated under the action of the Lorentz force and the Hall electric field $\\varepsilon_{y}$ includes the electron current component $J_{\\mathrm{n} y}$ and the hole current component $J_{\\mathrm{p} y}$. Find the ratio $f_{\\mathrm{c}}=J_{\\mathrm{ny}} / J_{x}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$f_{\\mathrm{c}}$": "ratio of transverse electron current to the original current", + "$J_{x}$": "original current density along the x axis", + "$J_{\\mathrm{ny}}$": "transverse electron current density", + "$J_{\\mathrm{py}}$": "transverse hole current density", + "$n$": "electron concentration", + "$p$": "hole concentration", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$\\sigma_{\\mathrm{n}}$": "electron conductivity", + "$\\sigma_{\\mathrm{p}}$": "hole conductivity", + "$\\varepsilon_{x}$": "electric field component along the x axis", + "$\\varepsilon_{y}$": "electric field component along the y axis", + "$B_{z}$": "magnetic field component along the z axis", + "$J_{y}$": "total transverse current density" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 225, + "topic": "Semiconductors", + "question": "Try to prove that in the Hall effect under the conditions of simultaneous presence of two types of charge carriers and a weak magnetic field, the Hall angle $\\theta$ and the Hall coefficient $R$ can be expressed as\n\n\\begin{aligned}\n& \\theta=\\arctan \\frac{p \\mu_{\\mathrm{p}}^{2}-n \\mu_{\\mathrm{n}}^{2}}{p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}}} B_{z} \\\\\n& R=\\frac{1}{q} \\frac{p \\mu_{\\mathrm{p}}^{2}-n \\mu_{\\mathrm{n}}^{2}}{(p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}})^{2}}\n\\end{aligned} If the Hall angle of a sample is measured to be $\\theta=0$, find the corresponding electrical conductivity;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\theta$": "Hall angle", + "$p$": "hole concentration", + "$\\mu_{\\text{p}}$": "hole mobility", + "$n$": "electron concentration", + "$\\mu_{\\text{n}}$": "electron mobility", + "$B_z$": "magnetic field component in the z-direction", + "$\\sigma$": "electrical conductivity", + "$q$": "elementary charge", + "$b$": "mobility ratio between electrons and holes" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 226, + "topic": "Semiconductors", + "question": "In the experiment of measuring the Hall coefficient of semiconductors, the current induced by the external electric field $\\mathscr{E}_{x}$ is called the primary current. Under the action of Lorentz force and Hall electric field, a transverse electron current $J_{\\mathrm{ey}}$ and a transverse hole current $J_{\\mathrm{py}}$ are generated. Find the ratio of the transverse electron current to the primary current at equilibrium $f_{\\mathrm{e}}=\\frac{J_{\\mathrm{ey}}}{J_{x}}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$f_{\\mathrm{e}}$": "ratio of transverse electron current to primary current", + "$J_{\\mathrm{ey}}$": "transverse electron current", + "$J_{x}$": "primary current", + "$n$": "electron concentration", + "$p$": "hole concentration", + "$\\mu_{\\mathrm{e}}$": "electron mobility", + "$\\mu_{\\mathrm{h}}$": "hole mobility", + "$\\sigma_{\\mathrm{e}}$": "electron conductivity", + "$\\sigma_{\\mathrm{h}}$": "hole conductivity", + "$\\mathscr{E}_{x}$": "external electric field along x-axis", + "$B$": "magnetic field", + "$\\mathscr{E}_{y}$": "transverse Hall electric field", + "$v_{e x}$": "electron drift velocity along x-axis", + "$R_{\\mathrm{H}}$": "Hall coefficient", + "$J_{y}$": "transverse current", + "$f_{\\mathrm{h}}$": "ratio of transverse hole current to primary current", + "$T$": "temperature" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 227, + "topic": "Others", + "question": "N atoms form a two-dimensional square lattice, with each atom contributing one electron to form a two-dimensional free electron gas. The electron energy expression is\n$$\nE(k)=\\frac{\\hbar^{2} k_{x}^{2}}{2 m}+\\frac{\\hbar^{2} k_{y}^{2}}{2 m}\n$$ Derive the formula for the density of st ates of a two-dimensional free gas.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E$": "energy", + "$k$": "wave vector magnitude", + "$k_x$": "wave vector component in x-direction", + "$k_y$": "wave vector component in y-direction", + "$m$": "mass", + "$\\hbar$": "reduced Planck's constant", + "$g(E)$": "density of states per unit area for a two-dimensional free electron gas" + }, + "chapter": "Movement of electrons in a crystal in electric and magnetic fields", + "section": "Movement of electrons in a crystal in electric and magnetic fields" + }, + { + "id": 228, + "topic": "Others", + "question": "A two-dimensional square lattice composed of N atoms, each contributing 1 electron to form a two-dimensional free electron gas. The expression for the electron energy is\n$$\nE(k)=\\frac{\\hbar^{2} k_{x}^{2}}{2 m}+\\frac{\\hbar^{2} k_{y}^{2}}{2 m}\n$$ At this time, a magnetic field B is applied perpendicular to the square lattice. The energy levels of the free electron gas will condense into Landau levels. What is the degeneracy of these levels?", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$B$": "magnetic field", + "$E$": "energy", + "$k$": "wave vector", + "$m$": "mass", + "$\\hbar$": "reduced Planck's constant", + "$\\omega_{c}$": "cyclotron frequency", + "$E_{n}$": "energy of the n-th Landau level", + "$n$": "quantum number", + "$D$": "degeneracy of the Landau levels", + "$e$": "elementary charge" + }, + "chapter": "Movement of electrons in a crystal in electric and magnetic fields", + "section": "Movement of electrons in a crystal in electric and magnetic fields" + }, + { + "id": 229, + "topic": "Theoretical Foundations", + "question": "A particle is incident with kinetic energy $E$, subjected to the following double $\\delta$ potential barriers:\n\nV(x)=V_{0}[\\delta(x)+\\delta(x-a)]\n\n\nFind the expression for the conditions under which complete transmission occurs. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$E$": "kinetic energy", + "$V_0$": "potential barrier height", + "$x$": "position", + "$a$": "distance between delta potential barriers", + "$k$": "wave number", + "$m$": "mass of the particle", + "$\\hbar$": "reduced Planck's constant", + "$C$": "dimensionless parameter related to potential and mass", + "$R$": "reflection coefficient", + "$D$": "transmission coefficient", + "$A$": "amplitude of wave function in region 0 < x < a", + "$B$": "amplitude of wave function in region 0 < x < a", + "$\\theta$": "dimensionless parameter related to wave number and potential" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 230, + "topic": "Theoretical Foundations", + "question": "For the energy eigenstate $|n\\rangle$ of the harmonic oscillator, calculate the expression for the uncertainty product $\\Delta x \\cdot \\Delta p$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$|n\\rangle$": "ket vector corresponding to state n", + "$\\Delta x$": "uncertainty in position", + "$\\Delta p$": "uncertainty in momentum", + "$\\langle n|$": "bra vector corresponding to state n", + "$\\delta_{n n^{\\prime}}$": "Kronecker delta function", + "$\\hbar$": "reduced Planck's constant", + "$m$": "mass", + "$\\omega$": "angular frequency", + "$a$": "annihilation operator", + "$a^{+}$": "creation operator", + "$n$": "quantum number", + "$\\bar{x}$": "average position", + "$\\bar{p}$": "average momentum", + "$\\overline{x^{2}}$": "average of the square of position", + "$\\overline{p^{2}}$": "average of the square of momentum", + "$\\hat{n}$": "number operator" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 231, + "topic": "Theoretical Foundations", + "question": "In the coherent state $|\\alpha\\rangle$, calculate the uncertainty product $\\Delta x \\cdot \\Delta p$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$|\\alpha\\rangle$": "coherent state represented by the parameter alpha", + "$\\Delta x$": "uncertainty in position", + "$\\Delta p$": "uncertainty in momentum", + "$\\alpha$": "parameter of the coherent state", + "$\\alpha^{*}$": "complex conjugate of alpha", + "$\\hbar$": "reduced Planck's constant", + "$m$": "mass", + "$\\omega$": "angular frequency", + "$\\bar{n}$": "average photon number", + "$\\bar{E}$": "average energy", + "$a$": "annihilation operator", + "$a^{+}$": "creation operator", + "$n$": "number operator" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 232, + "topic": "Theoretical Foundations", + "question": "Define the radial momentum operator\n\n\\begin{equation*}\n\\boldsymbol{p}_{r}=\\frac{1}{2}(\\frac{\\boldsymbol{r}}{r} \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot \\frac{\\boldsymbol{r}}{r}) \\tag{1}\n\\end{equation*}\n\n\nFind the commutation relation $[r, p_{r}]$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$r$": "radial distance", + "$p_{r}$": "radial momentum operator", + "$\\boldsymbol{r}$": "position vector", + "$\\boldsymbol{p}$": "momentum operator", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 233, + "topic": "Theoretical Foundations", + "question": "A particle with mass $\\mu$ moves in a central potential field\n\n\\begin{equation*}\nV(r)=\\lambda r^{\\nu}, \\quad-2<\\nu<\\infty \\tag{1}\n\\end{equation*}\n\n\nWe only discuss the case where bound states can exist, i.e., when $\\lambda \\nu > 0$. The radial wave function $u(r)=rR(r)$ satisfies the following radial Schrödinger equation:\n\n\\begin{equation*}\n\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\mathrm{~d}^{2} u}{\\mathrm{~d} r^{2}}+[E-\\lambda r^{\\nu}-l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}] u=0 \\tag{2}\n\\end{equation*}\n\nBy introducing the dimensionless radial distance $\\rho$ and energy $\\varepsilon$, and denoting the radial function as $w(\\rho)=u(r)$, the above equation can be non-dimensionalized. Please write down the dimensionless radial equation in terms of $w(\\rho)$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\mu$": "mass of the particle", + "$\\lambda$": "parameter of the potential field", + "$\\nu$": "exponent in the potential energy", + "$\\hbar$": "reduced Planck's constant", + "$E$": "energy", + "$l$": "angular momentum quantum number", + "$r$": "radial distance", + "$u$": "radial wave function", + "$\\rho$": "dimensionless radial distance", + "$\\varepsilon$": "dimensionless energy", + "$w$": "radial function in dimensionless form", + "$n$": "quantum number for energy levels", + "$N$": "quantum number for energy levels in harmonic oscillator potential" + }, + "chapter": "Schrödinger equation one-dimensional motion 5.4 p104", + "section": "" + }, + { + "id": 234, + "topic": "Theoretical Foundations", + "question": "For the hydrogen atom's s states $(n l m=n 00)$, calculate the expression for the uncertainty product $\\Delta x \\cdot \\Delta p_{x}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$n$": "principal quantum number", + "$l$": "azimuthal quantum number", + "$m$": "magnetic quantum number", + "$\\Delta x$": "position uncertainty", + "$\\Delta p_{x}$": "momentum uncertainty in the x-direction", + "$\\langle x\\rangle$": "average position", + "$\\langle p_{x}\\rangle$": "average momentum in the x-direction", + "$\\langle x^{2}\\rangle$": "average square of the position", + "$\\langle p_{x}^{2}\\rangle$": "average square of the momentum in the x-direction", + "$\\langle r^{2}\\rangle$": "average square of the radial distance", + "$a_{0}$": "Bohr radius", + "$\\mu$": "reduced mass", + "$E_{n}$": "energy at quantum number n", + "$e$": "elementary charge", + "$\\hbar$": "reduced Planck's constant", + "$\\boldsymbol{p}$": "momentum vector" + }, + "chapter": "Schrödinger equation one-dimensional motion 5.17 p121", + "section": "" + }, + { + "id": 235, + "topic": "Theoretical Foundations", + "question": "For electrons and other spin $1 / 2$ particles, the eigenstates of $s_{z}$ are often denoted by $\\alpha$ and $\\beta$, where $\\alpha$ is equivalent to $\\chi_{\\frac{1}{2}}$, and $\\beta$ is equivalent to $\\chi_{-\\frac{1}{2}}$. Given the electron wave function\n\n\\begin{equation*}\n\\psi(r, \\theta, \\varphi, s_{z})=\\alpha Y_{l 0}(\\theta, \\varphi) R(r) \n\\end{equation*}\n\n\nfind the only possible measurement value of the total angular momentum $j_{z}$ (taking $\\hbar=1$).", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$s_{z}$": "z-component of spin angular momentum", + "$\\alpha$": "spin-up eigenstate", + "$\\beta$": "spin-down eigenstate", + "$\\chi_{\\frac{1}{2}}$": "irreducible representation for spin-up state", + "$\\chi_{-\\frac{1}{2}}$": "irreducible representation for spin-down state", + "$r$": "radial coordinate", + "$\\theta$": "polar angle coordinate", + "$\\varphi$": "azimuthal angle coordinate", + "$Y_{l 0}$": "spherical harmonic (azimuthal quantum number zero)", + "$R(r)$": "radial wave function", + "$j_{z}$": "z-component of total angular momentum", + "$l$": "orbital angular momentum quantum number", + "$l^{2}$": "squared orbital angular momentum", + "$l_{z}$": "z-component of orbital angular momentum", + "$\\sigma_{z}$": "z-component of spin", + "$j^{2}$": "squared total angular momentum", + "$C_{1}$": "coefficient for state with total angular momentum j=l+1/2", + "$C_{2}$": "coefficient for state with total angular momentum j=l-1/2", + "$\\phi_{l, l+\\frac{1}{2}, \\frac{1}{2}}$": "eigenstate with total angular momentum quantum numbers (l, l+1/2, 1/2)", + "$\\phi_{l, l-\\frac{1}{2}, \\frac{1}{2}}$": "eigenstate with total angular momentum quantum numbers (l, l-1/2, 1/2)" + }, + "chapter": "Schrödinger equation one-dimensional motion 6.34 p180", + "section": "" + }, + { + "id": 236, + "topic": "Theoretical Foundations", + "question": "If the operator $\\hat{f}(x)$ commutes with $\\hat{D}_{x}(a)$, find the general solution for $\\hat{f}(x)$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\hat{f}(x)$": "a general operator function of x", + "$x$": "a variable representing position or an argument", + "$\\hat{D}_{x}(a)$": "a displacement operator acting on x", + "$a$": "the period of the function or displacement amount" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 237, + "topic": "Theoretical Foundations", + "question": "For a spin $1 / 2$ particle, find the effect of the operator $\\sigma_{r}=\\boldsymbol{\\sigma} \\cdot \\boldsymbol{r} / \\boldsymbol{r}$ acting on the common eigenfunctions $\\phi_{l j m_{j}}$ of $(l^{2}, j^{2}, j_{z})$ (taking $\\hbar=1$). You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\sigma_{r}$": "radial component of the Pauli spin operator", + "$\\boldsymbol{\\sigma}$": "Pauli spin operator", + "$\\boldsymbol{r}$": "position vector", + "$\\phi_{l j m_{j}}$": "common eigenfunction of angular momentum operators", + "$j$": "total angular momentum quantum number", + "$m_{j}$": "magnetic quantum number associated with total angular momentum", + "$l$": "orbital angular momentum quantum number", + "$j_{z}$": "z-component of total angular momentum", + "$\\hbar$": "reduced Planck's constant", + "$l_{z}$": "z-component of orbital angular momentum", + "$x$": "x-coordinate of the position vector", + "$y$": "y-coordinate of the position vector", + "$z$": "z-coordinate of the position vector", + "$C$": "transition coefficient between eigenstates for $\\sigma_{r}$ operator", + "$C^{\\prime}$": "inverse of the transition coefficient $C$", + "$\\theta$": "polar angle in spherical coordinates", + "$\\varphi$": "azimuthal angle in spherical coordinates" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 238, + "topic": "Theoretical Foundations", + "question": "For a system composed of two spin $1/2$ particles, where $\\boldsymbol{s}_{1}, ~ \\boldsymbol{\\sigma}_{1}$ and $\\boldsymbol{s}_{2}, ~ \\boldsymbol{\\sigma}_{2}$ represent the spin angular momentum and Pauli operators for particles 1 and 2, respectively, $\\boldsymbol{s}_{1}=\\frac{1}{2} \\boldsymbol{\\sigma}_{1}, \\boldsymbol{s}_{2}=\\frac{1}{2} \\boldsymbol{\\sigma}_{2}$ (taking $\\hbar=1$). Find the simplest algebraic equation satisfied by $\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\boldsymbol{s}_{1}$": "spin angular momentum for particle 1", + "$\\boldsymbol{\\sigma}_{1}$": "Pauli operators for particle 1", + "$\\boldsymbol{s}_{2}$": "spin angular momentum for particle 2", + "$\\boldsymbol{\\sigma}_{2}$": "Pauli operators for particle 2", + "$\\hbar$": "reduced Planck's constant", + "$\\boldsymbol{S}$": "total spin angular momentum", + "$\\boldsymbol{S}^{2}$": "square of the total spin angular momentum", + "$S$": "spin quantum number" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 239, + "topic": "Others", + "question": "Majorana fermions. One can write a relativistic equation for a massless 2 -component fermion field that transforms as the upper two components of a Dirac spinor $(\\psi_{L})$. Call such a 2-component field $\\chi_{a}(x), a=1,2.$ Let us write a 4 -component Dirac field as\n\n\\psi(x)=\\binom{\\psi_{L}}{\\psi_{R}}\n\nand recall that the lower components of $\\psi$ transform in a way equivalent by a unitary transformation to the complex conjugate of the representation $\\psi_{L}$. In this way, we can rewrite the 4 -component Dirac field in terms of two 2 -component spinors:\n\n\\psi_{L}(x)=\\chi_{1}(x), \\quad \\psi_{R}(x)=i \\sigma^{2} \\chi_{2}^{*}(x)\n\n\nFrom the Dirac Lagrangian $\\mathcal{L} = \\bar{\\psi}(\\mathrm{i} \\not \\partial-m) \\psi$ rewritten in terms of $\\chi_{1}$ and $\\chi_{2}$, identify and state the form of the mass term.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\psi$": "4-component Dirac field", + "$\\psi_L$": "2-component left-handed spinor", + "$\\psi_R$": "2-component right-handed spinor", + "$\\chi_1$": "2-component spinor associated with left-handed projection", + "$\\chi_2$": "2-component spinor associated with right-handed projection", + "$\\mathcal{L}$": "Lagrangian", + "$m$": "mass parameter", + "$\\sigma^2$": "second Pauli matrix" + }, + "chapter": "The Dirac Field", + "section": "Majorana fermions" + }, + { + "id": 240, + "topic": "Others", + "question": "Fierz transformations. Let $u_{i}, i=1, \\ldots, 4$, be four 4 -component Dirac spinors. In the text, we proved the Fierz rearrangement formulaes. The first of these formulae can be written in 4 -component notation as\n$$\n\\bar{u}_{1} \\gamma^{\\mu}(\\frac{1+\\gamma^{5}}{2}) u_{2} \\bar{u}_{3} \\gamma_{\\mu}(\\frac{1+\\gamma^{5}}{2}) u_{4}=-\\bar{u}_{1} \\gamma^{\\mu}(\\frac{1+\\gamma^{5}}{2}) u_{4} \\bar{u}_{3} \\gamma_{\\mu}(\\frac{1+\\gamma^{5}}{2}) u_{2} .\n$$\n\nIn fact, there are similar rearrangement formulae for any product\n$$\n(\\bar{u}_{1} \\Gamma^{A} u_{2})(\\bar{u}_{3} \\Gamma^{B} u_{4}),\n$$\nwhere $\\Gamma^{A}, \\Gamma^{B}$ are any of the 16 combinations of Dirac matrices. To begin, normalize the 16 matrices $\\Gamma^{A}$ to the convention\n$$\n\\operatorname{tr}[\\Gamma^{A} \\Gamma^{B}]=4 \\delta^{A B} .\n$$\n\nThis gives $\\Gamma^{A}={1, \\gamma^{0}, i \\gamma^{j}, \\ldots}$; write all 16 elements of this set. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\Gamma^{A}$": "general Dirac matrix", + "$\\Gamma^{B}$": "general Dirac matrix", + "$\\Gamma^{C}$": "general Dirac matrix", + "$\\Gamma^{D}$": "general Dirac matrix", + "$\\gamma^{0}$": "gamma matrix for time component (0)", + "$\\gamma^{\\mu}$": "gamma matrix for spacetime component (μ)", + "$\\gamma^{5}$": "gamma matrix (fifth gamma matrix in 5D)", + "$\\sigma^{\\mu \\nu}$": "antisymmetric spin tensor", + "$\\sigma^{0 i}$": "spin tensor for time and spatial component", + "$\\sigma^{i j}$": "spin tensor for spatial components", + "$\\delta^{A B}$": "Kronecker delta", + "$\\bar{u}_{1}$": "Dirac spinor (1)", + "$\\bar{u}_{3}$": "Dirac spinor (3)", + "$u_{2}$": "Dirac spinor (2)", + "$u_{4}$": "Dirac spinor (4)" + }, + "chapter": "The Dirac Field", + "section": "Fierz transformations" + }, + { + "id": 241, + "topic": "Others", + "question": "This problem concerns the discrete symmetries $P, C$, and $T$. Compute the transformation property under $C$ of the antisymmetric tensor fermion bilinear $\\bar{\\psi} \\sigma^{\\mu \\nu} \\psi$, with $\\sigma^{\\mu \\nu}=\\frac{i}{2}[\\gamma^{\\mu}, \\gamma^{\\nu}]$. This completes the table of the transformation properties of bilinears at the end of the chapter. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$C$": "charge conjugation transformation", + "$\\bar{\\psi}$": "Dirac adjoint of psi", + "$\\psi$": "Dirac spinor field", + "$\\sigma^{\\mu \\nu}$": "antisymmetric tensor", + "$\\gamma^{\\mu}$": "gamma matrices component (mu)", + "$\\gamma^{\\nu}$": "gamma matrices component (nu)", + "$\\gamma^{0}$": "gamma matrices (time component)", + "$\\gamma^{2}$": "gamma matrices (spatial component)", + "$\\gamma^{1}$": "gamma matrices (spatial component)", + "$\\gamma^{3}$": "gamma matrices (spatial component)" + }, + "chapter": "The Dirac Field", + "section": "The discrete symmetries $P, C$ and $T$" + }, + { + "id": 242, + "topic": "Others", + "question": "Exotic contributions to $\\boldsymbol{g} \\mathbf{- 2}$. Any particle that couples to the electron can produce a correction to the electron-photon form factors and, in particular, a correction to $g-2$. Because the electron $g-2$ agrees with QED to high accuracy, these corrections allow us to constrain the properties of hypothetical new particles.\n\nThe unified theory of weak and electromagnetic interactions contains a scalar particle $h$ called the Higgs boson, which couples to the electron according to\n\\begin{align*}\nH_{\\text{int}} = \\int d^3 x \\frac{\\lambda}{\\sqrt{2}} h \\bar{\\psi} \\psi.\n\\end{align*}\nOne can study the contribution of a virtual Higgs boson to the electron $(g - 2)$, in terms of $\\lambda$ and the mass $m_h$ of the Higgs boson. QED accounts extremely well for the electron's anomalous magnetic moment. If $a=(g-2) / 2$,\n$$\n|a_{\\text {expt. }}-a_{\\mathrm{QED}}|<1 \\times 10^{-10}\n$$\n\nWhat limits does this place on $\\lambda$ and $m_{h}$ ? In the simplest version of the electroweak theory, $\\lambda=3 \\times 10^{-6}$ and $m_{h}>60 \\mathrm{GeV}$. Show that these values are not excluded. \n\nHint: You can find the contribution of a virtual Higgs boson to the electron $(g - 2)$, in terms of $\\lambda$ and the mass $m_h$ of the Higgs boson and check its value with $1 \\times 10^{-10}$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$a$": "anomalous part of the magnetic moment", + "$g$": "gyromagnetic ratio", + "$\\lambda$": "coupling constant in electroweak theory", + "$m_{h}$": "mass of the Higgs boson", + "$m$": "mass of the electron or muon in context", + "$\\delta F_{2}$": "form factor contribution", + "$q$": "momentum transfer parameter" + }, + "chapter": "Radiative Corrections: Introduction", + "section": "Exotic contributions to $g-2$" + }, + { + "id": 243, + "topic": "Others", + "question": "Although we have discussed QED radiative corrections at length in the last two chapters, so far we have made no attempt to compute a full radiatively corrected cross section. The reason is of course that such calculations are quite lengthy. Nevertheless it would be dishonest to pretend that one understands radiative corrections after computing only isolated effects as we have done. This \"final project\" is an attempt to remedy this situation. The project is the computation of one of the simplest, but most important, radiatively corrected cross sections. \n\nStrongly interacting particles-pions, kaons, and protons-are produced in $e^{+} e^{-}$annihilation when the virtual photon creates a pair of quarks. If one ignores the effects of the strong interactions, it is easy to calculate the total cross section for quark pair production. In this final project, we will analyze the first corrections to this formula due to the strong interactions.\n\nLet us represent the strong interactions by the following simple model: Introduce a new massless vector particle, the gluon, which couples universally to quarks:\n\n\\Delta H=\\int d^{3} x g \\bar{\\psi}_{f i} \\gamma^{\\mu} \\psi_{f i} B_{\\mu}\n\n\nHere $f$ labels the type (\"flavor\") of the quark ( $u, d, s, c$, etc.) and $i=1,2,3$ labels the color. The strong coupling constant $g$ is independent of flavor and color. The electromagnetic coupling of quarks depends on the flavor, since the $u$ and $c$ quarks have charge $Q_{f}=+2 / 3$ while the $d$ and $s$ quarks have charge $Q_{f}=-1 / 3$. By analogy to $\\alpha$, let us define\n\n\\alpha_{g}=\\frac{g^{2}}{4 \\pi}\n\n\nIn this exercise, we will compute the radiative corrections to quark pair production proportional to $\\alpha_{g}$.\n\nThis model of the strong interactions of quarks does not quite agree with the currently accepted theory of the strong interactions, quantum chromodynamics (QCD). However, all of the results that we will derive here are also\ncorrect in QCD with the replacement\n$$\n\\alpha_{g} \\rightarrow \\frac{4}{3} \\alpha_{s} .\n$$\n\nThroughout this exercise, you may ignore the masses of quarks. You may also ignore the mass of the electron, and average over electron and positron polarizations. To control infrared divergences, it will be necessary to assume that the gluons have a small nonzero mass $\\mu$, which can be taken to zero only at the end of the calculation. However, it is consistent to sum over polarization states of this massive boson by the replacement:\n$$\n\\sum \\epsilon^{\\mu} \\epsilon^{\\nu *} \\rightarrow-g^{\\mu \\nu}\n$$\nthis also implies that we may use the propagator\n$$\n\\widehat{B^{\\mu} B^{\\nu}}=\\frac{-i g^{\\mu \\nu}}{k^{2}-\\mu^{2}+i \\epsilon}\n$$ In the analysis of the 3-body final state process $e^{+} e^{-} \\rightarrow \\bar{q} q g$, the total 4-momentum is $q$. The final quark (4-momentum $k_1$) and antiquark (4-momentum $k_2$) are massless, while the gluon (4-momentum $k_3$) has mass $\\mu$. Dimensionless energy fractions are defined as $x_i = \\frac{2 k_i \\cdot q}{q^{2}}$. The physical integration region for $x_1$ and $x_2$ is bounded. One of these boundaries corresponds to the kinematic configuration where the 3-momenta of the quark ($\\mathbf{k}_1$) and the antiquark ($\\mathbf{k}_2$) are parallel. Determine the equation for this specific boundary in terms of $x_1$, $x_2$, $\\mu^2$, and $q^2$ (the square of the total 4-momentum). You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$e^{+}$": "positron", + "$e^{-}$": "electron", + "$\\bar{q}$": "antiquark", + "$q$": "quark or total 4-momentum", + "$g$": "gluon", + "$k_1$": "4-momentum of the final quark", + "$k_2$": "4-momentum of the final antiquark", + "$k_3$": "4-momentum of the gluon", + "$\\mu$": "mass of the gluon", + "$x_1$": "dimensionless energy fraction for the quark", + "$x_2$": "dimensionless energy fraction for the antiquark", + "$x_3$": "dimensionless energy fraction for the gluon", + "$\\mathbf{k}_1$": "3-momentum of the final quark", + "$\\mathbf{k}_2$": "3-momentum of the final antiquark", + "$q^2$": "square of the total 4-momentum", + "$E_1$": "energy of the final quark", + "$E_2$": "energy of the final antiquark", + "$E_3$": "energy of the gluon" + }, + "chapter": "Final Project I", + "section": "Radiation of Gluon Jets" + }, + { + "id": 244, + "topic": "Others", + "question": "Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory with masses set to zero,\n$$\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{\\lambda}{4!} \\phi^{4}+\\bar{\\psi}(i \\not \\partial) \\psi-i g \\bar{\\psi} \\gamma^{5} \\psi \\phi,\n$$\ncompute the Callan-Symanzik $\\beta$ function for $g$:\n$$\n\\beta_{g}(\\lambda, g),\n$$\nto leading order in coupling constants, assuming that $\\lambda$ and $g^{2}$ are of the same order.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathcal{L}$": "Lagrangian", + "$\\phi$": "scalar field", + "$\\lambda$": "quartic coupling constant", + "$\\bar{\\psi}$": "Dirac adjoint spinor field", + "$\\psi$": "spinor field", + "$g$": "Yukawa coupling constant", + "$\\gamma^{5}$": "gamma matrix", + "$\\beta$": "beta function", + "$\\beta_{g}$": "beta function for coupling constant g", + "$\\delta_{\\psi}$": "wave function renormalization constant for the spinor field", + "$\\delta_{\\phi}$": "wave function renormalization constant for the scalar field", + "$\\delta_{g}$": "coupling constant renormalization", + "$\\Lambda$": "UV cutoff", + "$M$": "renormalization scale" + }, + "chapter": "The Renormalization Group", + "section": "Beta Function in Yukawa Theory" + }, + { + "id": 245, + "topic": "Others", + "question": "Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory with masses set to zero,\n$$\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{\\lambda}{4!} \\phi^{4}+\\bar{\\psi}(i \\not \\partial) \\psi-i g \\bar{\\psi} \\gamma^{5} \\psi \\phi,\n$$\ncompute the Callan-Symanzik $\\beta$ function for $\\lambda$:\n\n\\beta_{\\lambda}(\\lambda, g),\n\nto leading order in coupling constants, assuming that $\\lambda$ and $g^{2}$ are of the same order.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\beta$": "beta function", + "$\\lambda$": "coupling constant in the theory", + "$g$": "coupling constant in the theory", + "$\\phi$": "scalar field", + "$\\psi$": "fermionic field", + "$\\gamma^{5}$": "gamma matrix with pseudoscalar property", + "$\\delta_{\\phi}$": "field renormalization counterterm for the scalar field", + "$\\delta_{\\lambda}$": "renormalization counterterm for the coupling constant", + "$M$": "renormalization scale", + "$\\Lambda$": "ultraviolet cutoff" + }, + "chapter": "The Renormalization Group", + "section": "Beta Function in Yukawa Theory" + }, + { + "id": 246, + "topic": "Others", + "question": "Asymptotic symmetry. Consider the following Lagrangian, with two scalar fields $\\phi_{1}$ and $\\phi_{2}$ :\n\n\\mathcal{L}=\\frac{1}{2}((\\partial_{\\mu} \\phi_{1})^{2}+(\\partial_{\\mu} \\phi_{2})^{2})-\\frac{\\lambda}{4!}(\\phi_{1}^{4}+\\phi_{2}^{4})-\\frac{2 \\rho}{4!}(\\phi_{1}^{2} \\phi_{2}^{2}) .\n\n\nNotice that, for the special value $\\rho=\\lambda$, this Lagrangian has an $O(2)$ invariance rotating the two fields into one another. For a theory with two scalar fields $\\phi_1$ and $\\phi_2$ described by the Lagrangian:\n$$\n\\mathcal{L}=\\frac{1}{2}((\\partial_{\\mu} \\phi_{1})^{2}+(\\partial_{\\mu} \\phi_{2})^{2})-\\frac{\\lambda}{4!}(\\phi_{1}^{4}+\\phi_{2}^{4})-\\frac{2\\rho}{4!} \\phi_{1}^{2} \\phi_{2}^{2}\n$$\n\nWorking in four dimensions, what is the $\\beta$ function for the coupling constant $\\lambda$, denoted as $\\beta_{\\lambda}$, to leading order in the coupling constants?", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\phi_1$": "first scalar field", + "$\\phi_2$": "second scalar field", + "$\\lambda$": "coupling constant for self-interactions", + "$\\rho$": "coupling constant for interaction between fields", + "$\\beta$": "beta function symbol", + "$\\beta_{\\lambda}$": "beta function for the coupling constant \\lambda", + "$\\beta_{\\rho}$": "beta function for the coupling constant \\rho", + "$\\mu$": "renormalization scale", + "$\\delta_{\\lambda}$": "variation in coupling constant \\lambda", + "$\\delta_{\\rho}$": "variation in coupling constant \\rho" + }, + "chapter": "The Renormalization Group", + "section": "Asymptotic Symmetry" + }, + { + "id": 247, + "topic": "Others", + "question": "State the leading term in $\\gamma(\\lambda)$ for $\\phi^{4}$ theory.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\gamma$": "anomalous dimension", + "$\\lambda$": "coupling constant", + "$\\phi$": "field" + }, + "chapter": "Critical Exponents and Scalar Field Theory", + "section": "The exponent $\\eta$" + }, + { + "id": 248, + "topic": "Others", + "question": "Compute the anomalous dimension \\(\\gamma\\) in an \\(O(N)\\)-symmetric \\(\\phi^{4}\\) theory for \\(N = 3\\) and coupling constant \\(\\lambda = 0.5\\), using the formula\n\\[\\gamma = (N+2)\\,\\frac{\\lambda^{2}}{(4\\pi)^{4}}.\\]", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\gamma$": "anomalous dimension", + "$N$": "symmetry index of O(N) theory", + "$\\phi$": "field in \\(\\phi^{4}\\) theory", + "$\\lambda$": "coupling constant", + "$\\pi$": "mathematical constant pi" + }, + "chapter": "Critical Exponents and Scalar Field Theory", + "section": "The exponent $\\eta$" + }, + { + "id": 249, + "topic": "Others", + "question": "Brute-force computations in $\\boldsymbol{S U ( 3 )}$. The standard basis for the fundamental representation of $S U(3)$ is\n\n\\begin{array}{rlrl}\nt^{1} & =\\frac{1}{2}(\\begin{array}{lll}\n0 & 1 & 0 \\\\\n1 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}), \\quad t^{2}=\\frac{1}{2}(\\begin{array}{ccc}\n0 & -i & 0 \\\\\ni & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}), \\quad t^{3}=\\frac{1}{2}(\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & -1 & 0 \\\\\n0 & 0 & 0\n\\end{array}), \\\\\nt^{4} & =\\frac{1}{2}(\\begin{array}{lll}\n0 & 0 & 1 \\\\\n0 & 0 & 0 \\\\\n1 & 0 & 0\n\\end{array}), & t^{5}=\\frac{1}{2}(\\begin{array}{ccc}\n0 & 0 & -i \\\\\n0 & 0 & 0 \\\\\ni & 0 & 0\n\\end{array}), \\\\\nt^{6} & =\\frac{1}{2}(\\begin{array}{lll}\n0 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & 1 & 0\n\\end{array}), \\quad t^{7}=\\frac{1}{2}(\\begin{array}{ccc}\n0 & 0 & 0 \\\\\n0 & 0 & -i \\\\\n0 & i & 0\n\\end{array}), \\quad t^{8}=\\frac{1}{2 \\sqrt{3}}(\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & -2\n\\end{array}) .\n\\end{array} Write down the dimension $d$ of $S U(N)$ group. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$d$": "dimension of the $SU(N)$ group", + "$N$": "parameter of the $SU(N)$ group" + }, + "chapter": "Non-Abelian Gauge Invariance", + "section": "Brute-force computations in $S U(3)$" + }, + { + "id": 250, + "topic": "Others", + "question": "Matrix element for proton decay. Some advanced theories of particle interactions include heavy particles $X$ whose couplings violate the conservation of baryon number. Integrating out these particles produces an effective interaction that allows the proton to decay to a positron and a photon or a pion. This effective interaction is most easily written using the definite-helicity components of the quark and electron fields: If $u_{L}, d_{L}, u_{R}, e_{R}$ are two-component spinors, then this effective interaction is\n\n\\Delta \\mathcal{L}=\\frac{2}{m_{X}^{2}} \\epsilon_{a b c} \\epsilon^{\\alpha \\beta} \\epsilon^{\\gamma \\delta} e_{R \\alpha} u_{R a \\beta} u_{L b \\gamma} d_{L c \\delta} .\n\n\nA typical value for the mass of the $X$ boson is $m_{X}=10^{16} \\mathrm{GeV}$. Estimate, in order of magnitude, the value of the proton lifetime if the proton is allowed to decay through this interaction.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$p$": "proton", + "$e^{+}$": "positron", + "$\\pi^{0}$": "neutral pion", + "$\\mathcal{O}_{X}$": "operator for the decay process", + "$m_{X}$": "scale of the higher dimensional operator", + "$i$": "color index for quark", + "$j$": "color index for quark", + "$k$": "color index for quark", + "$\\alpha$": "spinor index", + "$\\beta$": "spinor index", + "$\\gamma$": "spinor index", + "$\\delta$": "spinor index", + "$e_{R \\alpha}$": "right-handed electron field with spinor index", + "$u_{R i \\beta}$": "right-handed up-quark field with color and spinor indices", + "$u_{L j \\gamma}$": "left-handed up-quark field with color and spinor indices", + "$d_{L k \\delta}$": "left-handed down-quark field with color and spinor indices", + "$\\mathcal{M}$": "amplitude of the decay process", + "$m_{p}$": "proton mass", + "$\\Gamma$": "decay width" + }, + "chapter": "Operator Products and Effective Vertices", + "section": "Matrix element for proton decay" + }, + { + "id": 251, + "topic": "Others", + "question": "A model with two Higgs fields. Assume that the two Higgs fields couple to quarks by the set of fundamental couplings\n\n\\mathcal{L}_{m}=-\\lambda_{d}^{i j} \\bar{Q}_{L}^{i} \\cdot \\phi_{1} d_{R}^{j}-\\lambda_{u}^{i j} \\epsilon^{a b} \\bar{Q}_{L a}^{i} \\phi_{2 b}^{\\dagger} u_{R}^{j}+\\text { h.c. }\n\n\nFind the couplings of the physical charged Higgs boson of part (c) to the mass eigenstates of quarks. These couplings depend only on the values of the quark masses and $\\tan \\beta$ and on the elements of the CKM matrix.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\lambda_{d}^{ij}$": "Yukawa coupling matrix for down-type quarks", + "$\\lambda_{u}^{ij}$": "Yukawa coupling matrix for up-type quarks", + "$\\bar{Q}_{L}^{i}$": "left-handed quark doublet", + "$\\phi_{1}$": "Higgs field component coupling to down-type quarks", + "$\\phi_{2}$": "Higgs field component coupling to up-type quarks", + "$d_{R}^{j}$": "right-handed down-type quark", + "$u_{R}^{j}$": "right-handed up-type quark", + "$\\tan \\beta$": "ratio of the two Higgs doublet vacuum expectation values", + "$\\phi^{+}$": "physical charged Higgs boson", + "$v_{1}$": "vacuum expectation value associated with $\\phi_{1}$", + "$v_{2}$": "vacuum expectation value associated with $\\phi_{2}$", + "$v$": "total vacuum expectation value", + "$U_{u}$": "unitary matrix for left-handed up-type quark fields", + "$U_{d}$": "unitary matrix for left-handed down-type quark fields", + "$W_{u}$": "unitary matrix for right-handed up-type quark fields", + "$W_{d}$": "unitary matrix for right-handed down-type quark fields", + "$D_{d}$": "diagonal mass matrix for down-type quarks", + "$D_{u}$": "diagonal mass matrix for up-type quarks", + "$V_{\\mathrm{CKM}}$": "Cabibbo-Kobayashi-Maskawa (CKM) matrix", + "$m_{u}$": "mass matrix for up-type quarks", + "$m_{d}$": "mass matrix for down-type quarks", + "$\\pi_{1}^{+}$": "charged Higgs boson component", + "$\\pi_{2}^{+}$": "charged Higgs boson component" + }, + "chapter": "Gauge Theories with Spontaneous Symmetry Breaking", + "section": "A model with two Higgs fields" + }, + { + "id": 252, + "topic": "Others", + "question": "Dependence of radiative corrections on the Higgs boson mass. In Feynman-'t Hooft gauge, compute the dependence of the vacuum polarization amplitude $\\Pi_{WW}(q^2)$ (specifically the part proportional to $g^{\\mu\\nu}$) on the Higgs boson mass $m_h$. Consider the diagrams involving the Higgs boson, work in the large $m_h$ limit, use dimensional regularization with $M$ as the subtraction scale, and fix the subtraction point at $M^2=m_W^2$. Provide the derivation steps and the final expression for $\\Pi_{WW}(q^2)$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\Pi_{WW}(q^2)$": "vacuum polarization amplitude for W bosons", + "$\\mu$": "index representing a spacetime dimension", + "$\\nu$": "index representing a spacetime dimension", + "$m_h$": "Higgs boson mass", + "$M$": "subtraction scale", + "$m_W$": "W boson mass", + "$g$": "coupling constant", + "$q^2$": "momentum transfer squared", + "$E$": "measure of divergence in dimensional regularization" + }, + "chapter": "Quantization of Spontaneously Broken Gauge Theories", + "section": "Dependence of radiative corrections on the Higgs boson mass" + }, + { + "id": 253, + "topic": "Others", + "question": "Dependence of radiative corrections on the Higgs boson mass. In Feynman-'t Hooft gauge, compute the dependence of the vacuum polarization amplitude $\\Pi_{ZZ}(q^2)$ (specifically the part proportional to $g^{\\mu\\nu}$) on the Higgs boson mass $m_h$. Consider the diagrams involving the Higgs boson, work in the large $m_h$ limit, use dimensional regularization with $M$ as the subtraction scale, and fix the subtraction point at $M^2=m_Z^2$. Provide the final expression for $\\Pi_{ZZ}(q^2)$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\Pi_{ZZ}(q^2)$": "vacuum polarization amplitude for Z bosons", + "$g$": "coupling constant", + "$\\theta_w$": "Weinberg angle", + "$m_h$": "Higgs boson mass", + "$m_Z$": "Z boson mass", + "$q^2$": "momentum transfer squared", + "$M$": "subtraction scale" + }, + "chapter": "Quantization of Spontaneously Broken Gauge Theories", + "section": "Dependence of radiative corrections on the Higgs boson mass" + }, + { + "id": 254, + "topic": "Theoretical Foundations", + "question": "Consider the wave equation $\\nabla^{2} u+k^{2} n(r)^{2} u=0$ with slowly varying $n(r)$. If we introduce the eikonal function $S(r)$ by substituting $u = \\mathrm{e}^{\\frac{2 \\pi i}{\\lambda} S(r)}$ (where $\\lambda=2 \\pi / k$) into the wave equation, what is the resulting differential equation for $S(r)$ before any series expansion of $S(r)$ is performed (this is known as the Riccati equation in this context)? You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$u$": "wave function", + "$k$": "wave number", + "$n(r)$": "refractive index as a function of position", + "$S(r)$": "eikonal function as a function of position", + "$\\lambda$": "wavelength", + "$\\hbar$": "reduced Planck's constant", + "$p$": "momentum", + "$m$": "mass", + "$E$": "energy", + "$V(r)$": "potential energy as a function of position" + }, + "chapter": "One-Body Problems without Spin", + "section": "OneDimensional Problems" + }, + { + "id": 255, + "topic": "Theoretical Foundations", + "question": "The tensor force between two particles 1 and 2 of spin $1/2$ is associated with the operator $T_{12}$ given by:\n\n T_{12}=\\frac{(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{r})(\\boldsymbol{\\sigma}_{2} \\cdot \\boldsymbol{r})}{r^{2}}-\\frac{1}{3}(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2})\n\nwhere $\\boldsymbol{\\sigma}_{1}$ and $\\boldsymbol{\\sigma}_{2}$ are the Pauli spin matrices for particle 1 and 2 respectively, and $\\boldsymbol{r}$ is the relative position vector between the particles.\nIf $\\chi_{0,0}$ represents the spin singlet state (an eigenfunction of total spin $S=0$, defined as $\\chi_{0,0}=\\frac{1}{\\sqrt{2}}(\\alpha_{1} \\beta_{2}-\\beta_{1} \\alpha_{2})$) for the two-particle system, calculate the result of applying the operator $T_{12}$ to $\\chi_{0,0}$. In other words, find $T_{12} \\chi_{0,0}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$T_{12}$": "tensor force operator between two particles", + "$\\boldsymbol{\\sigma}_{1}$": "Pauli spin matrix for particle 1", + "$\\boldsymbol{\\sigma}_{2}$": "Pauli spin matrix for particle 2", + "$\\boldsymbol{r}$": "relative position vector between the particles", + "$r$": "magnitude of the relative position vector", + "$\\chi_{0,0}$": "spin singlet state of total spin S=0", + "$\\alpha_{1}$": "spin-up state of particle 1", + "$\\beta_{1}$": "spin-down state of particle 1", + "$\\alpha_{2}$": "spin-up state of particle 2", + "$\\beta_{2}$": "spin-down state of particle 2", + "$\\chi_{1,1}$": "triplet state with total spin projection m_s=1", + "$\\chi_{1,0}$": "triplet state with total spin projection m_s=0", + "$\\chi_{1,-1}$": "triplet state with total spin projection m_s=-1", + "$\\vartheta$": "polar angle in spherical coordinates", + "$\\varphi$": "azimuthal angle in spherical coordinates", + "$Y_{2,0}$": "spherical harmonic function for l=2, m=0", + "$Y_{2,1}$": "spherical harmonic function for l=2, m=1", + "$Y_{2,-1}$": "spherical harmonic function for l=2, m=-1", + "$Y_{2,2}$": "spherical harmonic function for l=2, m=2", + "$Y_{2,-2}$": "spherical harmonic function for l=2, m=-2" + }, + "chapter": "Particles with Spin", + "section": "OneDimensional Problems" + }, + { + "id": 256, + "topic": "Theoretical Foundations", + "question": "In a neutral helium atom, one electron is in the $1s$ ground state and the other is in the $2p$ excited state ($n=2, l=1$). Using a theoretical model based on hydrogen-like wave functions with screening of one nuclear charge by the $1s$ electron, calculate the ionization energy (in eV) for the $2p$ electron if the atom is in the parahelium state (give a number and keep three decimal places).", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$1s$": "ground state orbital for helium electron", + "$2p$": "excited state orbital for helium electron", + "$n$": "principal quantum number", + "$l$": "azimuthal quantum number", + "$E_{1}$": "energy of the electron in the ground state 1s", + "$E_{n}$": "energy of the electron in the excited state np", + "$R_{n l}$": "normalized radial wave function for quantum numbers n and l", + "$r_{1}$": "distance of first electron from nucleus", + "$r_{2}$": "distance of second electron from nucleus", + "$r_{12}$": "distance between two electrons", + "$\\varepsilon$": "parameter for parahelium or orthohelium state", + "$\\psi$": "two-electron wave function", + "$\\mathscr{C}$": "classical integral for electron-electron interaction", + "$\\mathscr{E}$": "exchange integral for electron-electron interaction", + "$\\Omega_{1}$": "solid angle for first electron", + "$\\Omega_{2}$": "solid angle for second electron", + "$P_{\\lambda}$": "Legendre polynomial of degree lambda" + }, + "chapter": "Many-Body Problems", + "section": "OneDimensional Problems" + }, + { + "id": 257, + "topic": "Theoretical Foundations", + "question": "The function\n\\begin{equation*}\n\\tilde{\\varphi}(x)=\\frac{1}{(1+\\alpha x)^{2}} \\tag{176.1}\n\\end{equation*}\nwith a suitable value of $\\alpha$, independent of $Z$, may be used as a fair approximation to the Thomas-Fermi function $\\varphi_{0}(x)$ for a neutral atom. The constant $\\alpha$ shall be determined in such a way as to permit exact normalization of $\\tilde{\\varphi}$. Hint: the answer is a number.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\tilde{\\varphi}$": "approximated function to the Thomas-Fermi function", + "$x$": "variable in the function", + "$\\alpha$": "constant independent of $Z$ for approximation", + "$Z$": "atomic number", + "$\\varphi_{0}$": "Thomas-Fermi function for a neutral atom", + "$n(r)$": "electron density", + "$r$": "radial distance in atomic units", + "$V(r)$": "atomic potential", + "$a$": "constant related to the atomic number, defined as $a=0.88534 Z^{-\\frac{1}{3}}$", + "$y$": "integration variable, $y=\\alpha x$" + }, + "chapter": "Many-Body Problems", + "section": "OneDimensional Problems" + }, + { + "id": 258, + "topic": "Theoretical Foundations", + "question": "Calculate the numerical value for the mean lifetime (in seconds) of the $2P$ state in a hydrogen atom, which decays to the $1S$ state by emission of a photon. This requires first determining the total transition probability $P$ for an electron from a higher $P$ state to a lower $S$ state (summed over all photon directions and polarizations), and then specializing this for the hydrogen $2P \\rightarrow 1S$ transition.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$2P$": "2P state in a hydrogen atom", + "$1S$": "1S state in a hydrogen atom", + "$P$": "total transition probability", + "$m$": "magnetic quantum number", + "$e$": "elementary charge", + "$\\omega$": "angular frequency of light", + "$\\hbar$": "reduced Planck's constant", + "$c$": "speed of light in vacuum", + "$R$": "radial transition matrix element", + "$\\Theta$": "polar angle", + "$\\Phi$": "azimuthal angle", + "$\\tau$": "mean lifetime of the 2P state" + }, + "chapter": "Radiation Theory", + "section": "OneDimensional Problems" + }, + { + "id": 259, + "topic": "Theoretical Foundations", + "question": "To compare the intensities of emission of the two first Lyman lines of atomic hydrogen, Ly $\\alpha$ and Ly $\\beta$. Numerically calculate $I_\\alpha/I_\\beta$ You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$I_{\\alpha}$": "intensity of Lyman alpha line", + "$I_{\\beta}$": "intensity of Lyman beta line", + "$E_{\\alpha}$": "energy difference for Lyman alpha transition", + "$E_{\\beta}$": "energy difference for Lyman beta transition" + }, + "chapter": "Radiation Theory", + "section": "OneDimensional Problems" + }, + { + "id": 260, + "topic": "Others", + "question": "Consider a scalar field theory with interaction Lagrangian $\\mathcal{L}_{\\mathrm{I}}=-\\frac{g}{3!} \\phi^{3}-\\frac{\\lambda}{4!} \\phi^{4}$. Derive the identity that relates the number of loops ($n_L$), internal lines ($n_I$), trivalent vertices ($n_3$), and tetravalent vertices ($n_4$). You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\mathcal{L}_{\\mathrm{I}}$": "interaction Lagrangian", + "$g$": "coupling constant for the $\\phi^3$ interaction", + "$\\lambda$": "coupling constant for the $\\phi^4$ interaction", + "$\\phi$": "scalar field", + "$n_{L}$": "number of loops", + "$n_{I}$": "number of internal lines", + "$n_{3}$": "number of trivalent vertices", + "$n_{4}$": "number of tetravalent vertices" + }, + "chapter": "Perturbation Theory", + "section": "Perturbation Theory" + }, + { + "id": 261, + "topic": "Others", + "question": "For a massless spin-one particle with four-momentum $p$, its physical helicity polarization vectors are denoted by $\\epsilon_{\\pm}^{\\mu}(\\mathbf{p})$. Under a Lorentz transformation $\\Lambda$ (which transforms the momentum $p$ to $\\Lambda p$), these polarization vectors transform according to the following relation derived using little group properties:\n$$[\\Lambda^{-1}]_{\\nu}^{\\mu} \\epsilon_{ \\pm}^{v}(\\Lambda \\mathbf{p}) = X$$ \nWhat is the expression for $X$ in terms of the original polarization vector $\\epsilon_{ \\pm}^{\\mu}(\\mathbf{p})$, the original four-momentum $p^{\\mu}$, a phase factor $e^{\\mp i \\theta}$ (where $\\theta$ is the Wigner rotation angle), and coefficients $\\beta_{ \\pm}$ which depend on the parameters of the Lorentz transformation and the reference momentum?", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$p$": "four-momentum", + "$\\epsilon_{\\pm}^{\\mu}(\\mathbf{p})$": "physical helicity polarization vector for momentum $p$", + "$\\theta$": "Wigner rotation angle", + "$\\beta_{\\pm}$": "coefficients depending on Lorentz transformation parameters", + "$\\epsilon_{\\pm}^{\\mu}(\\mathbf{q})$": "helicity polarization vector for reference momentum $q$", + "$\\omega$": "reference energy component", + "$q^{\\mu}$": "reference four-momentum", + "$\\mathcal{M}_{\\mu}$": "matrix element component", + "$\\Lambda$": "Lorentz transformation", + "$R$": "element of the little group", + "$\\mathbf{p}$": "momentum vector", + "$\\mathcal{R}(\\widehat{\\mathbf{p}})$": "spatial rotation aligning axis 3 with vector $\\widehat{\\mathbf{p}}$" + }, + "chapter": "Quantum Electrodynamics", + "section": "Quantum Electrodynamics" + }, + { + "id": 262, + "topic": "Others", + "question": "Give the expression of the one-loop self-energy in a $\\phi^{4}$ theory in the Matsubara formalism. Calculate it in the limit $\\beta \\mathrm{m} \\ll 1$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\Sigma$": "self-energy", + "$\\lambda$": "coupling constant", + "$T$": "temperature", + "$n$": "index for summation", + "$p$": "momentum", + "$m$": "mass", + "$E_p$": "energy associated with momentum $p$", + "$n_B$": "Bose-Einstein distribution", + "$\\Lambda$": "momentum cutoff" + }, + "chapter": "Functional Quantization", + "section": "Functional Quantization" + }, + { + "id": 263, + "topic": "Others", + "question": "Consider two Grassmann variables $\\theta_\\pm$. For the operator $\\tau_3 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{-} \\frac{\\partial}{\\partial \\theta_{-}})$, find two linearly independent eigenfunctions corresponding to the eigenvalue $0$. Express them using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize constant terms or leading $\\theta$ terms to 1. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\tau_3$": "operator", + "$\\theta_{+}$": "theta plus", + "$\\theta_{-}$": "theta minus", + "$\\lambda$": "eigenvalue", + "$a$": "coefficient for constant term", + "$b$": "coefficient for theta plus term", + "$c$": "coefficient for theta minus term", + "$d$": "coefficient for theta plus theta minus term" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 264, + "topic": "Others", + "question": "Consider two Grassmann variables $\\theta_\\pm$. Consider the operator $\\tau_1 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$ acting on functions of two Grassmann variables $\\theta_{\\pm}$. Find all distinct eigenvalues of $\\tau_1$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\tau_1$": "operator", + "$\\theta_{+}$": "Grassmann variable (positive)", + "$\\theta_{-}$": "Grassmann variable (negative)", + "$\\theta_{\\pm}$": "Grassmann variables (positive/negative)", + "$a$": "constant term in function of Grassmann variables", + "$b$": "coefficient of $\\theta_{+}$ in function", + "$c$": "coefficient of $\\theta_{-}$ in function", + "$d$": "coefficient of $\\theta_{+}\\theta_{-}$ in function", + "$\\lambda$": "eigenvalue" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 265, + "topic": "Others", + "question": "Consider two Grassmann variables $\\theta_\\pm$. Consider the operator $\\tau_2 \\equiv \\frac{i}{2}(\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}})$ acting on functions of two Grassmann variables $\\theta_{\\pm}$. Find all distinct eigenvalues of $\\tau_2$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\tau_2$": "operator", + "$\\theta_{-}$": "Grassmann variable (negative)", + "$\\theta_{+}$": "Grassmann variable (positive)", + "$\\theta_{\\pm}$": "Grassmann variables (positive and negative)" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 266, + "topic": "Others", + "question": "Given the operators $\\tau_{1} \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$ and $\\tau_{2} \\equiv \\frac{i}{2}(\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}})$, calculate the action of the operator $(\\tau_1 - i\\tau_2)$ on the Grassmann variable $\\theta_-$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\tau_{1}$": "operator 1", + "$\\tau_{2}$": "operator 2", + "$\\theta_{+}$": "Grassmann variable (positive)", + "$\\theta_{-}$": "Grassmann variable (negative)" + }, + "chapter": "Path Integrals for Fermions and Photons", + "section": "Path Integrals for Fermions and Photons" + }, + { + "id": 267, + "topic": "Others", + "question": "Consider a non-Abelian gauge theory with the usual $\\mathfrak{s u}(N)$ gauge fields, $n_{s}$ complex scalar fields in the adjoint representation and $n_{f}$ Dirac fermions in the adjoint representation. Calculate the expression for $\\frac{1}{g_{\\mathrm{r}}^{2}(\\mu)}$ for this theory, analogous to the standard one-loop running coupling constant equation. The constants for fields in the adjoint representation are given as $\\mathrm{c}_{\\mathrm{adj}, 0}=\\frac{\\mathrm{N}}{3(4 \\pi)^{2}}$ for gauge fields/ghosts, $\\mathrm{c}_{\\mathrm{adj}, 1 / 2}=-\\frac{8 \\mathrm{~N}}{3(4 \\pi)^{2}}$ for Dirac fermions, and we can infer the scalar contribution from the context (scalars have bosonic statistics and contribute with an opposite sign to ghosts or fermions regarding their statistical nature's impact on the beta function coefficient).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathfrak{s u}(N)$": "special unitary Lie algebra of degree N", + "$n_{s}$": "number of complex scalar fields", + "$n_{f}$": "number of Dirac fermions", + "$g_{\\mathrm{r}}$": "renormalized coupling constant", + "$\\mu$": "energy scale", + "$\\mathrm{c}_{\\mathrm{adj}, 0}$": "coefficient for gauge fields and ghosts in adjoint representation", + "$N$": "dimension of the gauge group in adjoint representation", + "$\\mathrm{c}_{\\mathrm{adj}, 1 / 2}$": "coefficient for Dirac fermions in adjoint representation", + "$g_{\\mathrm{b}}$": "bare coupling constant", + "$\\kappa$": "reference scale", + "$n_{\\mathrm{s}}$": "number of scalars in calculation", + "$\\mathrm{n}_{s}$": "number of scalars", + "$n_{\\mathrm{f}}$": "number of Dirac fermions in calculation" + }, + "chapter": "Renormalization of Gauge Theories", + "section": "Renormalization of Gauge Theories" + }, + { + "id": 268, + "topic": "Others", + "question": "For a non-Abelian gauge theory with $\\mathfrak{s u}(N)$ gauge fields, $n_{s}$ complex scalar fields and $n_{f}$ Dirac fermions all in the adjoint representation, the one-loop running of the inverse squared coupling is given by $\\frac{1}{g_{\\mathrm{r}}^{2}(\\mu)}=\\frac{1}{\\mathrm{~g}_{\\mathrm{b}}^{2}}+\\frac{\\mathrm{N}}{3(4 \\pi)^{2}}(11-4 n_{\\mathrm{f}}-\\mathrm{n}_{\\mathrm{s}}) \\ln \\frac{\\mu^{2}}{\\kappa^{2}}$. Determine the condition on $n_s$ and $n_f$ for the gauge coupling to not be running at one loop. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\mathfrak{s u}(N)$": "special unitary group in N dimensions", + "$n_{s}$": "number of complex scalar fields", + "$n_{f}$": "number of Dirac fermions", + "$\\mu$": "renormalization scale", + "$\\kappa$": "arbitrary scale in field theory", + "$g_{\\mathrm{r}}$": "renormalized gauge coupling", + "$g_{\\mathrm{b}}$": "bare gauge coupling", + "$N$": "dimension of gauge group SU(N)" + }, + "chapter": "Renormalization of Gauge Theories", + "section": "Renormalization of Gauge Theories" + }, + { + "id": 269, + "topic": "Others", + "question": "Carry out explicitly the calculation of the functions $A$ and $B$ in \n\n$$B(q^{2})=-i\\mathrm{D}e\\int\\frac{d^{\\mathrm{D}}l}{(2\\pi)^{\\mathrm{D}}}\\int_{0}^{1}dx\\frac{2\\Delta(x)}{(l^{2}+\\Delta(x))^{2}}.$$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$A$": "function A related to the calculation", + "$B$": "function B related to the calculation", + "$q$": "momentum variable", + "$\\mathrm{D}$": "dimension of spacetime", + "$\\Delta(x)$": "function dependent on variable x", + "$e$": "mathematical constant in the expressions", + "$\\ell$": "integration variable" + }, + "chapter": "Quantum Anomalies", + "section": "Quantum Anomalies" + }, + { + "id": 270, + "topic": "Others", + "question": "Carry out explicitly the calculation of the functions $A$ and $B$ in \n$$A(q^{2})=-i\\mathrm{D}e\\int\\frac{d^{\\mathrm{D}}l}{(2\\pi)^{D}}\\int_{0}^{1}dx\\frac{\\Delta(x)+(\\frac{2}{\\mathrm{D}}-1)l^{2}}{(l^{2}+\\Delta(x))^{2}}, $$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$A$": "function A", + "$B$": "function B", + "$q$": "momentum", + "$e$": "mathematical constant e", + "$\\mathrm{D}$": "dimension in dimensional regularization", + "$x$": "integration variable" + }, + "chapter": "Quantum Anomalies", + "section": "Quantum Anomalies" + }, + { + "id": 271, + "topic": "Others", + "question": "For the time independent, $z^{3}$-dependent electrical field $E_{3}(z^{3}) \\equiv \\frac{E}{\\cosh ^{2}(k z^{3})}$ with gauge potential $A^{4}=-i \\frac{E}{k} \\tanh (k z^{3})$ (other $A^i=0$), the equations of motion for the stationary solutions $z^3(u)$ and $z^4(u)$, (assuming $z^1, z^2$ are constant) are given in first-order form. What are these equations, expressed in terms of $v = \\sqrt{(\\dot{z}^3)^2+(\\dot{z}^4)^2}$ and $\\gamma \\equiv mk/(eE)$? You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$z^3$": "coordinate in the third spatial dimension", + "$z^4$": "coordinate in the fourth spatial dimension", + "$E_3$": "electrical field component in the third dimension", + "$E$": "electric field magnitude", + "$k$": "wave number", + "$A^4$": "gauge potential in the fourth dimension", + "$A^i$": "gauge potential component", + "$v$": "velocity magnitude", + "$\\gamma$": "dimensionless parameter related to mass, wave number, charge, and electric field", + "$m$": "mass", + "$e$": "electric charge" + }, + "chapter": "Worldline Formalism", + "section": "Worldline Formalism" + }, + { + "id": 272, + "topic": "Others", + "question": "The Polyakov loop is defined as $\\mathrm{L}(x) \\equiv \\mathrm{N}^{-1} \\operatorname{tr} ( P \\exp \\int_{0}^{\\beta} d \\tau A^{0}(\\tau, x) )$. Under a center transformation, where the gauge transformation $\\Omega(\\tau,x)$ obeys $\\Omega(\\beta, x)=\\xi \\Omega(0, x)$ with $\\xi \\in \\mathbb{Z}_{N}$, how does $\\mathrm{L}(x)$ transform? Express the transformed Polyakov loop $\\mathrm{L}'(x)$ in terms of $\\mathrm{L}(x)$ and $\\xi$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\mathrm{L}(x)$": "Polyakov loop at position x", + "$\\mathrm{L}'(x)$": "transformed Polyakov loop at position x", + "$\\xi$": "N-th root of unity from the group \\mathbb{Z}_{N}", + "$N$": "number of colors or group rank in the context of SU(N)", + "$A^{0}(\\tau, x)$": "temporal component of the gauge field at position x and time τ" + }, + "chapter": "Quantum Field Theory at Finite Temperature", + "section": "Quantum Field Theory at Finite Temperature" + }, + { + "id": 273, + "topic": "Others", + "question": "A particle with mass $m$, constrained to move freely on a ring with radius $R$, with an added perturbation\n\\begin{equation*}\nH^{\\prime}=V(\\varphi)=\\left\\{\\begin{array}{ll}\nV_{1}, & -\\alpha<\\varphi<0 \\\\\nV_{2}, & 0<\\varphi<\\alpha \\\\\n0, & \\text { other angles }\n\\end{array} \\quad(\\alpha<\\pi)\\right.\n\\end{equation*}\n\nFind the first-order perturbation corrections for the three lowest energy levels. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$m$": "mass of the particle", + "$R$": "radius of the ring", + "$\\varphi$": "angle around the ring", + "$V_1$": "potential value for the angle range $-\\alpha < \\varphi < 0$", + "$V_2$": "potential value for the angle range $0 < \\varphi < \\alpha$", + "$\\alpha$": "angle constraint less than $\\pi$", + "$H_{nn}^{\\prime}$": "average value of perturbation for state $n$", + "$a$": "average potential value factor, defined as $\\frac{\\alpha}{2 \\pi}(V_{1}+V_{2})$", + "$E_0^{(0)}$": "ground state energy without perturbation", + "$E_0^{(1)}$": "first-order perturbation correction to the ground state energy", + "$H_{-1,1}^{\\prime}$": "perturbation matrix element for the first excited state", + "$b_1$": "perturbation factor for the first excited state", + "$E_1^{(1)}$": "first-order perturbation correction to the energy of the first excited state", + "$H_{-2.2}^{\\prime}$": "perturbation matrix element for the second excited state", + "$b_2$": "perturbation factor for the second excited state", + "$E_2^{(1)}$": "first-order perturbation correction to the energy of the second excited state" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 274, + "topic": "Others", + "question": "A small uniformly charged sphere acquires potential energy in an external electrostatic field\n$$\n\\begin{equation*}\nU(\\boldsymbol{r})=V(\\boldsymbol{r})+\\frac{1}{6} r_{0}^{2} \\nabla^{2} V(\\boldsymbol{r})+\\cdots \n\\end{equation*}\n$$\n\nwhere $r_{0}$ is the radius of the sphere, $\\boldsymbol{r}$ is the position of the sphere's center, and $V(\\boldsymbol{r})$ is the electrostatic potential energy acquired by the small charged sphere when approximated as a point charge. In a hydrogen atom, when the electron is treated as a point charge, the Coulomb potential energy between the electron and the nucleus is\n$$\n\\begin{equation*}\nV(\\boldsymbol{r})=-\\frac{e^{2}}{r} \n\\end{equation*}\n$$\n\nIf the electron is treated as a charged ($-e$) sphere, and $r_{0}=e^{2} / m_{e} c^{2}$ (classical electron radius) is used, the potential energy is modified by equation (1), treating the $r_{0}^{2}$ term as a perturbation. Find the perturbative correction for the 1s energy levels [equivalent to the Lamb shift].", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$U(\\boldsymbol{r})$": "potential energy of the sphere at position $\\boldsymbol{r}$", + "$V(\\boldsymbol{r})$": "electrostatic potential energy at position $\\boldsymbol{r}$", + "$r_0$": "radius of the sphere", + "$\\boldsymbol{r}$": "position of the sphere's center", + "$e$": "elementary charge", + "$m_e$": "electron mass", + "$c$": "speed of light in vacuum", + "$H^{\\prime}$": "perturbative correction term to the potential energy", + "$E^{(1)}$": "first-order perturbative correction to the energy level", + "$\\psi$": "wave function", + "$\\psi_{2 p}$": "wave function for the 2p state", + "$\\psi_{1 \\mathrm{~s}}$": "wave function for the 1s state", + "$a_0$": "Bohr radius", + "$\\hbar$": "reduced Planck's constant", + "$E_{1_{s}}^{(1)}$": "first-order energy correction for the 1s state", + "$\\alpha$": "fine structure constant", + "$m_{\\mathrm{e}}$": "electron rest mass" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 275, + "topic": "Others", + "question": "A small uniformly charged sphere acquires potential energy in an external electrostatic field\n$$\n\\begin{equation*}\nU(\\boldsymbol{r})=V(\\boldsymbol{r})+\\frac{1}{6} r_{0}^{2} \\nabla^{2} V(\\boldsymbol{r})+\\cdots \n\\end{equation*}\n$$\n\nwhere $r_{0}$ is the radius of the sphere, $\\boldsymbol{r}$ is the position of the sphere's center, and $V(\\boldsymbol{r})$ is the electrostatic potential energy acquired by the small charged sphere when approximated as a point charge. In a hydrogen atom, when the electron is treated as a point charge, the Coulomb potential energy between the electron and the nucleus is\n$$\n\\begin{equation*}\nV(\\boldsymbol{r})=-\\frac{e^{2}}{r} \n\\end{equation*}\n$$\n\nIf the electron is treated as a charged ($-e$) sphere, and $r_{0}=e^{2} / m_{e} c^{2}$ (classical electron radius) is used, the potential energy is modified by equation (1), treating the $r_{0}^{2}$ term as a perturbation. Find the perturbative correction for the 2p energy levels [equivalent to the Lamb shift].", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$r_{0}$": "radius of the sphere (classical electron radius)", + "$\\boldsymbol{r}$": "position of the sphere's center", + "$V(\\boldsymbol{r})$": "electrostatic potential energy", + "$e$": "elementary charge", + "$m_{e}$": "electron mass", + "$c$": "speed of light", + "$H^{\\prime}$": "perturbation term in potential energy", + "$E^{(1)}$": "first-order perturbative correction to the energy level", + "$\\psi$": "wave function", + "$\\psi_{2 p}(0)$": "wave function of the 2p state at the origin", + "$\\psi_{1 \\mathrm{~s}}(0)$": "wave function of the 1s state at the origin", + "$a_{0}$": "Bohr radius", + "$\\alpha$": "fine structure constant", + "$m_{\\mathrm{e}}$": "mass of the electron" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 276, + "topic": "Others", + "question": "Take the ground state wave function as\n$$\n\\psi_{0}(\\boldsymbol{r}_{1}, \\boldsymbol{r}_{2})=\\psi_{0}(r_{1}) \\psi_{0}(r_{2})\n$$\n\nwhere\n$$\n\\begin{equation*}\n\\psi_{0}(r)=(\\frac{\\lambda^{3}}{\\pi a_{0}^{3}})^{\\frac{1}{2}} \\mathrm{e}^{-x^{\\prime} / a_{0}}, \\quad a_{0}=\\frac{\\hbar^{2}}{m_{\\mathrm{e}} e^{2}}.\n\\end{equation*}\n$$\n\nCalculate the ground state magnetic susceptibility of a helium atom, in the unit of $\\mathrm{eV} /(\\mathrm{Gs})^{2}$", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\psi_{0}$": "ground state wave function", + "$\\boldsymbol{r}_{1}$": "position vector of electron 1", + "$\\boldsymbol{r}_{2}$": "position vector of electron 2", + "$r_{1}$": "radial coordinate of electron 1", + "$r_{2}$": "radial coordinate of electron 2", + "$\\lambda$": "variational parameter", + "$a_{0}$": "Bohr radius", + "$\\hbar$": "reduced Planck's constant", + "$m_{\\mathrm{e}}$": "electron mass", + "$e$": "elementary charge", + "$x^{\\prime}$": "scaled position variable", + "$c$": "speed of light", + "$x_{1}$": "x-coordinate of electron 1", + "$y_{1}$": "y-coordinate of electron 1", + "$x_{2}$": "x-coordinate of electron 2", + "$y_{2}$": "y-coordinate of electron 2", + "$\\boldsymbol{B}$": "magnetic field vector", + "$E^{(1)}$": "first-order energy correction", + "$\\alpha_{\\beta}$": "magnetic susceptibility", + "$\\mu_{\\mathrm{B}}$": "Bohr magneton", + "$Z$": "atomic number for helium" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 277, + "topic": "Others", + "question": "A hydrogen atom is situated within a certain ionic lattice, where the potential exerted by the surrounding ions on the electron in the hydrogen atom can be approximately represented as\n\n\\begin{equation*}\nH^{\\prime}=V_{0}(x^{4}+y^{4}+z^{4}-\\frac{3}{5} r^{4}) \n\\end{equation*}\n\n$H^{\\prime}$ can be considered a perturbation. If the 3d state wave functions of the hydrogen atom (orthonormalized) are taken as\n\n\\begin{align*}\n& \\psi_{1}=\\frac{1}{2}(y^{2}-z^{2}) f(r) \\\\\n& \\psi_{2}=\\frac{1}{2 \\sqrt{3}}(2 x^{2}-y^{2}-z^{2}) f(r) \\\\\n& \\psi_{3}=y z f(r) \\\\\n& \\psi_{4}=z x f(r) \\\\\n& \\psi_{5}=x y f(r)\n\\end{align*}\n\n\nUnder the influence of $H^{\\prime}$, what is the degeneracy of the first energy level after the 3d energy level splits (corresponding to $\\psi_1$ and $\\psi_2$)?", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$H^{\\prime}$": "perturbation potential", + "$V_{0}$": "constant potential coefficient", + "$x$": "Cartesian coordinate x", + "$y$": "Cartesian coordinate y", + "$z$": "Cartesian coordinate z", + "$r$": "radial distance from the origin", + "$\\psi_{1}$": "first wave function component", + "$\\psi_{2}$": "second wave function component", + "$\\psi_{3}$": "third wave function component", + "$\\psi_{4}$": "fourth wave function component", + "$\\psi_{5}$": "fifth wave function component", + "$E_{3 d}^{(0)}$": "unperturbed 3d energy level", + "$H_{11}^{\\prime}$": "perturbation matrix element (1,1)", + "$H_{33}^{\\prime}$": "perturbation matrix element (3,3)" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 278, + "topic": "Others", + "question": "A certain molecule is composed of three identical atoms ($\\alpha, \\beta, \\gamma$), with the three atoms located at the vertices of an equilateral triangle. There is one valence electron that can move among the three atoms. Denote the unperturbed Hamiltonian of this valence electron as $H_0$, and the atomic orbitals of the electron as $|\\alpha\\rangle, |\\beta\\rangle, |\\gamma\\rangle$ (which are mutually orthogonal and normalized). Assume the electron's atomic energy level is $\\langle k|H_0|k\\rangle = E_0$ (for $k=\\alpha,\\beta,\\gamma$). Between any two different atoms, the matrix element of $H_0$ is $\\langle j|H_0|k\\rangle = -a$ (where $j \\neq k, a>0$). Request to solve the molecular energy levels of the unperturbed Hamiltonian $H_0$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\alpha$": "atom alpha", + "$\\beta$": "atom beta", + "$\\gamma$": "atom gamma", + "$H_0$": "unperturbed Hamiltonian of the valence electron", + "$|\\alpha\\rangle$": "atomic orbital associated with atom alpha", + "$|\\beta\\rangle$": "atomic orbital associated with atom beta", + "$|\\gamma\\rangle$": "atomic orbital associated with atom gamma", + "$E_0$": "atomic energy level of the electron", + "$a$": "matrix element of the unperturbed Hamiltonian between different atoms" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 279, + "topic": "Others", + "question": "For the unperturbed system mentioned above (i.e., without an electric field), solve for the normalized ground state wave function $|\\psi_{GS,unperturbed}\\rangle$ corresponding to the lowest energy $E_0 - 2a$, and express it as a linear combination of $|\\alpha\\rangle, |\\beta\\rangle, |\\gamma\\rangle$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$|\\psi_{GS,unperturbed}\\rangle$": "normalized ground state wave function without an electric field", + "$E_0$": "initial lowest energy", + "$a$": "energy parameter", + "$|\\alpha\\rangle$": "basis state alpha", + "$|\\beta\\rangle$": "basis state beta", + "$|\\gamma\\rangle$": "basis state gamma", + "$C_{\\alpha}$": "coefficient for basis state alpha in linear combination", + "$C_{\\beta}$": "coefficient for basis state beta in linear combination", + "$C_{\\gamma}$": "coefficient for basis state gamma in linear combination" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 280, + "topic": "Others", + "question": "Apply a uniform weak electric field as a perturbation. Due to this electric field, the on-site energy level at atom $\\alpha$ decreases by $b$, becoming $E_0-b$, while the energy levels at atoms $\\beta$ and $\\gamma$ remain $E_0$. Assume $b \\ll a$. The hopping integral between atoms ($-a$) is not affected by the electric field. The perturbation matrix elements between different atomic orbitals are zero (i.e., $\\langle j|H'|k\\rangle = 0$ when $j \\neq k$, and $\\langle\\beta|H'|\\beta\\rangle = \\langle\\gamma|H'|\\gamma\\rangle = 0$, $\\langle\\alpha|H'|\\alpha\\rangle = -b$). Solve for the new molecular energy levels, with results approximated to first order in $b/a$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$E_0$": "initial energy level", + "$b$": "decrease in energy due to the electric field at atom $\\alpha$", + "$a$": "hopping integral between atoms", + "$\\alpha$": "first atom in the system", + "$\\beta$": "second atom in the system", + "$\\gamma$": "third atom in the system" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 281, + "topic": "Others", + "question": "Initially, the electrons are in the ground state $|\\psi_{GS,perturbed}\\rangle$ (corresponding to the situation where an electric field is perturbing atom $\\alpha$). If the field suddenly rotates so that the perturbation now acts on atom $\\beta$ (with the system's new ground state being $|\\psi'_{GS,perturbed}\\rangle$), what is the probability that the electrons are found in this new ground state $|\\psi'_{GS,perturbed}\\rangle$? Approximate the result to zero order of $(b/a)$, meaning terms containing $b$ should disregard terms of order $(b/a)$ or higher.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$|\\psi_{GS,perturbed}\\rangle$": "initial ground state of electrons under perturbation", + "$|\\psi'_{GS,perturbed}\\rangle$": "new ground state of electrons after perturbation rotation", + "$\\alpha$": "atom originally affected by the electric field", + "$\\beta$": "atom affected by the electric field after rotation", + "$C_{\\alpha}$": "coefficient for atom $\\alpha$ in the initial state", + "$C_{\\beta}$": "coefficient for atom $\\beta$ in the initial state", + "$C_{\\gamma}$": "coefficient for atom $\\gamma$ in the initial state", + "$C'_{\\alpha}$": "new coefficient for atom $\\alpha$ after rotation", + "$C'_{\\beta}$": "new coefficient for atom $\\beta$ after rotation", + "$C'_{\\gamma}$": "new coefficient for atom $\\gamma$ after rotation", + "$b$": "perturbation parameter to consider in approximation", + "$a$": "characteristic parameter of the system" + }, + "chapter": "Steady-state perturbation theory", + "section": "Steady-state perturbation theory" + }, + { + "id": 282, + "topic": "Theoretical Foundations", + "question": "Which powers among the operators $\\hat{s}$ (with any spin value $s$) are linearly independent? You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\hat{s}$": "spin operator", + "$s$": "spin value", + "$\\hat{s}_{z}$": "z-component of the spin operator" + }, + "chapter": "Spin", + "section": "Spin Operator" + }, + { + "id": 283, + "topic": "Theoretical Foundations", + "question": "The equation is\n\n\\begin{align}\\label{eq:57.4}\n(\\hat s_x \\psi)^1 &= \\tfrac12\\,\\psi^2, \n &(\\hat s_y \\psi)^1 &= -\\tfrac12\\,i\\,\\psi^2, \n &(\\hat s_z \\psi)^1 &= \\tfrac12\\,\\psi^1, \\\\\n(\\hat s_x \\psi)^2 &= \\tfrac12\\,\\psi^1, \n &(\\hat s_y \\psi)^2 &= \\tfrac12\\,i\\,\\psi^1, \n &(\\hat s_z \\psi)^2 &= -\\tfrac12\\,\\psi^2.\n\\end{align} Rewrite equation in the context, expressing the operators of spin $1 / 2$ in terms of the spinor components of the vector $\\hat{\\boldsymbol{S}}$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\hat{\\boldsymbol{S}}$": "vector of spin", + "$\\hat{s}^{\\lambda \\mu}$": "spinor component of the vector", + "$\\psi^{\\nu}$": "spinor component (nu)", + "$\\psi^{\\lambda}$": "spinor component (lambda)", + "$g^{\\mu \\nu}$": "metric tensor component (mu nu)", + "$g^{\\lambda \\nu}$": "metric tensor component (lambda nu)" + }, + "chapter": "Spin", + "section": "Wave functions for particles with arbitrary spin" + }, + { + "id": 284, + "topic": "Theoretical Foundations", + "question": "Determine all possible states of a three-nucleon system, where each nucleon has an angular momentum $j=3 / 2$ (with the same principal quantum number). You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$j$": "angular momentum of a nucleon", + "$m_{j}$": "magnetic quantum number of the nucleon's angular momentum", + "$\\tau_{\\zeta}$": "isospin projection quantum number of the nucleon", + "$M_{J}$": "total magnetic quantum number of the three-nucleon system", + "$T_{\\zeta}$": "total isospin projection quantum number of the three-nucleon system", + "$J$": "total angular momentum of the three-nucleon system", + "$T$": "total isospin quantum number of the three-nucleon system" + }, + "chapter": "Nuclear structure", + "section": "Shell Model" + }, + { + "id": 285, + "topic": "Others", + "question": "Obtain the density matrix $\\rho(q, q^{\\prime})=\\langle q| e^{-\\beta \\hat{H}}|q^{\\prime}\\rangle$ for the harmonic oscillator at finite temperature, $\\beta=1 / T(k_{\\mathrm{B}}=1)$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\rho$": "density matrix", + "$q$": "position coordinate", + "$q^{\\prime}$": "position coordinate prime", + "$\\beta$": "inverse temperature", + "$\\hat{H}$": "Hamiltonian operator", + "$m$": "mass", + "$\\omega$": "angular frequency", + "$\\hbar$": "reduced Planck's constant", + "$T$": "temperature", + "$E_{0}$": "ground state energy" + }, + "chapter": "Path Integral", + "section": "Density matrix" + }, + { + "id": 286, + "topic": "Theoretical Foundations", + "question": "A particle is moving in an infinite potential well $(-a0 \\tag{1}\n\\end{equation*} For the particle in the potential $V(r)=kr$ (s-wave), its exact ground state energy is $E_0 = C (\\frac{\\hbar^{2} k^{2}}{m})^{1 / 3}$. What is the value of the constant $C$ (accurate to four decimal places)? \n\nThe trial wave function is:\n\\begin{enumerate}\n \\item $\\psi \\sim e^{-\\lambda r}$.\n\\end{enumerate}", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$V(r)$": "potential energy as a function of distance $r$", + "$k$": "proportional constant in potential energy", + "$E_0$": "exact ground state energy", + "$C$": "constant to determine", + "$\\hbar$": "reduced Planck's constant", + "$m$": "particle mass", + "$\\lambda$": "variational parameter", + "$\\psi$": "trial wave function", + "$T$": "kinetic energy", + "$V$": "potential energy", + "$r$": "distance from the origin", + "$E(\\lambda)$": "energy as a function of $\\lambda$" + }, + "chapter": "Variational method", + "section": "" + }, + { + "id": 288, + "topic": "Theoretical Foundations", + "question": "Assume in the deuteron, the potential between the proton and neutron is expressed as\n\n\\begin{equation*}\nV(r)=-V_{0} \\mathrm{e}^{-r / a} \\tag{1}\n\\end{equation*}\n\n\nTake $V_{0}=32.7 \\mathrm{MeV}, a=2.16 \\mathrm{fm}$ (range of force). Use the variational method to find the ground state energy level of the deuteron.\n\nThe trial function is chosen as\n\n\\begin{equation*}\n\\psi(\\lambda, r)=N \\mathrm{e}^{-\\lambda r / 2 a}\n\\end{equation*}", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$V_{0}$": "potential energy constant", + "$a$": "range of force", + "$\\lambda$": "variational parameter", + "$N$": "normalization constant", + "$\\hbar$": "reduced Planck's constant", + "$\\mu$": "reduced mass of the proton-neutron system", + "$m_{\\mathrm{p}}$": "mass of the proton", + "$m_{\\mathrm{n}}$": "mass of the neutron" + }, + "chapter": "Variational method", + "section": "" + }, + { + "id": 289, + "topic": "Magnetism", + "question": "Two conductors with capacitances $C_{1}$ and $C_{2}$ respectively are separated by a distance $r$, where $r$ is greater than the dimensions of the conductors themselves. Determine the coefficient $C_{12}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$C_{1}$": "capacitance of conductor 1", + "$C_{2}$": "capacitance of conductor 2", + "$r$": "distance between the conductors", + "$C_{12}$": "coupling coefficient between the two conductors", + "$e_{1}$": "charge on conductor 1", + "$\\varphi_{1}$": "potential of conductor 1", + "$\\varphi_{2}$": "potential of conductor 2", + "$C_{11}$": "capacitance coefficient for conductor 1", + "$C_{22}$": "capacitance coefficient for conductor 2" + }, + "chapter": "Electrostatics of Conductors", + "section": "Electrostatic field energy of conductor" + }, + { + "id": 290, + "topic": "Magnetism", + "question": "Given two conductors with capacitances $C_{1}$ and $C_{2}$ separated by a distance $r$, where $r$ is much larger than the dimensions of the conductors. Try to determine the coefficient $C_{22}$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$C_{1}$": "capacitance of conductor 1", + "$C_{2}$": "capacitance of conductor 2", + "$r$": "distance between the conductors", + "$C_{22}$": "self-capacitance coefficient of conductor 2", + "$e_{1}$": "charge on conductor 1", + "$\\varphi_{1}$": "potential at conductor 1", + "$\\varphi_{2}$": "potential at conductor 2", + "$C_{11}$": "self-capacitance coefficient of conductor 1", + "$C_{12}$": "mutual capacitance coefficient between conductor 1 and conductor 2" + }, + "chapter": "Electrostatics of Conductors", + "section": "Electrostatic field energy of conductor" + }, + { + "id": 291, + "topic": "Magnetism", + "question": "Consider a conductor with a sharp conical tip on its surface. \n\nUsing spherical coordinates, place the origin at the vertex of the conical tip, with the cone axis as the polar axis. \n\nLet the cone's opening angle be $2 \\theta_{0} \\ll 1$, and the polar angle range corresponding to the external region of the conductor is $\\theta_{0} \\leqslant \\theta \\leqslant \\pi$. Assume that the potential $\\varphi$ has the form $\\varphi(r, \\theta) = r^{n} f(\\theta)$. \n\nBased on this setup, and using the boundary condition that the potential on the conductor's surface ($\\theta=\\theta_0$) is constant (i.e., $f(\\theta_0)=0$ for the angular dependence part of the potential), and that for small $\\theta_0$ and small $n \\ll 1$, the function $f(\\theta)$ can be approximately expressed as $f(\\theta) = \\mathrm{const} \\cdot (1+2 n \\ln \\sin (\\theta/2) )$, derive the expression for the exponent $n$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\theta_{0}$": "opening angle component of the cone", + "$\\theta$": "polar angle", + "$\\varphi$": "potential", + "$r$": "radial coordinate", + "$n$": "exponent in the potential function", + "$f(\\theta)$": "angular dependence part of the potential", + "$\\psi(\\theta)$": "correction term in the angular dependence" + }, + "chapter": "Electrostatics of Conductors", + "section": "Solutions to electrostatic problems" + }, + { + "id": 292, + "topic": "Magnetism", + "question": "Given that the conductor boundary is an infinite plane with a hemispherical protrusion whose radius is $R$, determine the charge distribution on the surface of the conductor at the hemispherical protrusion.\n\nHint: the potential is\n$$\n\\varphi=-4\\pi \\sigma_0 \\cdot z(1-\\frac{R^{3}}{r^{3}})\n$$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$R$": "radius of the hemispherical protrusion", + "$\\varphi$": "electric potential", + "$\\sigma_0$": "charge density far from the protruding part", + "$r$": "radial distance", + "$z$": "vertical coordinate" + }, + "chapter": "Electrostatics of Conductors", + "section": "Solutions to electrostatic problems" + }, + { + "id": 293, + "topic": "Magnetism", + "question": "Find the charge distribution on a non-charged conductor disk (with radius $a$) that is parallel to a uniform external electric field ${ }^{(1)}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the conductor disk", + "$c$": "short semi-axis of the rotational ellipsoid", + "$n^{(z)}$": "depolarization factor along the z-axis", + "$n^{(x)}$": "depolarization factor along the x-axis", + "$n^{(y)}$": "depolarization factor along the y-axis", + "$\\nu_{x}$": "component of the unit vector in the direction of the normal to the rotational ellipsoid surface", + "$x$": "coordinate along the x-axis", + "$y$": "coordinate along the y-axis", + "$z$": "coordinate along the z-axis", + "$\\sigma$": "charge density", + "$\\mathfrak{C}$": "constant related to the charge density", + "$\\rho$": "radial coordinate in polar coordinates", + "$\\varphi$": "angular coordinate (angle) in polar coordinates", + "$p$": "radial position in the plane of the disk" + }, + "chapter": "Electrostatics of Conductors", + "section": "Conductive ellipsoid" + }, + { + "id": 294, + "topic": "Magnetism", + "question": "For a conducting sphere placed in a uniform external electric field $\\mathfrak{C}$, determine its relative volume change $\\frac{\\Delta V}{V}$, the bulk modulus of the material is $K$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathfrak{C}$": "uniform external electric field", + "$\\Delta V$": "change in volume", + "$V$": "original volume", + "$K$": "bulk modulus", + "$\\Delta P$": "change in pressure", + "$\\alpha$": "polarizability", + "$\\delta_{ik}$": "Kronecker delta" + }, + "chapter": "Electrostatics of Conductors", + "section": "force on a conductor" + }, + { + "id": 295, + "topic": "Magnetism", + "question": "For a conducting sphere placed in a uniform external electric field $\\mathfrak{C}$, determine its shape deformation. Specifically, find the expression for the quantity $\\frac{a-b}{R}$ that describes its deformation, where $R$ is the original radius of the sphere, and $a$ and $b$ are the semi-axes of the ellipsoid along and perpendicular to the field direction, respectively. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\mathfrak{C}$": "external electric field", + "$R$": "original radius of the sphere", + "$a$": "semi-axis of the ellipsoid along the field direction", + "$b$": "semi-axis of the ellipsoid perpendicular to the field direction", + "$V$": "volume of the conductor", + "$n$": "depolarization factor term", + "$u_{x x}$": "strain tensor component in the x-direction", + "$u_{y y}$": "strain tensor component in the y-direction", + "$u_{z z}$": "strain tensor component in the z-direction", + "$u_{i i}$": "trace of the strain tensor", + "$\\sigma_{i k}$": "elastic stress tensor", + "$\\sigma_{x x}$": "elastic stress tensor component in the x-direction", + "$\\sigma_{y y}$": "elastic stress tensor component in the y-direction", + "$\\mu$": "shear modulus of the material" + }, + "chapter": "Electrostatics of Conductors", + "section": "force on a conductor" + }, + { + "id": 296, + "topic": "Magnetism", + "question": "Try to determine the volume change of a dielectric ellipsoid in a uniform electric field, assuming the direction of the electric field is parallel to one of the ellipsoid's axes. Specifically, determine $\\frac{V-V_0}{V}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$V$": "volume of the dielectric ellipsoid", + "$V_0$": "initial volume of the dielectric ellipsoid", + "$\\varepsilon$": "dielectric permittivity", + "$n$": "form factor of the ellipsoid, related to its shape", + "$\\mathfrak{E}$": "magnitude of the electric field", + "$K$": "compressibility coefficient", + "$P$": "pressure", + "$T$": "temperature" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Electrostriction of Isotropic Dielectrics" + }, + { + "id": 297, + "topic": "Magnetism", + "question": "Determine the electrothermal effect of a dielectric ellipsoid in a uniform electric field, assuming the direction of the field is parallel to one of the axes of the ellipsoid.\n\n\\footnotetext{\n(1) If the object is thermally insulated, the application of an electric field will cause a temperature change of $\\Delta T=-Q / \\mathscr{C}_{P}$, where $\\mathscr{C}_{P}$ is the constant pressure heat capacity of the object.\n}", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$Q$": "electrothermal effect", + "$T$": "temperature", + "$V$": "volume", + "$\\mathfrak{C}$": "electric field strength constant", + "$\\alpha$": "coefficient of thermal expansion", + "$\\varepsilon$": "dielectric constant", + "$n$": "depicts orientation related to field or geometry", + "$\\mathscr{C}_{P}$": "constant pressure heat capacity" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Electrostriction of Isotropic Dielectrics" + }, + { + "id": 298, + "topic": "Magnetism", + "question": "Assume the parallel plane plates are perpendicular to the electric field. Try to determine the difference between the heat capacity $\\mathscr{C}_{\\varphi}$ when the potential difference between the plates remains constant and the heat capacity $\\mathscr{C}_{D}$ when the electric displacement remains constant, while the external pressure is also maintained constant in both situations.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathscr{C}_{\\varphi}$": "heat capacity when the potential difference remains constant", + "$\\mathscr{C}_{D}$": "heat capacity when the electric displacement remains constant", + "$D$": "electric displacement", + "$\\mathbb{C}$": "external electric field", + "$\\varphi$": "potential difference between plates", + "$E$": "electric field", + "$l$": "thickness of the plates", + "$V$": "volume of the plates", + "$\\varepsilon$": "dielectric constant", + "$\\alpha$": "thermal expansion coefficient", + "$T$": "temperature", + "$P$": "pressure", + "$\\mathscr{S}$": "entropy of the plates", + "$\\mathscr{S}_{0}$": "reference entropy of the plates", + "$\\mathfrak{C}$": "constant related to the electric displacement" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Electrostriction of Isotropic Dielectrics" + }, + { + "id": 299, + "topic": "Magnetism", + "question": "Under the same conditions as the previous sub-question (the total volume of the parallel plane panel remains constant, $\\mathscr{C}_{\\varphi} \\equiv \\mathscr{C}_{E}$), consider representing the difference $\\mathscr{C}_{\\varphi}-\\mathscr{C}_{D}$ (i.e., $\\mathscr{C}_{E}-\\mathscr{C}_{D}$) using the external field $\\mathfrak{C}$.\nDetermine the corresponding mathematical expression for this difference.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathscr{C}_{\\varphi}$": "specific capacitance under condition \\(\\varphi\\)", + "$\\mathscr{C}_{E}$": "specific capacitance under electric field", + "$\\mathscr{C}_{D}$": "initial specific capacitance (reference)", + "$\\mathfrak{C}$": "external field", + "$E$": "electric field intensity", + "$T$": "temperature", + "$V$": "volume", + "$\\varepsilon$": "permittivity", + "$\\mathbb{C}$": "product of permittivity and electric field", + "$\\rho$": "constant density or charge density" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Electrostriction of Isotropic Dielectrics" + }, + { + "id": 300, + "topic": "Magnetism", + "question": "In an infinite anisotropic medium, there is a spherical cavity, and the uniform electric field far from the cavity within the medium is known to be $E^{(e)}$. Find the $x$ component of the electric field inside the cavity $E_x^{(i)}$, expressed in terms of $E_x^{(e)}$ and the medium and geometric parameters.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E^{(e)}$": "uniform electric field far from the cavity within the medium", + "$E_x^{(i)}$": "x component of the electric field inside the cavity", + "$E_x^{(e)}$": "x component of the electric field far from the cavity", + "$\\varepsilon^{(x)}$": "dielectric constant in the x direction", + "$\\varepsilon^{(y)}$": "dielectric constant in the y direction", + "$\\varepsilon^{(z)}$": "dielectric constant in the z direction", + "$a$": "radius of the sphere", + "$n^{(x)}$": "depolarization coefficient of the ellipsoid in the x direction", + "$n^{(y)}$": "depolarization coefficient of the ellipsoid in the y direction", + "$n^{(z)}$": "depolarization coefficient of the ellipsoid in the z direction", + "$\\varphi^{(i)}$": "potential inside the cavity", + "$\\varphi^{(e)}$": "potential outside the cavity" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric properties of crystals" + }, + { + "id": 301, + "topic": "Magnetism", + "question": "Assume there is a spherical cavity within an infinite anisotropic medium, and the uniform electric field far from the cavity inside the medium is known to be $E^{(e)}$. Try to find the $y$ component of the electric field inside the cavity $E_y^{(i)}$, expressed in terms of $E_y^{(e)}$ and the dielectric and geometric parameters.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E^{(e)}$": "uniform electric field far from the cavity", + "$E_y^{(i)}$": "y component of the electric field inside the cavity", + "$E_y^{(e)}$": "y component of the uniform electric field outside the cavity", + "$\\varepsilon^{(x)}$": "dielectric constant in the x-direction", + "$\\varepsilon^{(y)}$": "dielectric constant in the y-direction", + "$\\varepsilon^{(z)}$": "dielectric constant in the z-direction", + "$a$": "radius of the spherical cavity", + "$n^{(x)}$": "depolarization coefficient in the x-direction", + "$n^{(y)}$": "depolarization coefficient in the y-direction", + "$n^{(z)}$": "depolarization coefficient in the z-direction" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric properties of crystals" + }, + { + "id": 302, + "topic": "Magnetism", + "question": "Consider a spherical cavity in an infinite anisotropic medium, with a known uniform electric field $E^{(e)}$ far away from the cavity. Determine the $z$ component $E_z^{(i)}$ of the electric field inside the cavity, expressed in terms of $E_z^{(e)}$ and the medium and geometric parameters.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E^{(e)}$": "electric field in the external medium", + "$E_z^{(i)}$": "z component of the electric field inside the cavity", + "$E_z^{(e)}$": "z component of the external electric field", + "$\\varepsilon^{(x)}$": "dielectric constant in the x direction", + "$\\varepsilon^{(y)}$": "dielectric constant in the y direction", + "$\\varepsilon^{(z)}$": "dielectric constant in the z direction", + "$a$": "radius of the spherical cavity", + "$n^{(x)}$": "depolarization factor in the x direction", + "$n^{(y)}$": "depolarization factor in the y direction", + "$n^{(z)}$": "depolarization factor in the z direction" + }, + "chapter": "Electrostatics of Dielectrics", + "section": "Dielectric properties of crystals" + }, + { + "id": 303, + "topic": "Magnetism", + "question": "For antiferromagnetic ferrous carbonate (whose structure belongs to the magnetic class $\\boldsymbol{D}_{3 d}$), derive the expression for the x-component of magnetization $M_x$ under applied stress based on its magnetoelastic effect.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$M_x$": "x-component of magnetization", + "$\\lambda_{1}$": "magnetoelastic coupling coefficient 1", + "$\\lambda_{2}$": "magnetoelastic coupling coefficient 2", + "$\\sigma_{x x}$": "stress tensor component (xx)", + "$\\sigma_{y y}$": "stress tensor component (yy)", + "$\\sigma_{y z}$": "stress tensor component (yz)" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Piezomagnetic and magnetoelectric effects" + }, + { + "id": 304, + "topic": "Magnetism", + "question": "For the antiferromagnet iron carbonate (whose structure belongs to the magnetic class $\\boldsymbol{D}_{3 d}$), derive the expression for the magnetization component $M_y$ under applied stress based on its piezomagnetic effect. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$M_y$": "y component of magnetization", + "$\\lambda_1$": "piezomagnetic coefficient 1", + "$\\lambda_2$": "piezomagnetic coefficient 2", + "$\\sigma_{xy}$": "stress tensor component (xy)", + "$\\sigma_{xz}$": "stress tensor component (xz)", + "$\\boldsymbol{D}_{3 d}$": "magnetic class D3d", + "$H_x$": "x component of magnetic field", + "$H_y$": "y component of magnetic field", + "$\\widetilde{\\Phi}_{\\mathrm{pm}}$": "piezomagnetic potential" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Piezomagnetic and magnetoelectric effects" + }, + { + "id": 305, + "topic": "Magnetism", + "question": "For a crystal belonging to the magnetic crystal class $\\boldsymbol{D}_{4 h}(\\boldsymbol{D}_{2 h})$, determine the magnetization $x$ component $M_x$ induced by the magnetoelastic effect.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$M_x$": "magnetization x component", + "$\\lambda_1$": "magnetoelastic coupling constant", + "$\\sigma_{yz}$": "shear stress component (yz)" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Piezomagnetic and magnetoelectric effects" + }, + { + "id": 306, + "topic": "Magnetism", + "question": "For crystal belonging to the magnetic crystal class $\\boldsymbol{D}_{4 h}(\\boldsymbol{D}_{2 h})$, find the $y$ component of the magnetization $M_y$ induced by the piezomagnetic effect.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$M_y$": "y component of the magnetization", + "$\\lambda_1$": "piezomagnetic coefficient", + "$\\sigma_{xz}$": "stress component (xz)", + "$\\sigma_{yz}$": "stress component (yz)", + "$\\sigma_{xy}$": "stress component (xy)", + "$H_x$": "x component of the magnetic field", + "$H_y$": "y component of the magnetic field", + "$H_z$": "z component of the magnetic field" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Piezomagnetic and magnetoelectric effects" + }, + { + "id": 307, + "topic": "Magnetism", + "question": "For crystals belonging to the magnetic crystal class $\\boldsymbol{D}_{4 h}(\\boldsymbol{D}_{2 h})$, determine the $z$ component $M_z$ of the magnetization induced by magnetoelastic effects.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$M_z$": "z component of the magnetization", + "$\\widetilde{\\Phi}_{\\mathrm{pm}}$": "thermodynamic potential related to magnetoelastic effects", + "$\\lambda_{1}$": "magnetoelastic coupling coefficient 1", + "$\\lambda_{2}$": "magnetoelastic coupling coefficient 2", + "$\\sigma_{xz}$": "component of the stress tensor (xz)", + "$\\sigma_{yz}$": "component of the stress tensor (yz)", + "$\\sigma_{xy}$": "component of the stress tensor (xy)", + "$H_x$": "magnetic field component along x-axis", + "$H_y$": "magnetic field component along y-axis", + "$H_z$": "magnetic field component along z-axis" + }, + "chapter": "Ferromagnetism and Antiferromagnetism", + "section": "Piezomagnetic and magnetoelectric effects" + }, + { + "id": 308, + "topic": "Magnetism", + "question": "Ttry to find the magnetic susceptibility $\\alpha$ of a conductive cylinder (with radius $a$) in a uniform periodic external magnetic field perpendicular to its axis.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\alpha$": "magnetic susceptibility", + "$a$": "radius of the cylinder", + "$\\boldsymbol{r}$": "radial vector in the plane perpendicular to the cylinder axis", + "$r$": "radial distance in the plane", + "$k$": "wave number", + "$V$": "volume per unit length of the cylinder", + "$\\mathfrak{H}$": "external magnetic field vector", + "$\\boldsymbol{n}$": "unit vector in radial direction", + "$\\delta$": "skin depth", + "$\\sigma$": "conductivity", + "$\\omega$": "angular frequency", + "$c$": "speed of light" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Depth of magnetic field penetration into a conductor" + }, + { + "id": 309, + "topic": "Magnetism", + "question": "Ttry to find the magnetic susceptibility $\\alpha$ of a conductive cylinder (with radius $a$) in a uniform periodic external magnetic, but the magnetic field is parallel to the cylinder axis.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\alpha$": "magnetic susceptibility of the cylinder", + "$a$": "radius of the cylinder", + "$\\mathfrak{H}$": "external magnetic field", + "$f$": "symmetric solution of the two-dimensional equation", + "$k$": "wave number", + "$r$": "radial distance in the cylindrical coordinate system", + "$\\boldsymbol{j}$": "current density vector", + "$j_{\\varphi}$": "azimuthal component of the current density vector", + "$H$": "magnetic field inside the cylinder", + "$H_{z}$": "component of the magnetic field in the z-direction", + "$\\mathscr{M}$": "magnetic moment per unit length of the cylinder", + "$c$": "speed of light in vacuum" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Depth of magnetic field penetration into a conductor" + }, + { + "id": 310, + "topic": "Magnetism", + "question": "The surface of a uniaxial metallic crystal is cut so that its normal forms an angle $\\theta$ with the crystal's main axis of symmetry. Considering the thermoelectric effect, under isothermal boundary conditions ($\\tau=0$) and assuming $a \\ll 1$, find the $xx$ component of the surface impedance $\\zeta_{x x}^{(\\mathrm{is})}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\theta$": "angle between the crystal's main axis of symmetry and the surface normal", + "$\\tau$": "temperature fluctuation increment", + "$a$": "a parameter related to the thermoelectric effect", + "$\\zeta_{x x}^{(\\mathrm{is})}$": "xx component of the surface impedance under isothermal boundary conditions", + "$H_{y}$": "magnetic field component parallel to the y-axis", + "$H_{0}$": "amplitude of the magnetic field", + "$\\omega$": "angular frequency", + "$t$": "time", + "$j_{x}$": "current density component along the x-axis", + "$j_{y}$": "current density component along the y-axis", + "$j_{z}$": "current density component along the z-axis", + "$E_{x}$": "electric field component along the x-axis", + "$E_{y}$": "electric field component along the y-axis", + "$E_{z}$": "electric field component along the z-axis", + "$C$": "heat capacity per unit volume", + "$T$": "temperature", + "$q_{z}$": "heat flux density component along the z-axis", + "$\\rho_{x x}$": "xx component of the resistivity tensor", + "$\\rho_{y y}$": "yy component of the resistivity tensor", + "$\\rho_{z z}$": "zz component of the resistivity tensor", + "$\\rho_{x y}$": "xy component of the resistivity tensor", + "$\\rho_{y z}$": "yz component of the resistivity tensor", + "$\\rho_{x z}$": "xz component of the resistivity tensor", + "$\\rho_{\\|}$": "resistivity component along the crystal axis", + "$\\rho_{\\perp}$": "resistivity component perpendicular to the crystal axis", + "$\\varkappa_{x z}$": "xz component of the thermal conductivity tensor", + "$\\varkappa_{z z}$": "zz component of the thermal conductivity tensor", + "$\\alpha_{x z}$": "xz component of the thermoelectric tensor", + "$k$": "wave number", + "$b$": "a parameter related to the thermoelectric effect", + "$k_{1}$": "first modified wave number", + "$k_{2}$": "second modified wave number", + "$A$": "coefficient of electric field in the solution", + "$B$": "coefficient of electric field in the solution", + "$E_{T}$": "electric field component related to temperature gradient", + "$\\zeta_{0}$": "surface impedance without thermoelectric effect" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Depth of magnetic field penetration into a conductor" + }, + { + "id": 311, + "topic": "Magnetism", + "question": "Determine the x-component of the magnetic moment $\\mathscr{M}_{x}$ of a conducting sphere ($\\mu=1$), rotating uniformly in a uniform constant magnetic field whose components are $(\\mathfrak{H}_{x}, 0, \\mathfrak{H}_{z})$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathscr{M}_{x}$": "x-component of the magnetic moment", + "$\\mu$": "magnetic permeability", + "$\\mathfrak{H}_{x}$": "x-component of the uniform constant magnetic field", + "$\\mathfrak{H}_{z}$": "z-component of the uniform constant magnetic field", + "$\\boldsymbol{\\Omega}$": "angular velocity vector", + "$\\Omega$": "angular velocity", + "$\\xi$": "rotating reference frame axis", + "$\\eta$": "rotating reference frame axis", + "$V$": "volume of the sphere", + "$\\alpha$": "complex magnetic polarizability", + "$\\alpha^{\\prime}$": "real part of the magnetic polarizability relating to x-component", + "$\\alpha^{\\prime \\prime}$": "imaginary part of the magnetic polarizability relating to y-component", + "$\\mathscr{M}_{y}$": "y-component of the magnetic moment", + "$\\mathscr{M}_{z}$": "z-component of the magnetic moment" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Movement of a conductor in a magnetic field" + }, + { + "id": 312, + "topic": "Magnetism", + "question": "Determine the y-component $\\mathscr{M}_{y}$ of the magnetic moment of a conducting sphere ($\\mu=1$) uniformly rotating in a uniform constant magnetic field whose components are $(\\mathfrak{H}_{x}, 0, \\mathfrak{H}_{z})$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathscr{M}_{y}$": "y-component of the magnetic moment", + "$\\mu$": "magnetic permeability", + "$\\mathfrak{H}_{x}$": "x-component of the magnetic field", + "$\\mathfrak{H}_{z}$": "z-component of the magnetic field", + "$\\boldsymbol{\\Omega}$": "angular velocity vector", + "$t$": "time", + "$\\Omega$": "angular velocity", + "$V$": "volume of the sphere", + "$\\alpha$": "complex magnetic susceptibility", + "$\\alpha^{\\prime}$": "real part of the magnetic susceptibility", + "$\\alpha^{\\prime \\prime}$": "imaginary part of the magnetic susceptibility" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "Movement of a conductor in a magnetic field" + }, + { + "id": 313, + "topic": "Magnetism", + "question": "Try to determine the magnetic moment of a non-uniformly rotating sphere (sphere radius is $a$). Assume the rotation speed is very low so that the penetration depth $\\delta \\gg a$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "sphere radius", + "$\\delta$": "penetration depth", + "$\\mathfrak{H}$": "magnetic field", + "$V$": "volume", + "$\\hat{\\alpha}$": "operator acting on Fourier components of magnetic field", + "$\\omega$": "frequency", + "$\\mathscr{M}$": "magnetic moment", + "$m$": "mass", + "$\\sigma$": "conductivity", + "$c$": "speed of light", + "$e$": "elementary charge", + "$\\Omega$": "angular velocity" + }, + "chapter": "Quasi-static electromagnetic field", + "section": "The excitation of current by acceleration" + }, + { + "id": 314, + "topic": "Magnetism", + "question": "The tangential magnetic field before the shock wave $\\boldsymbol{H}_{t 1}=0$, while it is $\\boldsymbol{H}_{t 2} \\neq 0$ after the shock wave (this kind of shock wave is called a switch-on shock wave) . Determine the range of values $v_{n 1}$ for such a shock wave in a ideal gas with thermodynamic properties $(c_p/c_v = 5/3)$. You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion.", + "final_answer": [], + "answer_type": "Interval", + "answer": "", + "symbol": { + "$\\boldsymbol{H}_{t 1}$": "tangential magnetic field before the shock wave", + "$\\boldsymbol{H}_{t 2}$": "tangential magnetic field after the shock wave", + "$v_{n 1}$": "velocity of the shock wave relative to the gas in front of it", + "$v_{n 2}$": "velocity of the shock wave relative to the gas behind it", + "$H_{n}$": "normal magnetic field component", + "$\\rho_{2}$": "density of the gas behind the shock wave", + "$\\rho_{1}$": "density of the gas in front of the shock wave", + "$u_{\\mathrm{A} 2}$": "Alfvén velocity of the gas behind the shock wave", + "$u_{\\mathrm{A} 1}$": "Alfvén velocity of the gas in front of the shock wave", + "$H_{t 2}$": "tangential magnetic field component after the shock wave", + "$u_{01}$": "characteristic velocity of the gas" + }, + "chapter": "Magnetohydrodynamics", + "section": "Magnetohydrodynamics" + }, + { + "id": 315, + "topic": "Magnetism", + "question": "Determine the direction of the extraordinary ray when light is refracted from vacuum into the surface of a uniaxial crystal, assuming the crystal surface is perpendicular to its optical axis. Hint: calculate $\\tan(\\vartheta^{\\prime})$ where $\\vartheta^{\\prime}$ is the refraction angle.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\vartheta^{\\prime}$": "refraction angle", + "$\\vartheta$": "incidence angle", + "$n_{x}$": "x component of the refraction wave vector", + "$n_{z}$": "z component of the refracted wave", + "$\\varepsilon_{\\perp}$": "perpendicular dielectric constant", + "$\\varepsilon_{\\|}$": "parallel dielectric constant" + }, + "chapter": "Electromagnetic waves in anisotropic media", + "section": "Optical properties of uniaxial crystals" + }, + { + "id": 316, + "topic": "Magnetism", + "question": "Determine the direction of the extraordinary ray when vertically incident on the surface of a uniaxial crystal, assuming the optical axis of the uniaxial crystal is in an arbitrary direction. Hint: calculate $\\tan(\\vartheta^{\\prime})$ where $\\vartheta^{\\prime}$ is the refraction angle.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\vartheta^{\\prime}$": "refraction angle", + "$\\alpha$": "angle between the optical axis and the normal", + "$\\boldsymbol{s}$": "ray vector", + "$\\boldsymbol{n}$": "wave vector", + "$\\boldsymbol{l}$": "unit vector in the direction of the optical axis", + "$\\varepsilon_{\\|}$": "permittivity parallel to the optical axis", + "$\\varepsilon_{\\perp}$": "permittivity perpendicular to the optical axis", + "$s_{x}$": "component of the ray vector in the x-direction", + "$s_{z}$": "component of the ray vector in the z-direction" + }, + "chapter": "Electromagnetic waves in anisotropic media", + "section": "Optical properties of uniaxial crystals" + }, + { + "id": 317, + "topic": "Magnetism", + "question": "Determine the asymptotic form of the gyration vector's frequency dependence in the high-frequency regime.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\boldsymbol{H}$": "external constant magnetic field", + "$\\boldsymbol{E}$": "electric field", + "$m$": "mass of the electron", + "$e$": "charge of the electron", + "$c$": "speed of light in vacuum", + "$\\omega$": "angular frequency", + "$\\varepsilon(\\omega)$": "permittivity as a function of frequency", + "$N$": "concentration of electrons" + }, + "chapter": "Electromagnetic waves in anisotropic media", + "section": "Magneto-optical effect" + }, + { + "id": 318, + "topic": "Magnetism", + "question": "Determine the intensity distribution within the diffraction spot around the main maximum when diffraction occurs on a spherical crystal with radius $a$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "radius of the spherical crystal", + "$\\varkappa$": "wave vector difference", + "$e$": "elementary charge", + "$m$": "mass", + "$c$": "speed of light", + "$\\vartheta$": "angle between direction vectors", + "$n_{b}$": "refractive index component" + }, + "chapter": "Diffraction of X-rays in a crystal", + "section": "General Theory of X-ray Diffraction" + }, + { + "id": 319, + "topic": "Theoretical Foundations", + "question": "Assume energy levels depend only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\\prime} l^{\\prime} m^{\\prime})$, where $n, ~ n^{\\prime}, ~ l, ~ m$ are given. Find the branching ratios for transitions to $l^{\\prime}=l+1$, $m^{\\prime}=m+1, m, m-1$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$n$": "principal quantum number", + "$n^{\\prime}$": "final state's principal quantum number", + "$l$": "orbital quantum number", + "$l^{\\prime}$": "final state's orbital quantum number", + "$m$": "magnetic quantum number", + "$m^{\\prime}$": "final state's magnetic quantum number" + }, + "chapter": "Quantum Leap", + "section": "" + }, + { + "id": 320, + "topic": "Theoretical Foundations", + "question": "Assume the energy level depends only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\\prime} l^{\\prime} m^{\\prime})$, with $n, ~ n^{\\prime}, ~ l, ~ m$ all given. Find the branching ratios for transitions to $l^{\\prime}=l-1$, $m^{\\prime}=m+1, m, m-1$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$n$": "principal quantum number", + "$n^{\\prime}$": "principal quantum number in the final state", + "$l$": "azimuthal quantum number", + "$l^{\\prime}$": "azimuthal quantum number in the final state", + "$m$": "magnetic quantum number", + "$m^{\\prime}$": "magnetic quantum number in the final state" + }, + "chapter": "Quantum Leap", + "section": "" + }, + { + "id": 321, + "topic": "Theoretical Foundations", + "question": "Assume the energy level depends only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\\prime} l^{\\prime} m^{\\prime})$, with $n, ~ n^{\\prime}, ~ l, ~ m$ all given. Calculate the branching ratio for transitions to $l^{\\prime}=l+1$ and $l^{\\prime}=l-1$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$n$": "principal quantum number for initial state", + "$n^{\\prime}$": "principal quantum number for final state", + "$l$": "orbital angular momentum quantum number for initial state", + "$m$": "magnetic quantum number for initial state", + "$l^{\\prime}$": "orbital angular momentum quantum number for final state", + "$m^{\\prime}$": "magnetic quantum number for final state" + }, + "chapter": "Quantum Leap", + "section": "" + }, + { + "id": 322, + "topic": "Theoretical Foundations", + "question": "Irradiate atoms with right circularly polarized light propagating along the positive $z$ direction, causing stimulated transitions of electrons in the atom ($E_n < E_{n'}$). Find the selection rules. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$z$": "direction of light propagation", + "$E_n$": "initial energy level", + "$E_{n'}$": "final energy level", + "$n$": "initial quantum state", + "$n'$": "final quantum state", + "$l$": "initial azimuthal quantum number", + "$l'$": "final azimuthal quantum number", + "$m$": "initial magnetic quantum number", + "$m'$": "final magnetic quantum number", + "$x$": "x-coordinate", + "$y$": "y-coordinate", + "$\\omega$": "angular frequency", + "$t$": "time" + }, + "chapter": "Quantum Leap", + "section": "" + }, + { + "id": 323, + "topic": "Theoretical Foundations", + "question": "According to experimental measurements, the energy level of the hydrogen atom's $2\\mathrm{s}_{1 / 2}$ is higher than the $2 \\mathrm{p}_{1 / 2}$ level by 1058 MHz (Lamb shift). Find the average lifetime of the electron's spontaneous transition from the $2 \\mathrm{s}_{1 / 2}$ level to the $2 \\mathrm{p}_{1 / 2}$ level, in years.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$e$": "elementary charge", + "$\\omega$": "angular frequency of transition", + "$\\hbar$": "reduced Planck's constant", + "$c$": "speed of light", + "$|i\\rangle$": "initial state ket", + "$|f\\rangle$": "final state ket", + "$n$": "principal quantum number", + "$l$": "orbital angular momentum quantum number", + "$j$": "total angular momentum quantum number", + "$m_{j}$": "magnetic quantum number", + "$m_{s}$": "spin magnetic quantum number", + "$a_{0}$": "Bohr radius" + }, + "chapter": "Quantum Leap", + "section": "" + }, + { + "id": 324, + "topic": "Theoretical Foundations", + "question": "For two electrons at the $n \\mathrm{p}$ energy level $(l=1)$ of an atom, attempt to determine the number of all possible total angular momentum eigenstates using both $L-S$ coupling and $j-j$ coupling.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n$": "principal quantum number", + "$l$": "azimuthal quantum number (orbital angular momentum)", + "$L$": "total orbital angular momentum", + "$S$": "total spin angular momentum", + "$J$": "total angular momentum", + "$j$": "single electron total angular momentum quantum number", + "$J_2$": "2nd component of total angular momentum" + }, + "chapter": "symmetry", + "section": "" + }, + { + "id": 325, + "topic": "Theoretical Foundations", + "question": "Estimate the mass of a Uranium nucleus in micrograms, knowing that it contains 92 protons and 143 neutrons.\nHint: $m_{p} c^{2} \\simeq m_{n} c^{2} \\simeq 939 \\mathrm{MeV}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$m_{p}$": "mass of a proton", + "$m_{n}$": "mass of a neutron", + "$M$": "mass of the uranium nucleus", + "$c$": "speed of light", + "$M_{\\text{Planck}}$": "Planck mass", + "$G_{\\text{Newton}}$": "Newton's gravitational constant", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "Maxwell's Equations", + "section": "Maxwell's Equations" + }, + { + "id": 326, + "topic": "Theoretical Foundations", + "question": "Calculate the electric field in volt per meter that a muon feels in the $1 s$-state of muonic lead.\nHints Bohr radius $a_{B}=\\hbar c /(Z \\alpha m_{\\mu} c^{2}), m_{\\mu} c^{2}=105.6 \\mathrm{MeV}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$a_{B}$": "Bohr radius", + "$\\hbar$": "reduced Planck's constant", + "$c$": "speed of light", + "$Z$": "atomic number", + "$\\alpha$": "fine structure constant", + "$m_{\\mu}$": "muon mass", + "$m_{e}$": "electron mass", + "$a_{\\mathrm{B}}^{(\\mu)}$": "muonic Bohr radius", + "$r$": "radius at position of the muon", + "$e$": "elementary charge", + "$\\mathbf{E}$": "electric field magnitude" + }, + "chapter": "Maxwell's Equations", + "section": "Maxwell's Equations" + }, + { + "id": 327, + "topic": "Theoretical Foundations", + "question": "For the case of decay into two identical particles, when $V\\frac{v_{0}}{\\sqrt{1-v_{0}^{2}}}$, find the range of the angle between the two decay particles in the $L$ frame (their separation angle). You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion.", + "final_answer": [], + "answer_type": "Interval", + "answer": "", + "symbol": { + "$V$": "velocity parameter", + "$v_{0}$": "initial decay velocity", + "$L$": "laboratory frame", + "$C$": "center-of-mass frame", + "$\\theta_{10}$": "angle of first particle in C frame", + "$\\theta_{20}$": "angle of second particle in C frame", + "$\\theta_{0}$": "initial angle in C frame", + "$\\theta_{1}$": "angle of first particle in L frame", + "$\\theta_{2}$": "angle of second particle in L frame", + "$\\Theta$": "separation angle in L frame" + }, + "chapter": "Relativistic mechanics", + "section": "Particle decay" + }, + { + "id": 330, + "topic": "Theoretical Foundations", + "question": "Identify appropriately normalized coefficients in the expansion of the fields in terms of plane wave solutions with annihilation and/or creation operators. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\psi$": "field operator", + "$\\psi'$": "adjoint field operator", + "$\\mathbf{x}$": "position vector", + "$t$": "time", + "$\\mathbf{p}$": "momentum vector", + "$a_{\\mathbf{p}}$": "annihilation operator for momentum \\(\\mathbf{p}\\)", + "$a_{\\mathbf{p}}^{\\dagger}$": "creation operator for momentum \\(\\mathbf{p}\\)", + "$\\omega_{\\mathbf{p}}$": "angular frequency for momentum \\(\\mathbf{p}\\)", + "$m$": "mass", + "$f(\\mathbf{p})$": "function to be determined related to annihilation", + "$g(\\mathbf{p})$": "function to be determined related to creation" + }, + "chapter": "Introduction to perturbation theory and scattering", + "section": "Introduction to perturbation theory and scattering" + }, + { + "id": 331, + "topic": "Theoretical Foundations", + "question": "Although we won't use coherent states much in this course, coherent states do have applications in all sorts of odd corners of physics, and working out their properties is an instructive exercise in manipulating annihilation and creation operators.\n\nIt suffices to study a single harmonic oscillator; the generalization to a free field (= many oscillators) is trivial. Let\n\nH=\\frac{1}{2}(p^{2}+q^{2})\n\nand, as usual, let us define\n\na=\\frac{1}{\\sqrt{2}}(q+i p) \\quad a^{\\dagger}=\\frac{1}{\\sqrt{2}}(q-i p)\n\n\nDefine the coherent state $|z\\rangle$ by\n\n\\begin{equation*}\n|z\\rangle=N e^{z a^{\\dagger}}|0\\rangle \\tag{P4.2}\n\\end{equation*}\n\nwhere $z$ is a complex number and $N$ is a real, positive normalization factor (dependent on $z$ ), chosen such that $\\langle z \\mid z\\rangle=1$. The set of all coherent states for all values of $z$ is obviously complete. Indeed, it is overcomplete: The energy eigenstates can all be constructed by taking successive derivatives at $z=0$, so the coherent states\n\n\\footnotetext{\n${ }^{1}$ [Eds.] Roy J. Glauber, \"Photon correlations\", Phys. Rev. Lett. 10 (1963) 83-86. Glauber won the 2005 Nobel Prize in Physics for research in optical coherence.\n}\nwith $z$ in some small, real interval around the origin are already enough. Show that, despite this, there is an equation that looks something like a completeness relation, namely\n\n\\begin{equation*}\n1=\\alpha \\int d(\\operatorname{Re} z) d(\\operatorname{Im} z) e^{-\\beta z^{*} z}|z\\rangle\\langle z| \\tag{P4.3}\n\\end{equation*}\n\nand find the real constants $\\alpha$ and $\\beta$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$z$": "complex variable related to coherent states", + "$\\alpha$": "real constant in completeness relation", + "$\\beta$": "real constant in completeness relation", + "$z^{*}$": "complex conjugate of the variable z", + "$|z\\rangle$": "coherent state associated with the complex variable z", + "$|n\\rangle$": "energy eigenstate", + "$x$": "real part of the complex number z", + "$y$": "imaginary part of the complex number z", + "$r$": "radius in polar coordinates", + "$\\theta$": "angle in polar coordinates" + }, + "chapter": "Perturbation theory I. Wick diagrams", + "section": "Perturbation theory I. Wick diagrams" + }, + { + "id": 332, + "topic": "Theoretical Foundations", + "question": "The Lagrangian of Model 3 is:\n\\begin{align*}\n\\mathcal{L} = \\frac{1}{2}(\\partial^\\mu \\phi)(\\partial_\\mu \\phi) - \\frac{1}{2}\\mu^2\\phi^2 + \\partial^\\mu \\psi^* \\partial_\\mu \\psi - m^2 \\psi^* \\psi - g \\phi \\psi^* \\psi.\n\\end{align*} In Model 3, compute, to lowest non-vanishing order in $g$, the center-of-momentum differential cross-section and the total cross section for \"nucleon\"-\"antinucleon\" elastic scattering.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$g$": "coupling constant", + "$\\sigma$": "cross-section", + "$d\\sigma/d\\Omega$": "differential cross-section", + "$E_T$": "total energy", + "$\\mathbf{p}_f$": "final momentum vector", + "$\\mathbf{p}_i$": "initial momentum vector", + "$\\mathcal{A}_{fi}$": "amplitude from initial to final state", + "$p_1$": "four-momentum of initial particle 1", + "$p_2$": "four-momentum of initial particle 2", + "$p_3$": "four-momentum of final particle 1", + "$p_4$": "four-momentum of final particle 2", + "$\\mu$": "mass parameter", + "$\\epsilon$": "infinitesimal positive number", + "$\\theta$": "scattering angle", + "$\\phi$": "azimuthal angle" + }, + "chapter": "Scattering II. Applications", + "section": "Scattering II. Applications" + }, + { + "id": 333, + "topic": "Theoretical Foundations", + "question": "The two-particle density of states factor, $D$, in the center-of-momentum frame, $\\mathbf{P}_{T}=\\mathbf{0}$:\n\n\\begin{equation*}\nD=\\frac{1}{16 \\pi^{2}} \\frac{|\\mathbf{p}_{f}| d \\Omega_{f}}{E_{T}}.\n\\end{equation*}\nwhere we have used the notation: the final particles' momenta $|\\mathbf{p_f|$ in the center-of-momentum frame The factor $d\\Omega_f$ describes the solid angle associated with $d^3\\mathbf{p}_f$.}\n\n\nFind the formula that replaces this one if $\\mathbf{P}_{T} \\neq \\mathbf{0}$. Comment: Although the center-of-momentum frame is certainly the simplest one in which to work, sometimes we want to do calculations in other frames, for example, the \"lab frame\", in which one of the two initial particles is at rest.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$D$": "two-particle density of states factor", + "$\\mathbf{P}_{T}$": "total momentum", + "$E_{T}$": "total energy", + "$\\mathbf{p}_{f}$": "final particles' momenta", + "$d\\Omega_{f}$": "solid angle associated with the final momentum", + "$\\mathbf{k}$": "3-momentum of a final particle", + "$\\mathbf{q}$": "3-momentum of a final particle", + "$m_{k}$": "mass of the particle with momentum $\\mathbf{k}$", + "$E_{k}$": "energy of the particle with momentum $\\mathbf{k}$", + "$m_{q}$": "mass of the particle with momentum $\\mathbf{q}$", + "$E_{q}$": "energy of the particle with momentum $\\mathbf{q}$", + "$\\theta$": "angle between $\\mathbf{k}$ and $\\mathbf{P}_{T}$", + "$\\phi$": "azimuthal angle of $\\mathbf{k}$ about the $\\mathbf{P}_{T}$ axis", + "$\\theta_{k}$": "value of $\\theta$ at which energy is conserved" + }, + "chapter": "Green's functions and Heisenberg fields", + "section": "Green's functions and Heisenberg fields" + }, + { + "id": 334, + "topic": "Theoretical Foundations", + "question": "When we attempted to quantize the free Dirac theory\n\n\\begin{equation*}\n\\mathscr{L}= \\pm \\bar{\\psi}(i \\partial \\!\\!\\!/-m) \\psi \\tag{P12.1}\n\\end{equation*}\n\nwith canonical commutation relations, we encountered a disastrous contradiction with the positivity of energy. We succeeded when we used canonical anticommutators (if we chose ( $\\pm$ ) to be + ). Much earlier we were able to quantize the free charged scalar field,\n\n\\begin{equation*}\n\\mathscr{L}= \\pm(\\partial_{\\mu} \\phi^{*} \\partial^{\\mu} \\phi-\\mu^{2} \\phi^{*} \\phi) \\tag{P12.2}\n\\end{equation*}\n\nwith canonical commutators (if we chose $( \\pm)$ to be + ). Attempt to quantize the free charged scalar field with (nearly) canonical anticommutators:\n\n\\begin{align*}\n\\{\\phi(\\mathbf{x}, t), \\phi(\\mathbf{y}, t)\\} & =\\{\\dot{\\phi}(\\mathbf{x}, t), \\dot{\\phi}(\\mathbf{y}, t)\\}=0 \\\\\n{\\phi(\\mathbf{x}, t), \\phi^{*}(\\mathbf{y}, t)} & ={\\dot{\\phi}(\\mathbf{x}, t), \\dot{\\phi}^{*}(\\mathbf{y}, t)}=0 \\tag{P12.3}\\\\\n{\\phi(\\mathbf{x}, t), \\dot{\\phi}^{*}(\\mathbf{y}, t)} & =\\lambda \\delta^{(3)}(\\mathbf{x}-\\mathbf{y})\n\\end{align*}\n\nwhere $\\lambda$ is a (possibly complex) constant.\nShow that one reaches a disastrous contradiction with the positivity of the norm in Hilbert space; that is to say:\n\n\\begin{equation*}\n\\langle\\phi|{\\theta, \\theta^{\\dagger}}|\\phi\\rangle \\geq 0 \\tag{21.20}\n\\end{equation*}\n\nfor any operator $\\theta$ and any state $|\\phi\\rangle$.\n\nHints: (1) Canonical anticommutation implies that, even on the classical level, $\\phi$ and $\\phi^{*}$ are Grassmann variables. If you don't take proper account of this (especially in ordering terms when deriving the canonical momenta), you'll get hopelessly confused. (2) Dirac theory is successfully quantized using anticommutators; the sign of the Lagrangian is fixed by appealing to the positivity of the inner product in Hilbert space. If we attempt to quantize the theory using commutators, we get into trouble with the positivity of the energy. The Klein-Gordon theory is successfully quantized using commutators; the sign of the Lagrangian is fixed by appealing to the positivity of energy. So it's to be expected that we'd get into trouble, if we attempted to quantize the Klein-Gordon theory with anticommutators, with the positivity of the inner product. (3) You should do the expansion\n\\begin{align*}\n\\phi(x) = \\int \\frac{d^3 \\mathbf{p}}{(2\\pi)^{3/2}\\sqrt{2\\omega_{\\mathbf{p}}}} [b_{\\mathbf{p}} e^{-i\\mathbf{p}\\cdot x} + c_{\\mathbf{p}}^\\dagger e^{i\\mathbf{p}\\cdot x}],\n\\end{align*}\nand output $\\{b_p,b_p^\\dagger\\}, \\{c_p,c_p^\\dagger\\}$.\n\nHint: You can use $a = \\mathrm{Re}\\lambda,b=\\mathrm{Im}\\lambda$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\mathscr{L}$": "Lagrangian", + "$\\bar{\\psi}$": "Dirac adjoint spinor field", + "$\\psi$": "Dirac spinor field", + "$m$": "mass of the particle", + "$\\partial_{\\mu}$": "partial derivative with respect to space-time coordinates", + "$\\phi$": "scalar field", + "$\\phi^{*}$": "complex conjugate of the scalar field", + "$\\mu^{2}$": "mass term for the scalar field", + "$\\lambda$": "a complex constant related to canonical anticommutation", + "$\\lambda^{*}$": "complex conjugate of the constant \\lambda", + "$b_{\\mathbf{p}}$": "annihilation operator for a particle with momentum \\mathbf{p}", + "$b_{\\mathbf{p}}^{\\dagger}$": "creation operator for a particle with momentum \\mathbf{p}", + "$c_{\\mathbf{p}}$": "annihilation operator for an antiparticle with momentum \\mathbf{p}", + "$c_{\\mathbf{p}}^{\\dagger}$": "creation operator for an antiparticle with momentum \\mathbf{p}", + "$\\omega_{\\mathbf{p}}$": "frequency associated with momentum \\mathbf{p}" + }, + "chapter": "The Dirac Equation III. Quantization and Feynman Rules", + "section": "The Dirac Equation III. Quantization and Feynman Rules" + }, + { + "id": 335, + "topic": "Theoretical Foundations", + "question": "Let $\\psi_{A}, \\psi_{B}, \\psi_{C}$ and $\\psi_{D}$ be four Dirac spinor fields. These fields interact with each other (and possibly with unspecified scalar and pseudoscalar fields) in some way that is invariant under $P, C$, and $T$, where these operations are defined in the \"standard way\":\n\n\\begin{equation*}\nU_{P}^{\\dagger} \\psi(\\mathbf{x}, t) U_{P}=\\beta \\psi(-\\mathbf{x}, t) \\tag{22.8}\n\\end{equation*}\n\n\nLikewise,\n\n\\begin{equation*}\nU_{C}^{\\dagger} \\psi(x) U_{C}=\\psi'(x) \\tag{22.49}\n\\end{equation*}\n\nin a Majorana basis (one in which $\\gamma^{\\mu}=-\\gamma^{\\mu *}$ ). Finally,\n\n\\begin{equation*}\n\\Omega_{P T}^{-1} \\psi(x) \\Omega_{P T}=i \\gamma_{5} \\psi(-x) \\tag{22.80}\n\\end{equation*}\n\nagain in a Majorana basis. Now let us consider adding a term to the Hamiltonian density,\n\n\\begin{align*}\n\\mathscr{H}^{\\prime}= & g_{1}(\\bar{\\psi}_{A} \\gamma^{\\mu} \\psi_{B})(\\bar{\\psi}_{C} \\gamma_{\\mu} \\psi_{D})+g_{2}(\\bar{\\psi}_{A} \\gamma^{\\mu} \\psi_{B})(\\bar{\\psi}_{C} \\gamma_{\\mu} \\gamma_{5} \\psi_{D})+g_{3}(\\bar{\\psi}_{A} \\gamma^{\\mu} \\gamma_{5} \\psi_{B})(\\bar{\\psi}_{C} \\gamma_{\\mu} \\psi_{D}) \\\\\n& +g_{4}(\\bar{\\psi}_{A} \\gamma^{\\mu} \\gamma_{5} \\psi_{B})(\\bar{\\psi}_{C} \\gamma_{\\mu} \\gamma_{5} \\psi_{D})+\\text { Hermitian conjugate } \\tag{P14.1}\n\\end{align*}\n\nwhere the $g_{i}$ 's are (possibly complex) numbers. Under what conditions on the $g^{\\prime}$ s is $\\mathscr{H}^{\\prime}(0)$ invariant under $P$. Hint: $\\Omega_{P T}$ is anti-unitary. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$g^{\\prime}$": "a generic symbol referring to one of the coupling constants", + "$\\mathscr{H}^{\\prime}(0)$": "the perturbed Hamiltonian at time zero", + "$P$": "parity transformation", + "$\\Omega_{P T}$": "a specific operator representing a combined parity and time-reversal transformation", + "$C$": "charge conjugation transformation", + "$T$": "time reversal transformation", + "$\\gamma^{\\mu}$": "gamma matrices in the context of quantum field theory, representing one of the space-time directions", + "$\\psi_{A}$": "a fermion field with an index A", + "$\\psi_{B}$": "a fermion field with an index B", + "$\\psi_{C}$": "a fermion field with an index C", + "$\\psi_{D}$": "a fermion field with an index D", + "$\\gamma_{\\mu}$": "gamma matrices in the context of quantum field theory, with lower index representing contracted space-time directions", + "$\\gamma^{0}$": "gamma matrix corresponding to the time direction", + "$\\gamma^{i}$": "gamma matrices corresponding to spatial directions", + "$\\gamma_{5}$": "gamma five matrix used in quantum field theory, related to chirality", + "$g_{1}$": "a specific coupling constant in the Hamiltonian", + "$g_{2}$": "a specific coupling constant in the Hamiltonian", + "$g_{3}$": "a specific coupling constant in the Hamiltonian", + "$g_{4}$": "a specific coupling constant in the Hamiltonian", + "$g_{1}^{*}$": "the complex conjugate of the coupling constant $g_{1}$", + "$U_{P}$": "unitary operator for parity transformation", + "$U_{C}$": "unitary operator for charge conjugation", + "$U_{P C}$": "unitary operator for combined parity and charge conjugation", + "$U_{P T}$": "unitary operator for combined parity and time reversal" + }, + "chapter": "Coping with infinities: regularization and renormalization", + "section": "Coping with infinities: regularization and renormalization" + }, + { + "id": 336, + "topic": "Theoretical Foundations", + "question": "Even in quantum electrodynamics, it is possible (though not usual) to work in a gauge where ghost fields are needed. For example, this is a valid form of the electrodynamic Lagrangian:\n\n\\mathscr{L}=\\mathscr{L}_{\\mathrm{em}}-\\frac{1}{2} \\lambda(\\partial_{\\mu} A^{\\mu}+\\sigma A_{\\mu} A^{\\mu})^{2}+\\mathscr{L}_{\\mathrm{ghost}}\n\n\nHere $\\mathscr{L}_{\\mathrm{em}}$ is the standard Lagrangian, with neither gauge-fixing nor ghost terms, and $\\lambda$ and $\\sigma$ are arbitrary real numbers.\n\nWhat are the vertices involving ghost fields?", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathscr{L}_{\\mathrm{em}}$": "standard electrodynamic Lagrangian", + "$\\lambda$": "arbitrary real number", + "$\\sigma$": "arbitrary real number", + "$A_{\\mu}$": "vector field", + "$A^{\\mu}$": "contravariant component of the vector field", + "$\\chi$": "ghost field", + "$k^{\\mu}$": "momentum of the ghost field", + "$\\bar{\\eta}$": "complex conjugate ghost field", + "$\\eta$": "ghost field" + }, + "chapter": "The renormalization of QED", + "section": "The renormalization of QED" + }, + { + "id": 337, + "topic": "Theoretical Foundations", + "question": "Even in quantum electrodynamics, it is possible (though not usual) to work in a gauge where ghost fields are needed. For example, this is a valid form of the electrodynamic Lagrangian:\n$$\n\\mathscr{L}=\\mathscr{L}_{\\mathrm{em}}-\\frac{1}{2} \\lambda(\\partial_{\\mu} A^{\\mu}+\\sigma A_{\\mu} A^{\\mu})^{2}+\\mathscr{L}_{\\mathrm{ghost}}\n$$\n\nHere $\\mathscr{L}_{\\mathrm{em}}$ is the standard Lagrangian, with neither gauge-fixing nor ghost terms, and $\\lambda$ and $\\sigma$ are arbitrary real numbers.\n\nWhat is $\\mathscr{L}_{\\text {ghost }}$ ?", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathscr{L}$": "Lagrangian", + "$\\mathscr{L}_{\\mathrm{em}}$": "standard electromagnetic Lagrangian", + "$\\lambda$": "arbitrary real number (gauge-fixing parameter)", + "$\\sigma$": "arbitrary real number", + "$A_{\\mu}$": "vector potential", + "$A^{\\mu}$": "vector potential (contravariant component)", + "$F(A)$": "gauge-fixing function", + "$\\delta \\chi$": "variation of the gauge parameter", + "$\\eta$": "ghost field", + "$\\bar{\\eta}$": "conjugate ghost field", + "$\\mathcal{S}_{\\mathrm{ghost}}$": "ghost action", + "$\\square$": "d'Alembertian operator", + "$x$": "position (spacetime coordinate)", + "$x^{\\prime}$": "position (spacetime coordinate)", + "$k$": "momentum", + "$k^{\\mu}$": "momentum (contravariant component)" + }, + "chapter": "The renormalization of QED", + "section": "The renormalization of QED" + }, + { + "id": 338, + "topic": "Theoretical Foundations", + "question": "$\\mathrm{SU}(3)$ allows only one possible coupling of the electromagnetic current to a quark and an antiquark. Thus (by the same reasoning used for the decuplet in the previous problem), in the limit of perfect $\\mathrm{SU}(3)$ symmetry, if quarks are observable, their magnetic moments would be proportional to their charges. In the non-relativistic limit,\n\n\\begin{equation*}\n\\boldsymbol{\\mu}=\\kappa q \\boldsymbol{\\sigma}, \\tag{P21.2}\n\\end{equation*}\n\nwhere $\\kappa$ is an unknown constant, $q$ is the electric charge of the quark in question, and $\\boldsymbol{\\sigma}$ is the vector of Pauli spin matrices.\n\n\\footnotetext{\n${ }^{1}$ [Eds.] \"An Introduction to Unitary Symmetry\", the Erice notes from the summer of 1966, originally published in Strong and Weak Interactions - Present Problems, Academic Press, 1966, and reprinted in Coleman Aspects.\n}\n\nIn the naive quark model discussed in class, the baryons are considered as non-relativistic three-quark bound states with no spin-dependent interactions. Thus, as in atomic physics, we can compute the baryon moments in terms of the quark moments, that is, in terms of the single unknown constant $\\kappa$, if we know the baryon wave function. For the lightest baryon octet, the one that contains the proton and the neutron, there is no orbital contribution to the magnetic moments because each quark has zero orbital angular momentum. Thus all we need is the spin-flavor-color part of the wave function. Of course, since the assumption of perfect $\\mathrm{SU}(3)$ already gives all the baryon moments in terms of the proton and neutron moments, the only new information we get from this analysis is the ratio of these moments. Compute the ratio and compare it to experiment.\n\nRemark. It's clear from the way the calculation is set up that it's the total moment you will be computing, not the anomalous moment. Be careful that you don't use the anomalous moments when you make the computation.\n\nHint: You will need the spin-flavor part of the wave functions for both the proton and the neutron to do this problem. Here is an easy way to construct them without resorting to tables of $3 j$ symbols. It is trivial to construct the wave function for the $I_{z}=J_{z}=\\frac{3}{2}$ state of the $\\Delta$; it is $|u \\uparrow, u \\uparrow, u \\uparrow\\rangle$, with all three quarks being up quarks, and all three spinning up. If we apply both an isospin lowering operator and a spin lowering operator to this, we obtain the $I_{z}=J_{z}=\\frac{1}{2}$ state of the $\\Delta$. The $J_{z}=\\frac{1}{2}$ state of the proton must be orthogonal to this. The $J_{z}=\\frac{1}{2}$ state of the neutron (up to an irrelevant phase) is obtained from the proton state by exchanging $u$ and $d$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\kappa$": "unknown constant relating magnetic moment, electric charge, and spin", + "$q$": "electric charge of the quark", + "$\\boldsymbol{\\sigma}$": "vector of Pauli spin matrices", + "$\\mu_{B}$": "magnetic moment of the baryon", + "$\\mu_{1 z}$": "magnetic moment related to the first quark", + "$\\mu_{2 z}$": "magnetic moment related to the second quark", + "$\\mu_{3 z}$": "magnetic moment related to the third quark", + "$\\mu_{p}$": "magnetic moment of the proton", + "$\\mu_{n}$": "magnetic moment of the neutron", + "$|B\\rangle$": "state of the baryon", + "$|p\\rangle$": "state of the proton", + "$|n\\rangle$": "state of the neutron", + "$\\Delta^{++}$": "state of the Delta baryon with charge ++", + "$\\Delta^{+}$": "state of the Delta baryon with charge +" + }, + "chapter": "Broken $\\mathrm{SU}(3)$ and the naive quark model", + "section": "Broken $\\mathrm{SU}(3)$ and the naive quark model" + }, + { + "id": 339, + "topic": "Theoretical Foundations", + "question": "The model\n\\begin{equation*}\n\\mathscr{L}=\\frac{1}{2}(\\partial_{\\mu} \\boldsymbol{\\Phi}) \\cdot(\\partial^{\\mu} \\boldsymbol{\\Phi})-U(\\boldsymbol{\\Phi}) \n\\end{equation*}\nwas a theory with spontaneous breakdown of $\\mathrm{U}(1)$ internal symmetry. The particle spectrum of the theory consisted of a massless Goldstone boson and a massive neutral scalar. Furthermore, this term in the Lagrangian\n\n\\begin{equation*}\n\\frac{1}{2} \\rho^{2}(\\partial_{\\mu} \\theta)^{2}=\\frac{1}{2} a^{2}(\\partial_{\\mu} \\theta)^{2}+a \\rho^{\\prime}(\\partial_{\\mu} \\theta)^{2}+\\frac{1}{2} \\rho^{\\prime 2}(\\partial_{\\mu} \\theta)^{2} \\tag{P24.5}\n\\end{equation*}\n\ngives rise to the decay of the massive meson into two Goldstone bosons, with an invariant Feynman amplitude proportional to $a^{-1}$. (This is not a misprint: before reading the decay amplitude from the Lagrangian, we must first rescale $\\theta$ to put the free Lagrangian in standard form.) Now consider the theory minimally coupled instead to a massive photon with mass $\\mu_{0}$ (before symmetry breaking). What is the photon mass after the symmetry breaks? \n\nComment: The Abelian Higgs model is the same theory minimally coupled to a massless photon.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mu_{0}$": "mass of the photon before symmetry breaking", + "$a$": "parameter related to symmetry breaking", + "$e$": "gauge coupling constant", + "$\\Phi$": "complex scalar field", + "$\\phi_{1}$": "real part of the scalar field", + "$\\phi_{2}$": "imaginary part of the scalar field", + "$\\rho$": "magnitude of the scalar field", + "$\\rho^{\\prime}$": "shifted field component after symmetry breaking", + "$\\theta$": "phase angle of the scalar field", + "$\\theta^{\\prime}$": "redefined phase angle after symmetry breaking", + "$m_{\\rho^{\\prime}}$": "mass of the shifted field component", + "$B_{\\mu}$": "redefined gauge field", + "$F_{\\mu \\nu}$": "field strength tensor", + "$\\vartheta$": "redefined Goldstone boson field" + }, + "chapter": "Topics in spontaneous symmetry breaking", + "section": "Topics in spontaneous symmetry breaking" + }, + { + "id": 340, + "topic": "Superconductivity", + "question": "Attempt to derive the elementary excitation spectrum in a nearly ideal Bose gas, where the elementary excitation spectrum is considered as the dispersion relation of the collective wave function fluctuations. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$n$": "constant mean value (density)", + "$A$": "complex small amplitude", + "$B^{*}$": "complex conjugate of small amplitude", + "$\\hbar$": "reduced Planck's constant", + "$\\omega$": "angular frequency", + "$k$": "wave vector", + "$r$": "position vector", + "$p$": "momentum", + "$m$": "mass", + "$U_{0}$": "interaction potential" + }, + "chapter": "Superfluidity", + "section": "Inhomogeneous Bose gas" + }, + { + "id": 341, + "topic": "Superconductivity", + "question": "Calculate the probability of a quasiparticle with momentum $p$ (close to the threshold $p_{\\mathrm{c}}$) emitting a phonon, when the quasiparticle speed reaches the speed of sound.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$p$": "momentum of the quasiparticle", + "$p_{\\mathrm{c}}$": "threshold momentum", + "$\\boldsymbol{p}$": "momentum vector of the quasiparticle", + "$p^{\\prime}$": "momentum after the emission", + "$\\boldsymbol{q}$": "momentum vector of the phonon", + "$q$": "magnitude of the momentum of the phonon", + "$u$": "speed of sound", + "$A$": "expressed parameter related to momentum and energy", + "$\\rho$": "density", + "$\\varepsilon$": "energy function", + "$\\theta$": "angle related to momentum vectors", + "$w$": "probability of phonon emission", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "Superfluidity", + "section": "Fission of quasiparticles" + }, + { + "id": 342, + "topic": "Superconductivity", + "question": "There is a planar film with thickness $d \\ll \\xi, \\delta$. Find the critical value of the magnetic field parallel to the planar film, which can destroy superconductivity.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$d$": "thickness of the planar film", + "$\\xi$": "coherence length", + "$\\delta$": "penetration depth", + "$B$": "magnetic field along the x-axis", + "$B_{x}$": "component of the magnetic field along the x-axis", + "$\\theta$": "normalized order parameter ratio", + "$\\psi$": "order parameter", + "$\\psi_{0}$": "critical order parameter", + "$a$": "coefficient in the Landau free energy expansion", + "$b$": "coefficient in the Landau free energy expansion", + "$j$": "current density", + "$j_{z}$": "z-component of the current density", + "$\\mathfrak{F}$": "amplitude factor for the magnetic field", + "$H_{\\mathrm{c}}$": "critical magnetic field of a large-scale superconductor", + "$H_{\\mathrm{c}}^{f}$": "critical magnetic field of the film", + "$H_{\\mathrm{o}}$": "critical field of large-scale superconductors", + "$\\mathfrak{S}$": "parameter related to magnetic field strength", + "$e$": "elementary charge", + "$\\hbar$": "reduced Planck constant", + "$m$": "electron mass", + "$\\overline{j^{2}}$": "average of the square of the current density", + "$\\mathfrak{K}$": "parameter related to the current distribution", + "$A_{2}$": "potential along the y-axis" + }, + "chapter": "Superconductivity", + "section": "Ginzburg-Landau equation" + }, + { + "id": 343, + "topic": "Superconductivity", + "question": "If the average magnetic induction intensity of the cylindrical sample cross-section is $\\bar{B}$, and in the mixed state with an external magnetic field $\\mathfrak{S}$, all vortices are distributed at distances $d \\gg \\delta$ from each other, forming an equilateral triangular lattice in the sample cross-section. Try to determine the relationship between the external field $\\mathfrak{S}$, lower critical field $H_{\\mathrm{cl}}$, and the dimensionless vortex spacing $a = d/\\delta$ at thermodynamic equilibrium under the condition $1/a \\ll 1$. (Hint: the thermodynamic potential per unit volume $\\tilde{f}$ reaches its minimum).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\bar{B}$": "average magnetic induction intensity", + "$\\mathfrak{S}$": "external magnetic field", + "$H_{\\mathrm{cl}}$": "lower critical field", + "$a$": "dimensionless vortex spacing", + "$d$": "distance between vortices", + "$\\delta$": "characteristic length scale", + "$\\tilde{f}$": "thermodynamic potential per unit volume", + "$\\tilde{f}_{\\mathrm{s}}$": "thermodynamic potential per unit volume in superconducting state", + "$\\phi_{0}$": "magnetic flux quantum", + "$\\nu$": "number of vortices per unit area", + "$\\varepsilon_{12}$": "interaction energy between two vortices", + "$i$": "vortex index", + "$k$": "vortex index", + "$\\varepsilon_{i k}$": "interaction energy between vortices indexed by $i$ and $k$" + }, + "chapter": "Superconductivity", + "section": "Mixed structure" + }, + { + "id": 344, + "topic": "Superconductivity", + "question": "If the average magnetic induction intensity of the cross-section of a cylindrical sample is $\\bar{B}$, and in the mixed state with an applied magnetic field of $\\mathfrak{S}$, each vortex line is distributed at a distance $d \\gg \\delta$ from each other and forms an equilateral triangular lattice within the sample cross-section. It is known that the average magnetic induction intensity $\\bar{B} = \\nu \\phi_0$ (where $\\nu$ is the number of vortices per unit area and $\\phi_0$ is the magnetic flux quantum) and the dimensionless vortex spacing $a = d/\\delta$ (where $d$ is the vortex spacing, and $\\delta$ is the London penetration depth). Try to derive the relationship between the dimensionless vortex spacing $a$ and the average magnetic induction intensity $\\bar{B}$ under the condition $1/a \\ll 1$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\bar{B}$": "average magnetic induction intensity", + "$\\mathfrak{S}$": "applied magnetic field", + "$\\nu$": "number of vortices per unit area", + "$\\phi_0$": "magnetic flux quantum", + "$a$": "dimensionless vortex spacing", + "$d$": "vortex spacing", + "$\\delta$": "London penetration depth" + }, + "chapter": "Superconductivity", + "section": "Mixed structure" + }, + { + "id": 345, + "topic": "Magnetism", + "question": "Ignoring interactions between spins, calculate the magnetization of a paramagnet when the ratio of $\\beta \\mathscr{G}$ to $T$ is arbitrary.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\beta$": "inverse temperature", + "$\\mathscr{G}$": "magnetic field", + "$T$": "temperature", + "$S_{\\mathrm{z}}$": "spin component along z-axis", + "$S$": "spin quantum number", + "$\\mathfrak{S}$": "spin parameter", + "$Z$": "partition function", + "$M$": "magnetization", + "$N$": "number of particles", + "$V$": "volume" + }, + "chapter": "Magnetic", + "section": "Spin Hamiltonian" + }, + { + "id": 346, + "topic": "Theoretical Foundations", + "question": "If the interaction energy of a crystal can be expressed as $u(r)=-\\frac{\\alpha}{r^{m}}+\\frac{\\beta}{r^{n}}$ Taking $m=2, n=10, r_{0}=0.3 \\mathrm{~nm}, W=4 \\mathrm{eV}$, calculate the value of $\\beta$, unit $\\mathrm{eV} \\cdot \\mathrm{m}^{10}$", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$m$": "constant parameter (value: 2)", + "$n$": "constant parameter (value: 10)", + "$r_{0}$": "given radius (value: 0.3 nm)", + "$W$": "energy (value: 4 eV)", + "$\\beta$": "quantity to be calculated", + "$\\alpha$": "auxiliary parameter" + }, + "chapter": "Chapter 2", + "section": "" + }, + { + "id": 347, + "topic": "Theoretical Foundations", + "question": "For $\\mathrm{H}_{2}$, the Lennard-Jones potential parameters obtained from gas measurements are $\\varepsilon=50 \\times 10^{-6} J, \\sigma=2.96 \\stackrel{\\circ}{\\mathrm{~A}}$. Calculate the binding energy of $\\mathrm{H}_{2}$ when it forms a face-centered cubic solid molecular hydrogen (in units of $\\mathbf{K J} / \\mathrm{mol}$), considering each hydrogen molecule as spherical. The experimental value of the binding energy is $0.751 \\mathrm{~kJ} / \\mathrm{mo1}$. Compare it with the calculated value.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\varepsilon$": "depth of the potential well", + "$\\sigma$": "finite distance at which the inter-particle potential is zero", + "$N$": "Avogadro's number", + "$R$": "intermolecular distance", + "$R_{0}$": "equilibrium intermolecular distance", + "$U$": "total interaction energy of the crystal", + "$\\mathrm{H}_{2}$": "hydrogen molecule" + }, + "chapter": "Chapter 2", + "section": "" + }, + { + "id": 348, + "topic": "Theoretical Foundations", + "question": "Consider the lattice vibrations of a diatomic chain where the force constants between nearest neighbor atoms alternate as $c$ and $10c$. The two types of atoms have the same mass, and the nearest neighbor distance is $\\frac{a}{2}$. Find the vibrational frequency $\\omega(k)$ at $k=0$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$c$": "force constant", + "$a$": "lattice constant", + "$\\omega(k)$": "vibrational frequency at wave number k", + "$k$": "wave number", + "$M$": "mass of atom" + }, + "chapter": "Chapter 3", + "section": "" + }, + { + "id": 349, + "topic": "Theoretical Foundations", + "question": "Consider the lattice vibrations of a diatomic chain where the force constants alternate as $c$ and $10c$ between nearest neighbor atoms on the chain. The two kinds of atoms have the same mass and the nearest neighbor distance is $\\frac{a}{2}$. Find the vibration frequency $\\omega(k)$ at $k=\\frac{\\pi}{a}$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$c$": "force constant", + "$a$": "lattice constant", + "$\\omega(k)$": "vibration frequency", + "$k$": "wave vector", + "$\\pi$": "mathematical constant pi", + "$M$": "mass of atom" + }, + "chapter": "Chapter 3", + "section": "" + }, + { + "id": 350, + "topic": "Theoretical Foundations", + "question": "One-dimensional Compound Lattice\n$m=5 \\times 1.67 \\times 10^{-24} g, \\frac{M}{m}=4, \\beta=1.5 \\times 10^{1} \\mathrm{~N} / \\mathrm{m}$ (i.e., $1.51 \\times 10^{4} \\mathrm{dyn} / \\mathrm{cm}$), Find the optical frequencies $\\omega_{\\max }^{0}, \\omega_{\\min }^{0}$, and present the answer in tuple form. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\omega_{\\max }^{0}$": "maximum optical frequency (0)", + "$\\omega_{\\min }^{0}$": "minimum optical frequency (0)", + "$\\omega_{\\max }^{A}$": "maximum optical frequency (A)", + "$\\beta$": "parameter related to the stiffness or elastic constant", + "$M$": "mass of the larger component", + "$m$": "mass of the smaller component", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "Chapter 3", + "section": "" + }, + { + "id": 351, + "topic": "Theoretical Foundations", + "question": "One-dimensional complex lattice\n$m=5 \\times 1.67 \\times 10^{-24} g, \\frac{M}{m}=4, \\beta=1.5 \\times 10^{1} \\mathrm{~N} / \\mathrm{m}$ (i.e., $1.51 \\times 10^{4} \\mathrm{dyn} / \\mathrm{cm}$), What is the corresponding phonon energy in electron volts for the optical wave $\\omega_{\\min }^{o}$?", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\hbar$": "reduced Planck's constant", + "$\\omega_{\\min }^{o}$": "optical wave minimum angular frequency" + }, + "chapter": "Chapter 3", + "section": "" + }, + { + "id": 352, + "topic": "Theoretical Foundations", + "question": "One-dimensional complex lattice\n$m=5 \\times 1.67 \\times 10^{-24} g, \\frac{M}{m}=4, \\beta=1.5 \\times 10^{1} \\mathrm{~N} / \\mathrm{m}$ (i.e., $1.51 \\times 10^{4} \\mathrm{dyn} / \\mathrm{cm}$), Determine the wavelength band of the electromagnetic wave corresponding to the optical wave $\\omega_{\\max }^{0}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\omega_{\\max }^{0}$": "maximum optical wave frequency", + "$\\lambda$": "wavelength", + "$c$": "speed of light", + "$\\omega$": "angular frequency" + }, + "chapter": "Chapter 3", + "section": "" + }, + { + "id": 353, + "topic": "Theoretical Foundations", + "question": "For a one-dimensional lattice with a lattice constant of $2.5 A$, estimate the time required for an electron to move from the bottom of the energy band to the top under an external electric field of $10^{7} \\mathrm{~V} / \\mathrm{m}$", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$a$": "lattice constant", + "$e$": "elementary charge", + "$\\hbar$": "reduced Planck constant", + "$\\vec{E}$": "external electric field", + "$t$": "time required for electron transition" + }, + "chapter": "Chapter 5", + "section": "" + }, + { + "id": 354, + "topic": "Theoretical Foundations", + "question": "The spin of $\\mathrm{He}^{3}$ is $1 / 2$, making it a fermion. The density of liquid $\\mathrm{He}^{3}$ near absolute zero is $0.081 \\mathrm{gcm}^{-3}$. Calculate the Fermi temperature $\\mathbf{T}^{\\mathbf{F}}$", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$V$": "volume", + "$k_{F}$": "Fermi wavevector", + "$N$": "number of particles", + "$N_{A}$": "Avogadro's number", + "$E_{F}$": "Fermi energy", + "$\\hbar$": "reduced Planck's constant", + "$m$": "mass of $^3\\mathrm{He}$ atom", + "$T_{F}$": "Fermi temperature", + "$k_{B}$": "Boltzmann constant" + }, + "chapter": "Chapter 6", + "section": "" + }, + { + "id": 355, + "topic": "Theoretical Foundations", + "question": "If silver is considered to be a monovalent metal with a spherical Fermi surface, calculate the following quantities Find the Fermi energy and Fermi temperature, and provide the answer as a tuple You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$E_{F}^{0}$": "Fermi energy", + "$m$": "mass", + "$n$": "number density", + "$k_{F}^{0}$": "Fermi wave vector", + "$N_{A}$": "Avogadro's number", + "$\\hbar$": "reduced Planck's constant", + "$T_{F}$": "Fermi temperature", + "$k_{B}$": "Boltzmann constant" + }, + "chapter": "Chapter 6", + "section": "" + }, + { + "id": 356, + "topic": "Theoretical Foundations", + "question": "If silver is considered a single-valence metal with a spherical Fermi surface, calculate the following quantities Find the average free path of electrons at room temperature and low temperature, represented as a tuple You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\sigma$": "conductivity", + "$\\rho$": "resistivity", + "$n$": "electron density", + "$q$": "elementary charge", + "$\\tau(E_{F}^{0})$": "relaxation time at Fermi energy", + "$m$": "electron mass", + "$l$": "average free path", + "$v_{F}$": "Fermi velocity", + "$\\hbar$": "reduced Planck's constant", + "$k_{F}$": "Fermi wavevector", + "$k_{F}^{0}$": "Fermi wavevector at 0 K", + "$\\rho_{T=295 K}$": "resistivity at 295 K", + "$\\rho_{T=20 K}$": "resistivity at 20 K", + "$l_{T=295 \\mathrm{~K}}$": "average free path at 295 K", + "$l_{T=20 \\mathrm{~K}}$": "average free path at 20 K" + }, + "chapter": "Chapter 6", + "section": "" + }, + { + "id": 357, + "topic": "Theoretical Foundations", + "question": "InSb effective electron mass $m_{e}=0.015 m$, dielectric constant $\\varepsilon=18$, lattice constant $a=6.49 A$. Find the ground state orbital radius;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$a_{0}$": "ground state orbital radius", + "$\\hbar$": "reduced Planck's constant", + "$\\varepsilon_{0}$": "vacuum permittivity", + "$\\varepsilon$": "relative permittivity", + "$m^{*}$": "effective mass", + "$e$": "elementary charge", + "$m_{0}$": "electron rest mass" + }, + "chapter": "Chapter 7", + "section": "" + }, + { + "id": 358, + "topic": "Others", + "question": "Given the expression for the free energy of an object, how can the average kinetic energy of the object's particles be calculated?", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E$": "energy", + "$p$": "momentum", + "$q$": "position", + "$U$": "interaction potential energy", + "$K$": "kinetic energy", + "$m$": "mass", + "$F$": "free energy", + "$r$": "parameter related to constraint in the system", + "$v$": "parameter related to constraint in the system" + }, + "chapter": "Thermodynamic quantities", + "section": "0" + }, + { + "id": 359, + "topic": "Others", + "question": "Find the expression for heat capacity $C_{\\nu}$ when variables are $T, \\mu, V$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$C_{\\nu}$": "heat capacity at constant volume", + "$T$": "temperature", + "$\\mu$": "chemical potential", + "$V$": "volume", + "$S$": "entropy", + "$N$": "number of particles" + }, + "chapter": "Thermodynamic quantities", + "section": "Dependence of thermodynamic quantities on the number of particles" + }, + { + "id": 360, + "topic": "Others", + "question": "Find the probability distribution of atomic kinetic energy.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$w_{\\varepsilon}$": "probability distribution component related to energy", + "$\\varepsilon$": "kinetic energy", + "$T$": "temperature" + }, + "chapter": "Gibbs distribution", + "section": "Maxwell distribution" + }, + { + "id": 361, + "topic": "Others", + "question": "Find the probability distribution of molecular rotational angular velocity.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon_{\\mathrm{rot}}$": "rotational kinetic energy", + "$I_{1}$": "principal moment of inertia (1)", + "$I_{2}$": "principal moment of inertia (2)", + "$I_{3}$": "principal moment of inertia (3)", + "$\\Omega_{1}$": "angular velocity projection on principal axis (1)", + "$\\Omega_{2}$": "angular velocity projection on principal axis (2)", + "$\\Omega_{3}$": "angular velocity projection on principal axis (3)", + "$M_{1}$": "angular momentum component (1)", + "$M_{2}$": "angular momentum component (2)", + "$M_{3}$": "angular momentum component (3)", + "$T$": "temperature" + }, + "chapter": "Gibbs distribution", + "section": "Maxwell distribution" + }, + { + "id": 362, + "topic": "Others", + "question": "Determine the coordinate density matrix of the harmonic oscillator.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\rho$": "coordinate density matrix", + "$q$": "coordinate", + "$q^{\\prime}$": "coordinate prime", + "$a$": "normalization constant", + "$\\epsilon_{n}$": "energy level", + "$T$": "temperature", + "$\\psi_{n}$": "wave function", + "$\\omega$": "angular frequency", + "$\\hbar$": "reduced Planck's constant", + "$q_{n, n+1}$": "transition element", + "$r$": "midpoint", + "$s$": "displacement variable", + "$A(r)$": "function of midpoint" + }, + "chapter": "Gibbs distribution", + "section": "Probability distribution of oscillator" + }, + { + "id": 363, + "topic": "Others", + "question": "Find the distribution of particles by momentum for a relativistic ideal gas.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon$": "energy of a relativistic particle", + "$c$": "speed of light", + "$m$": "mass of the particle", + "$p$": "momentum of the particle", + "$N$": "number of particles", + "$V$": "volume", + "$T$": "temperature", + "$K_{0}$": "Macdonald function of order 0", + "$K_{1}$": "Macdonald function of order 1", + "$p_{x}$": "momentum component in x-direction", + "$p_{y}$": "momentum component in y-direction", + "$p_{z}$": "momentum component in z-direction" + }, + "chapter": "Ideal Gas", + "section": "Boltzmann Distribution in Classical Statistics" + }, + { + "id": 364, + "topic": "Others", + "question": "Find the number of gas molecules that collide with the unit area of the vessel wall per unit time, with the angle between the velocity direction and the surface normal of the vessel wall located between $\\theta$ and $\\theta+\\mathrm{d} \\theta$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\theta$": "angle between the velocity direction and the surface normal", + "$N$": "number of gas molecules", + "$V$": "volume", + "$T$": "temperature", + "$m$": "mass of a molecule" + }, + "chapter": "Ideal Gas", + "section": "Molecular collision" + }, + { + "id": 365, + "topic": "Others", + "question": "Find the number of gas molecules with speeds between $v$ and $v+\\mathrm{d} v$ that collide with a unit area of the wall per unit time.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$v$": "speed", + "$N$": "number of gas molecules", + "$V$": "volume", + "$m$": "mass of a molecule", + "$T$": "temperature" + }, + "chapter": "Ideal Gas", + "section": "Molecular collision" + }, + { + "id": 366, + "topic": "Others", + "question": "Find the work and heat obtained in the process of gas under constant pressure (isobaric process), expressed as a tuple. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$R$": "work done by the gas", + "$P$": "pressure", + "$V_{2}$": "final volume", + "$V_{1}$": "initial volume", + "$Q$": "heat added to the gas", + "$W_{2}$": "final energy", + "$W_{1}$": "initial energy", + "$N$": "amount of substance", + "$T_{1}$": "initial temperature", + "$T_{2}$": "final temperature", + "$c_{p}$": "specific heat at constant pressure" + }, + "chapter": "Ideal Gas", + "section": "Ideal Gas with Constant Heat Capacity" + }, + { + "id": 367, + "topic": "Others", + "question": "If a gas is compressed from volume $V_{1}$ to volume $V_{2}$ following the law $P V^{n}=a$ (polytropic process), calculate the work done on it and the heat received by it, expressed as a tuple. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$V_{1}$": "initial volume", + "$V_{2}$": "final volume", + "$P$": "pressure", + "$n$": "polytropic index", + "$a$": "constant", + "$Q$": "heat received", + "$R$": "work done", + "$N$": "number of moles", + "$c_{v}$": "specific heat at constant volume", + "$c_{\\nu}$": "specific heat capacity at constant volume, alternate notation", + "$T_{1}$": "initial temperature", + "$T_{2}$": "final temperature" + }, + "chapter": "Ideal Gas", + "section": "Ideal Gas with Constant Heat Capacity" + }, + { + "id": 368, + "topic": "Others", + "question": "Find the average value $\\langle\\exp (\\alpha_{i} x_{i})\\rangle$, where $\\alpha_{i}$ is a constant and $x_{i}$ is a fluctuation following a Gaussian distribution.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\langle\\exp (\\alpha_{i} x_{i})\\rangle$": "average value of the exponential function", + "$\\alpha_{i}$": "constant associated with index i", + "$x_{i}$": "fluctuation following a Gaussian distribution", + "$\\beta_{i k}$": "component of the beta matrix", + "$a_{i k}$": "component of the transformation matrix related to inverse beta", + "$a_{k m}^{-1}$": "inverse of the matrix component of a", + "$\\beta_{m i}^{-1}$": "inverse of the beta matrix component", + "$\\alpha_{l}$": "constant associated with index l", + "$\\langle x_{i} x_{k}\\rangle$": "expected value of the product of fluctuations" + }, + "chapter": "Fluctuation", + "section": "Gaussian distribution of multiple thermodynamic quantities" + }, + { + "id": 369, + "topic": "Others", + "question": "This is a thermodynamics problem. Try to find $\\langle(\\Delta W)^{2}\\rangle$ (with $P,V,C_p,T$ and $S$ as variables).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\langle(\\Delta W)^{2}\\rangle$": "variance of work", + "$P$": "pressure", + "$V$": "volume", + "$C_p$": "heat capacity at constant pressure", + "$T$": "temperature", + "$S$": "entropy" + }, + "chapter": "Fluctuation", + "section": "Fluctuation of basic thermodynamic quantities" + }, + { + "id": 370, + "topic": "Others", + "question": "This is a thermodynamics problem. Try to find $\\langle\\Delta T \\Delta P\\rangle$ (where $V,C_v,P$ and $T$ are variables).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\langle\\Delta T \\Delta P\\rangle$": "correlation between changes in temperature and pressure", + "$V$": "volume", + "$C_v$": "heat capacity at constant volume", + "$P$": "pressure", + "$T$": "temperature" + }, + "chapter": "Fluctuation", + "section": "Fluctuation of basic thermodynamic quantities" + }, + { + "id": 371, + "topic": "Others", + "question": "This is a thermodynamics problem. Try to find $\\langle\\Delta S \\Delta V\\rangle$ ( $V$ and $T$ are variables).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\langle\\Delta S \\Delta V\\rangle$": "expectation of product of changes in entropy and volume", + "$V$": "volume", + "$T$": "temperature", + "$P$": "pressure" + }, + "chapter": "Fluctuation", + "section": "Fluctuation of basic thermodynamic quantities" + }, + { + "id": 372, + "topic": "Others", + "question": "The response function is :\n\\begin{align*}\n\\alpha(\\omega) = \\frac{1}{\\hbar} \\sum_m |x_{mn}|^2 \\left[ \\frac{1}{\\omega_{mn} - \\omega - i0} + \\frac{1}{\\omega_{mn} + \\omega + i0} \\right].\n\\end{align*} Find the asymptotic behavior of $\\alpha(\\omega)$ as $\\omega \\rightarrow \\infty$ (assuming $\\alpha(\\infty)=0$).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\alpha(\\omega)$": "asymptotic behavior function of frequency", + "$\\omega$": "frequency", + "$\\alpha(\\infty)$": "asymptotic limit of function as frequency goes to infinity", + "$t$": "time", + "$\\hbar$": "reduced Planck's constant", + "$\\dot{\\hat{x}}$": "time derivative of operator x", + "$\\hat{x}$": "operator x", + "$\\alpha^{\\prime}(\\omega)$": "real part of asymptotic function at frequency omega", + "$\\alpha^{\\prime \\prime}(\\xi)$": "imaginary part of asymptotic function at frequency xi", + "$\\xi$": "dummy variable for frequency", + "$\\langle\\dot{\\hat{x}} \\hat{x}-\\hat{x} \\dot{\\hat{x}}\\rangle$": "expectation value of the commutator of operator x and its time derivative" + }, + "chapter": "Fluctuation", + "section": "Operator Form of Generalized Response Rate" + }, + { + "id": 373, + "topic": "Others", + "question": "Consider a system composed of two independent oscillators (i.e., two types of phonons), with $n_{1}, ~ n_{2}$ representing their quantum numbers (phonon numbers), and $a_{1}^{+}, ~ a_{1}, ~ a_{2}^{+}, ~ a_{2}$ representing the quantum number raising and lowering operators (i.e., the creation and annihilation operators of the two types of phonons), $\\hat{n}_{1}=a_{1}^{+} a_{1}$ and $\\hat{n}_{2}=a_{2}^{+} a_{2}$ represent the particle number operators. The normalized eigenstate in the particle number representation is denoted as $|n_{1} n_{2}\\rangle$. Let\n\n\\begin{gathered}\na=\\binom{a_{1}}{a_{2}}, \\quad a^{+}=(a_{1}^{+} a_{2}^{+}) \\\\\nJ=\\frac{1}{2} a^{+} \\sigma a \\quad(\\sigma \\text { is the Pauli matrix })\n\\end{gathered}\n\n\nThat is,\n\n\\begin{align*}\n& J_{x}=\\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\\begin{array}{ll}\n0 & 1 \\\\\n1 & 0\n\\end{array})\\binom{a_{1}}{a_{2}}=\\frac{1}{2}(a_{1}^{+} a_{2}+a_{2}^{+} a_{1}) \\\\\n& J_{y}=\\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\\begin{array}{cc}\n0 & -\\mathrm{i} \\\\\n\\mathrm{i} & 0\n\\end{array})\\binom{a_{1}}{a_{2}}=\\frac{1}{2 \\mathrm{i}}(a_{1}^{+} a_{2}-a_{2}^{+} a_{1}) \\tag{1}\\\\\n& J_{z}=\\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\\begin{array}{cc}\n1 & 0 \\\\\n0 & -1\n\\end{array})\\binom{a_{1}}{a_{2}}=\\frac{1}{2}(a_{1}^{+} a_{1}-a_{2}^{+} a_{2})=\\frac{1}{2}(\\hat{n}_{1}-\\hat{n}_{2})\n\\end{align*}\n\n\nAlso,\n\n\\begin{align*}\n& J_{+}=J_{x}+\\mathrm{i} J_{y}=a_{1}^{+} a_{2} \\tag{2}\\\\\n& J_{-}=J_{x}-\\mathrm{i} J_{y}=a_{2}^{+} a_{1}=(J_{+})^{+}\n\\end{align*}\n\n\nFind the eigenvalues of $J_z$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$n_{1}$": "quantum number of the first type of phonon", + "$n_{2}$": "quantum number of the second type of phonon", + "$a_{1}^{+}$": "raising operator (creation operator) for the first type of phonon", + "$a_{1}$": "lowering operator (annihilation operator) for the first type of phonon", + "$a_{2}^{+}$": "raising operator (creation operator) for the second type of phonon", + "$a_{2}$": "lowering operator (annihilation operator) for the second type of phonon", + "$\\hat{n}_{1}$": "particle number operator for the first type of phonon", + "$\\hat{n}_{2}$": "particle number operator for the second type of phonon", + "$a$": "column vector of lowering operators for two types of phonons", + "$a^{+}$": "row vector of raising operators for two types of phonons", + "$J$": "angular momentum operator", + "$J_{x}$": "x-component of angular momentum operator", + "$J_{y}$": "y-component of angular momentum operator", + "$J_{z}$": "z-component of angular momentum operator", + "$J_{+}$": "angular momentum raising operator", + "$J_{-}$": "angular momentum lowering operator", + "$m$": "magnetic quantum number", + "$j$": "total angular momentum quantum number" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 374, + "topic": "Others", + "question": "Denote the creation and annihilation operators of a single-particle state in a fermion system as $a^{+}$ and $a$, respectively, satisfying the fundamental anti-commutation relations\n\n\\begin{align*}\n& {[a, a^{+}]_{+} \\equiv a a^{+}+a^{+} a=1} \\tag{1}\\\\\n& a^{2}=0, \\quad(a^{+})^{2}=0 \\tag{2}\n\\end{align*}\n\n\nLet $\\hat{n}=a^{+} a$ be the particle number operator on this single-particle state. Calculate the commutator $[\\hat{n}, a^{+}]$ and $[\\hat{n}, a]$, represent it as a tuple. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$a^{+}$": "creation operator in a fermion system", + "$a$": "annihilation operator in a fermion system", + "$\\hat{n}$": "particle number operator in a single-particle state" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 375, + "topic": "Others", + "question": "Assume a Fermi particle system moves in a central force field. The single-particle energy level is related to the total angular momentum of the particle, denoted as $\\varepsilon_j$, its degeneracy is $(2j+1)$, corresponding to the single-particle state, $|jm\\rangle = a_{jm}^\\dagger |0\\rangle$, where $m=\\pm j, \\pm(j-1), \\dots, \\pm \\frac{1}{2}$. $|0\\rangle$ represents the vacuum state; $a_{jm}^\\dagger$ is the Fermi particle creation operator for the $|jm\\rangle$ state. Consider the state on the energy level $\\varepsilon_j$ where a pair of Fermi particles have their angular momentum coupled to $0$, denoted as (coupled representation, $J=M=0$) $|jj00\\rangle$. Introduce the total particle number operator on the energy level $\\varepsilon_{j}$\n\n\\begin{align*}\n\\hat{N}_{j} & =\\sum_{m>0}(a_{j m}^{+} a_{j m}+a_{j-m}^{+} a_{j-m}) \\\\\n& =\\sum_{m>0}(a_{j m}^{+} a_{j m}+a_{j \\bar{m}}^{+} a_{j \\bar{m}}) \\tag{1}\n\\end{align*}\n\nCalculate $[\\hat{N}_{j}, C_{j}^{+}], ~[C_{j}, C_{j}^{+}], ~[C_{j}^{+} C_{j}, C_{j}^{+}]$. You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$\\hat{N}_{j}$": "total particle number operator on level $\\varepsilon_{j}$", + "$a_{j m}^{+}$": "creation operator for a particle in state $m$ on level $\\varepsilon_{j}$", + "$a_{j m}$": "annihilation operator for a particle in state $m$ on level $\\varepsilon_{j}$", + "$a_{j \\bar{m}}^{+}$": "creation operator for a particle in state $\\bar{m}$ on level $\\varepsilon_{j}$", + "$a_{j \\bar{m}}$": "annihilation operator for a particle in state $\\bar{m}$ on level $\\varepsilon_{j}$", + "$C_{j}^{+}$": "creation operator for a pair of fermions with angular momentum coupling of 0 on level $\\varepsilon_{j}$", + "$C_{j}$": "annihilation operator for a pair of fermions with angular momentum coupling of 0 on level $\\varepsilon_{j}$", + "$\\Omega_{j}$": "degree of degeneracy for level $\\varepsilon_{j}$", + "$\\varepsilon_{j}$": "energy level index $j$" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 376, + "topic": "Others", + "question": "Starting from the Dirac equation describing the motion of electrons in an electromagnetic field, derive the electronic current flow density.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\hbar$": "reduced Planck's constant", + "$t$": "time", + "$\\psi$": "wave function", + "$c$": "speed of light", + "$\\boldsymbol{\\alpha}$": "Dirac matrices (alpha)", + "$\\boldsymbol{p}$": "momentum operator", + "$e$": "electron charge", + "$\\boldsymbol{A}$": "vector potential", + "$m$": "mass of the electron", + "$\\beta$": "Dirac matrix (beta)", + "$\\phi$": "scalar potential", + "$\\psi^{+}$": "conjugate transpose of wave function", + "$\\rho$": "probability density", + "$\\boldsymbol{j}$": "probability current density", + "$\\varphi$": "large component of wave function", + "$\\chi$": "small component of wave function", + "$\\boldsymbol{\\sigma}$": "Pauli matrices", + "$\\boldsymbol{j}_{e}$": "electronic current flow density", + "$\\boldsymbol{\\mu}$": "intrinsic magnetic moment of the electron", + "$\\boldsymbol{s}$": "spin operator", + "$\\boldsymbol{B}$": "magnetic field" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 377, + "topic": "Others", + "question": "Consider the second-order approximation of the stationary Dirac equation under the non-relativistic limit, determining the specific form of the Hamiltonian operator when an electron is moving in a central force field.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\phi$": "scalar potential", + "$e$": "elementary charge", + "$V$": "potential energy", + "$c$": "speed of light", + "$m$": "electron mass", + "$E$": "total energy", + "$E^{\\prime}$": "energy above rest energy", + "$\\psi$": "Dirac wave function", + "$\\varphi$": "large component of wave function", + "$\\chi$": "small component of wave function", + "$\\boldsymbol{p}$": "momentum operator", + "$\\boldsymbol{\\sigma}$": "Pauli matrices", + "$\\hbar$": "reduced Planck's constant", + "$\\boldsymbol{l}$": "orbital angular momentum operator", + "$\\Psi$": "wave function in non-relativistic approximation", + "$\\boldsymbol{s}$": "spin angular momentum operator", + "$\\boldsymbol{r}$": "position vector", + "$Z$": "atomic number", + "$\\delta(\\boldsymbol{r})$": "Dirac delta function" + }, + "chapter": "Schrödinger equation one-dimensional motion", + "section": "" + }, + { + "id": 378, + "topic": "Semiconductors", + "question": "Consider a two-dimensional square lattice. The maximum energy value is at the corners of the first Brillouin zone.\nTry to find the number of states $N(E) \\mathrm{d} E$ in the unit area crystal within the energy range $E \\sim(E+\\mathrm{d} E)$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E$": "energy", + "$N(E)$": "number of states as a function of energy", + "$k_{x}$": "x-component of wave vector", + "$k_{y}$": "y-component of wave vector", + "$k_{0}$": "central wave vector component", + "$E_{\\mathrm{v}}$": "extremum energy", + "$\\hbar$": "reduced Planck's constant", + "$m_{\\mathrm{p}}$": "particle mass", + "$k^{\\prime}$": "shifted wave vector", + "$N(k^{\\prime})$": "number of states as a function of shifted wave vector" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 379, + "topic": "Semiconductors", + "question": "The energy $E$ near the valence band top of a certain semiconductor crystal can be expressed as: $E(k)=E_{\\text {max }}-10^{26} k^{2}(\\mathrm{erg})$. Now, removing an electron with wave vector $k=10^{7} \\mathrm{i} / \\mathrm{cm}$, calculate the effective mass of the hole left by this electron.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$E$": "energy", + "$E_{\\text{max}}$": "maximum energy at the valence band top", + "$k$": "wave vector", + "$m_{\\mathrm{n}}^{*}$": "effective mass of an electron", + "$m_{\\mathrm{p}}^{*}$": "effective mass of a hole", + "$h$": "Planck's constant", + "$k_{x}$": "wave vector component in the x-direction", + "$k_{y}$": "wave vector component in the y-direction", + "$k_{z}$": "wave vector component in the z-direction", + "$v_{x}$": "velocity component in the x-direction", + "$v_{y}$": "velocity component in the y-direction", + "$v_{z}$": "velocity component in the z-direction", + "$k_{\\mathrm{p}}$": "wave vector of the hole", + "$k_{\\mathrm{n}}$": "wave vector of the electron" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 380, + "topic": "Semiconductors", + "question": "In an anisotropic crystal, its energy $E$ can be expressed in terms of the components of wave vector $\\boldsymbol{k}$ as $E(k)=A k_{x}^{2}+B k_{y}^{2}+C k_{z}^{2}$. Try to derive the equation of motion of electrons where the left-hand-side is $\\frac{dv{dt}$.} You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$E$": "energy", + "$A$": "anisotropy coefficient for x-direction", + "$B$": "anisotropy coefficient for y-direction", + "$C$": "anisotropy coefficient for z-direction", + "$k_{x}$": "wave vector component in x-direction", + "$k_{y}$": "wave vector component in y-direction", + "$k_{z}$": "wave vector component in z-direction", + "$F_{x}$": "force component in x-direction", + "$F_{y}$": "force component in y-direction", + "$F_{z}$": "force component in z-direction", + "$v_{x}$": "velocity component in x-direction", + "$v_{y}$": "velocity component in y-direction", + "$v_{z}$": "velocity component in z-direction", + "$m_{\\mathrm{n} x}^{*}$": "effective mass for x-direction", + "$m_{\\mathrm{n} y}^{*}$": "effective mass for y-direction", + "$m_{\\mathrm{n} z}^{*}$": "effective mass for z-direction" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 381, + "topic": "Semiconductors", + "question": "For a one-dimensional lattice with a lattice constant of $2.5 \\AA$, when an external electric field of $10^{2} \\mathrm{~V} / \\mathrm{m}$ is applied, calculate the time required for an electron to move from the bottom of the energy band to the top. $(1 \\AA=10 \\mathrm{~nm}=10^{-10} \\mathrm{~m})$", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$E$": "electric field strength", + "$a$": "lattice constant", + "$h$": "Planck's constant", + "$q$": "elementary charge", + "$t$": "time" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 382, + "topic": "Semiconductors", + "question": "A one-dimensional lattice with a lattice constant of $2.5 \\AA$, calculate the time required for an electron to move from the bottom to the top of the energy band when an external electric field of $10^{7} \\mathrm{~V} / \\mathrm{m}$ is applied. $(1 \\AA=10 \\mathrm{~nm}=10^{-10} \\mathrm{~m})$", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$E$": "electric field", + "$t$": "time", + "$h$": "Planck's constant", + "$q$": "elementary charge", + "$a$": "lattice constant" + }, + "chapter": "Electronic states in semiconductors", + "section": "Electronic states in semiconductors" + }, + { + "id": 383, + "topic": "Semiconductors", + "question": "The dielectric constant of semiconductor silicon single crystal $\\varepsilon_{\\mathrm{r}}=11.8$, the effective masses of electrons and holes are $m_{\\mathrm{n} 1}=$ $0.97 m_{0}, m_{\\mathrm{nt}}=0.19 m_{0}$ and $m_{\\mathrm{pl}}=0.16 m_{0}, m_{\\mathrm{ph}}=0.53 m_{0}$, using the hydrogen-like model estimate: Donor ionization energy;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", + "$m_{\\mathrm{p}}^{*}$": "effective mass of holes", + "$m_{\\mathrm{nl}}$": "longitudinal effective mass of electrons", + "$m_{\\mathrm{nt}}$": "transverse effective mass of electrons", + "$m_{\\mathrm{pl}}$": "longitudinal effective mass of holes", + "$m_{\\mathrm{ph}}$": "heavy holes effective mass", + "$m_{0}$": "rest mass of an electron", + "$\\Delta E_{\\mathrm{D}}$": "ionization energy of donor impurities", + "$\\Delta E_{\\mathrm{A}}$": "ionization energy of acceptor impurities", + "$E_{0}$": "constant energy value 13.6 eV", + "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity" + }, + "chapter": "Impurity and defect levels in semiconductors", + "section": "Impurity and defect levels in semiconductors" + }, + { + "id": 384, + "topic": "Semiconductors", + "question": "The dielectric constant of semiconductor silicon single crystal $\\varepsilon_{\\mathrm{r}}=11.8$, and the effective masses of electrons and holes are $m_{\\mathrm{n} 1}=$ $0.97 m_{0}, m_{\\mathrm{nt}}=0.19 m_{0}$ and $m_{\\mathrm{pl}}=0.16 m_{0}, m_{\\mathrm{ph}}=0.53 m_{0}$ respectively. Using a hydrogen-like model estimate: Acceptor ionization energy;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$m_{\\mathrm{n}}^{*}$": "effective mass of electrons in the conduction band", + "$m_{\\mathrm{p}}^{*}$": "effective mass of holes in the valence band", + "$m_{\\mathrm{nl}}$": "longitudinal effective mass of electrons", + "$m_{\\mathrm{nt}}$": "transverse effective mass of electrons", + "$m_{\\mathrm{pl}}$": "longitudinal effective mass of holes", + "$m_{\\mathrm{ph}}$": "heavy hole mass", + "$m_{0}$": "free electron rest mass", + "$\\Delta E_{\\mathrm{D}}$": "ionization energy of donor impurities", + "$\\Delta E_{\\mathrm{A}}$": "ionization energy of acceptor impurities", + "$E_{0}$": "hydrogen ionization energy", + "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity" + }, + "chapter": "Impurity and defect levels in semiconductors", + "section": "Impurity and defect levels in semiconductors" + }, + { + "id": 385, + "topic": "Semiconductors", + "question": "The dielectric constant of semiconductor silicon single crystal $\\varepsilon_{\\mathrm{r}}=11.8$, the effective masses of electrons and holes are $m_{\\mathrm{n} 1}=$ $0.97 m_{0}, m_{\\mathrm{nt}}=0.19 m_{0}$ and $m_{\\mathrm{pl}}=0.16 m_{0}, m_{\\mathrm{ph}}=0.53 m_{0}$, using a hydrogen-like model to estimate: Ground state electron orbital radius $r_{1}$; You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": { + "$r_{1}$": "ground state electron orbital radius", + "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity", + "$m_{\\mathrm{c}}$": "carrier effective mass", + "$r_{\\mathrm{B} 1}$": "Bohr radius", + "$m_{\\mathrm{p}}^{*}$": "effective mass of hole (p-type)", + "$m_{\\mathrm{n}}^{*}$": "effective mass of electron (n-type)" + }, + "chapter": "Impurity and defect levels in semiconductors", + "section": "Impurity and defect levels in semiconductors" + }, + { + "id": 386, + "topic": "Semiconductors", + "question": "The dielectric constant of semiconductor silicon single crystal $\\varepsilon_{\\mathrm{r}}=11.8$, and the effective masses of electrons and holes are $m_{\\mathrm{n} 1}=$ $0.97 m_{0}, m_{\\mathrm{nt}}=0.19 m_{0}$ and $m_{\\mathrm{pl}}=0.16 m_{0}, m_{\\mathrm{ph}}=0.53 m_{0}$, respectively. Using the hydrogen-like model to estimate: What is the acceptor concentration when the electron orbitals of adjacent impurity atoms overlap significantly?", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$r_{1, n}$": "radius related to donor impurity atoms", + "$N_{\\mathrm{D}}$": "donor impurity concentration", + "$r_{1, p}$": "radius related to acceptor impurity atoms", + "$N_{\\mathrm{A}}$": "acceptor impurity concentration" + }, + "chapter": "Impurity and defect levels in semiconductors", + "section": "Impurity and defect levels in semiconductors" + }, + { + "id": 387, + "topic": "Semiconductors", + "question": "For silicon material doped with n-type impurity phosphorus, try to calculate the concentration of phosphorus when weak degeneracy occurs at room temperature.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n_{\\mathrm{D}}^{+}$": "positive donor concentration", + "$n$": "electron concentration", + "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", + "$F_{1 / 2}$": "Fermi-Dirac integral of order 1/2", + "$E_{\\mathrm{F}}$": "Fermi energy", + "$E_{\\mathrm{c}}$": "conduction band edge energy", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$N_{\\mathrm{D}}$": "donor concentration", + "$E_{\\mathrm{D}}$": "donor energy level", + "$\\Delta E_{\\mathrm{D}}$": "ionization energy of phosphorus impurity" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 388, + "topic": "Semiconductors", + "question": "For the germanium material doped with n-type impurity phosphorus, try to calculate the numerical value of its phosphorus doping concentration at room temperature when weak degeneracy occurs.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n_{\\mathrm{D}}^{+}$": "donor ion concentration", + "$n$": "electron concentration", + "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", + "$F_{1 / 2}$": "Fermi-Dirac integral of order 1/2", + "$E_{\\mathrm{F}}$": "Fermi energy", + "$E_{\\mathrm{c}}$": "conduction band edge energy", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$N_{\\mathrm{D}}$": "donor impurity concentration", + "$E_{\\mathrm{D}}$": "donor energy level", + "$\\Delta E_{\\mathrm{D}}$": "ionization energy of donor impurity" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 389, + "topic": "Semiconductors", + "question": "For silicon materials doped with n-type impurity phosphorus, try to calculate the dopant concentration when weak degeneracy occurs at room temperature.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n_{\\mathrm{D}}^{+}$": "doped donor concentration in ionized states", + "$n$": "electron concentration", + "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", + "$E_{\\mathrm{F}}$": "Fermi energy level", + "$E_{\\mathrm{c}}$": "conduction band energy level", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$N_{\\mathrm{D}}$": "dopant concentration", + "$E_{\\mathrm{D}}$": "dopant ionization energy", + "$\\Delta E_{\\mathrm{D}}$": "ionization energy difference of dopant" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 390, + "topic": "Semiconductors", + "question": "For a semiconductor silicon sample with a donor impurity concentration of $10^{12} \\mathrm{~cm}^{-3}$, calculate the equation that the temperature value (in K) satisfies when its intrinsic carrier concentration $n_i$ equals the donor impurity concentration $N_d$. Assume $E_{\\mathrm{g}}=1 \\mathrm{eV}, m_{\\mathrm{n}}^{*}=m_{\\mathrm{p}}^{*}=0.2 m_{0}$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$n_i$": "intrinsic carrier concentration", + "$N_d$": "donor impurity concentration", + "$E_{\\mathrm{g}}$": "band gap energy", + "$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", + "$m_{\\mathrm{p}}^{*}$": "effective mass of holes", + "$m_{0}$": "rest mass of an electron", + "$N_{\\mathrm{c}}$": "effective density of states for conduction band", + "$N_{\\mathrm{v}}$": "effective density of states for valence band", + "$T$": "temperature" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 391, + "topic": "Semiconductors", + "question": "Given a silicon sample with a donor concentration $N_{\\mathrm{D}}=2 \\times 10^{14} \\mathrm{~cm}^{-3}$ and an acceptor concentration $N_{\\mathrm{A}}=10^{14} \\mathrm{~cm}^{-3}$, where the donor ionization energy $\\Delta E_{\\mathrm{D}}=E_{\\mathrm{c}}-E_{\\mathrm{D}}=0.05 \\mathrm{eV}$, find the equation that the temperature value (in K) satisfies when $99\\%$ of the donor impurities are ionized. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$N_{\\mathrm{D}}$": "donor concentration", + "$N_{\\mathrm{A}}$": "acceptor concentration", + "$\\Delta E_{\\mathrm{D}}$": "donor ionization energy", + "$E_{\\mathrm{c}}$": "conduction band energy", + "$E_{\\mathrm{D}}$": "donor energy level", + "$n_{\\mathrm{D}}^{+}$": "concentration of ionized donors", + "$n_{0}$": "electron concentration", + "$n_{\\mathrm{A}}^{-}$": "concentration of ionized acceptors", + "$P_{0}$": "hole concentration", + "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", + "$E_{\\mathrm{F}}$": "Fermi energy", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$f(E_{\\mathrm{D}})$": "Fermi-Dirac occupation probability at donor level" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 392, + "topic": "Semiconductors", + "question": "In a boron-doped non-degenerate p-type silicon, containing a certain concentration of indium, the hole concentration at room temperature is measured as $p_{0}=1.1 \\times 10^{16} \\mathrm{~cm}^{-3}$. Given that the boron doping concentration $N_{\\mathrm{A} 1}=10^{16} \\mathrm{~cm}^{-3}$ and its ionization energy $\\Delta E_{\\mathrm{A} 1}=E_{\\mathrm{A} 1}-E_{\\mathrm{v}}=0.046 \\mathrm{eV}$, and indium's ionization energy $\\Delta E_{\\mathrm{A} 2}=E_{\\mathrm{A} 2}-E_{\\mathrm{v}}=0.16 \\mathrm{eV}$, determine the concentration of indium in this semiconductor. At room temperature, silicon's $N_{\\mathrm{v}}=1.04 \\times 10^{19}$ $\\mathrm{cm}^{-3}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$p_{0}$": "hole concentration at room temperature", + "$N_{\\mathrm{A} 1}$": "boron doping concentration", + "$\\Delta E_{\\mathrm{A} 1}$": "ionization energy of boron", + "$E_{\\mathrm{A} 1}$": "energy level of boron", + "$E_{\\mathrm{v}}$": "energy of the valence band", + "$\\Delta E_{\\mathrm{A} 2}$": "ionization energy of indium", + "$E_{\\mathrm{A} 2}$": "energy level of indium", + "$N_{\\mathrm{v}}$": "effective density of states in the valence band", + "$E_{\\mathrm{F}}$": "Fermi level energy" + }, + "chapter": "Statistical Distribution of Charge Carriers in Semiconductors", + "section": "Statistical Distribution of Charge Carriers in Semiconductors" + }, + { + "id": 393, + "topic": "Semiconductors", + "question": "At room temperature, the resistivity of intrinsic germanium is $47 \\Omega \\cdot \\mathrm{~cm}$. If antimony impurities are added so that there is one impurity atom per $10^{6}$ germanium atoms. Assume all impurities are ionized. The concentration of germanium atoms is $4.4 \\times 10^{22} / \\mathrm{cm}^{3}$. Calculate the resistivity of this doped germanium material. Assume $\\mu_{\\mathrm{n}}=3600 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s}), ~ \\mu_{\\mathrm{p}}$ $=1700 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and are unchanged by doping. $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\rho$": "resistivity", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$N_{\\mathrm{D}}$": "donor impurity concentration", + "$n_{0}$": "electron concentration in doped semiconductor", + "$p_{0}$": "hole concentration in doped semiconductor", + "$\\rho_{\\mathrm{n}}$": "resistivity of n-type semiconductor" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 394, + "topic": "Semiconductors", + "question": "When $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3}, \\mu_{\\mathrm{p}}=1900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{n}}=3800 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, try to find the intrinsic conductivity of germanium.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$\\mu_{\\mathrm{p}}$": "mobility of holes", + "$\\mu_{\\mathrm{n}}$": "mobility of electrons", + "$q$": "elementary charge" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 395, + "topic": "Semiconductors", + "question": "When $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3}, \\mu_{\\mathrm{p}}=1900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{n}}=3800 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, calculate the minimum conductivity of germanium.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$\\mu_{\\mathrm{p}}$": "mobility of holes", + "$\\mu_{\\mathrm{n}}$": "mobility of electrons", + "$\\sigma_{\\min }$": "minimum conductivity" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 396, + "topic": "Semiconductors", + "question": "At room temperature, the electron mobility of high-purity germanium $\\mu_{\\mathrm{n}}=3900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. Given the effective mass of electrons $m_{\\mathrm{n}}=0.3 m \\approx 3 \\times 10^{-28} \\mathrm{~g}$, try to calculate: Mean free time $\\tau$;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\tau$": "mean free time", + "$\\mu_{\\mathrm{n}}$": "mobility", + "$e$": "elementary charge", + "$m_{\\mathrm{n}}$": "effective mass of charge carriers" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 397, + "topic": "Semiconductors", + "question": "At room temperature, the electron mobility of high-purity germanium is $\\mu_{\\mathrm{n}}=3900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. Given that the effective mass of the electron $m_{\\mathrm{n}}=0.3 m \\approx 3 \\times 10^{-28} \\mathrm{~g}$, try to calculate: Average free path $l$;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$l$": "average free path", + "$\\bar{v}$": "average velocity", + "$\\tau$": "mean free time" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 398, + "topic": "Semiconductors", + "question": "At room temperature, we have a a p-type silicon wafer, and want to convert a p-type silicon wafer with a resistivity of $0.2 \\Omega \\cdot \\mathrm{~cm}$. What should be the density of impurities to achieve a resistivity of $0.2 \\Omega \\cdot \\mathrm{~cm}$ for n-type silicon?", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$N_{\\mathrm{d}}$": "donor impurity density", + "$N_{\\mathrm{a}}$": "acceptor impurity density", + "$\\mu_{\\mathrm{n}}$": "electron mobility in silicon", + "$\\mu_{\\mathrm{p}}$": "hole mobility in silicon", + "$\\rho$": "resistivity" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 399, + "topic": "Semiconductors", + "question": "A boron-doped non-degenerate p-type silicon sample contains a certain concentration of indium, with a measured resistivity at room temperature (300K) of $\\rho=2.84 \\Omega \\cdot \\mathrm{~cm}^{2}$. Given that the doped boron concentration is $N_{\\mathrm{a} 1}=10^{16} / \\mathrm{cm}^{3}$, with boron ionization energy $E_{\\mathrm{a} 1}-E_{\\mathrm{v}}=0.045 \\mathrm{eV}$ and indium ionization energy $E_{\\mathrm{a} 2}-E_{\\mathrm{v}}=0.16 \\mathrm{eV}$, determine the concentration of indium $N_{\\mathrm{a} 2}$ in the sample [At room temperature, $N_{\\mathrm{v}}=$ $1.04 \\times 10^{19} / \\mathrm{cm}^{3}, ~ \\mu_{\\mathrm{p}}=200 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})]$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\rho$": "resistivity", + "$N_{\\mathrm{a} 1}$": "doped boron concentration", + "$E_{\\mathrm{a} 1}$": "boron ionization energy", + "$E_{\\mathrm{v}}$": "valence band energy", + "$E_{\\mathrm{a} 2}$": "indium ionization energy", + "$N_{\\mathrm{v}}$": "density of states in the valence band", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$p$": "hole concentration", + "$E_{\\mathrm{f}}$": "Fermi level energy", + "$k$": "Boltzmann constant", + "$T$": "temperature", + "$N_{\\overline{\\mathrm{a} 2}}$": "ionized indium concentration" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 400, + "topic": "Semiconductors", + "question": "Given that the conductivity of intrinsic germanium is $3.56 \\times 10^{-2} \\mathrm{~S} / \\mathrm{cm}$ at 310 K and $0.42 \\times$ $10^{-2} \\mathrm{~S} / \\mathrm{cm}_{\\circ}$ at 273 K, an n-type germanium sample has a donor impurity concentration of $N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}$ at these two temperatures. Calculate the conductivity of the doped germanium at the above temperatures. [Assume $\\mu_{\\mathrm{n}}=3600 \\mathrm{~cm} /(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{p}}=1700 \\mathrm{~cm} /(\\mathrm{V} \\cdot \\mathrm{s})$]", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$N_{\\mathrm{D}}$": "donor impurity concentration", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$\\sigma_{\\mathrm{i}}$": "intrinsic conductivity", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$n_{0}$": "majority carrier (electron) concentration", + "$p_{0}$": "minority carrier (hole) concentration", + "$\\sigma$": "conductivity" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 401, + "topic": "Semiconductors", + "question": "A thermistor made of intrinsic silicon material has a resistance value of $500 \\Omega$ at 290 K. Assuming the band gap of silicon $E_{\\mathrm{q}}=1.12 \\mathrm{eV}$ and does not change with temperature, if we assume the carrier mobility remains unchanged, try to estimate the approximate value of the thermistor at 325 K.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$E_{\\mathrm{q}}$": "band gap of silicon", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$q$": "elementary charge", + "$\\mu_{\\mathrm{n}}$": "mobility of electrons", + "$\\mu_{\\mathrm{p}}$": "mobility of holes", + "$T$": "temperature", + "$N_{\\mathrm{c}}$": "effective density of states in conduction band", + "$N_{\\mathrm{v}}$": "effective density of states in valence band", + "$k_{0}$": "Boltzmann constant", + "$R$": "resistance", + "$C$": "proportionality constant related to resistance" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 402, + "topic": "Semiconductors", + "question": "The resistivity of intrinsic germanium material with temperature $T$ can be tabulated as follows:\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$T(\\mathrm{~K})$ & 385 & 458 & 556 & 714 \\\\\n\\hline$\\rho(\\Omega \\cdot \\mathrm{~cm})$ & 0.028 & 0.0061 & 0.0013 & 0.00027 \\\\\n\\hline\n\\end{tabular}\n\nAssume $E_{\\mathrm{g}}$ is independent of temperature $T$, and the mobilities of electrons and holes $\\mu_{\\mathrm{n}} \\mu_{\\mathrm{p}}$ both vary as $T^{-\\frac{3}{2}}$. Find the band gap $E_{\\mathrm{g}}$ of germanium.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$T$": "temperature", + "$E_{\\mathrm{g}}$": "band gap of germanium", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$\\sigma_{\\mathrm{i}}$": "intrinsic conductivity", + "$\\rho_{\\mathrm{i}}$": "intrinsic resistivity", + "$k_{0}$": "Boltzmann constant", + "$C$": "constant" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 403, + "topic": "Semiconductors", + "question": "Calculate the resistivity of intrinsic silicon at room temperature (unit $\\Omega \\cdot \\mathrm{~cm}$). It is known that the electron mobility of intrinsic silicon at room temperature is $1350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, the hole mobility is $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the intrinsic carrier concentration is $n_{\\mathrm{i}}=1.5 \\times 10^{10} / \\mathrm{cm}^{3}$, elementary charge $q=1.6 \\times 10^{-19} \\mathrm{C}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$q$": "elementary charge", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$\\sigma_{\\mathrm{i}}$": "intrinsic conductivity", + "$N_{\\mathrm{D}}$": "concentration of the dopant", + "$n_{0}$": "majority carrier concentration", + "$N_{\\mathrm{i}}$": "impurity concentration", + "$\\sigma$": "conductivity" + }, + "chapter": "Conductivity of semiconductors", + "section": "Conductivity of semiconductors" + }, + { + "id": 404, + "topic": "Semiconductors", + "question": "For a certain n-type semiconductor silicon with a doping concentration $N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}$, and a minority carrier lifetime $\\tau_{\\mathrm{p}}=5 \\mu \\mathrm{~s}$, if due to external influences all minority carriers are removed (such as near a reverse-biased pn junction), what is the electron-hole generation rate at this time (let $n_{\\mathrm{i}}=1.5 \\times 10^{10} \\mathrm{~cm}^{-3}$)?", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$N_{\\mathrm{D}}$": "doping concentration", + "$\\tau_{\\mathrm{p}}$": "minority carrier lifetime", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$p$": "minority carrier concentration", + "$p_{0}$": "equilibrium minority carrier concentration", + "$\\Delta_{p}$": "change in minority carrier concentration", + "$R$": "recombination rate" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 405, + "topic": "Semiconductors", + "question": "The concentration of copper in a copper-doped germanium sample is $10^{15} \\mathrm{~cm}^{-3}$, and the concentration of antimony is $10^{17} \\mathrm{~cm}^{-3}$. Its minority carrier lifetime measured under small injection conditions is $10^{-7} \\mathrm{~s}^{-1}$. Given that $N_{\\mathrm{c}}=1.04 \\times 10^{19} \\mathrm{~cm}^{-1}$. If the effective mass of holes in germanium $m_{\\mathrm{p}}^{*}=0.30 m_{0}(m_{0}$ is the free electron mass), find the hole capture cross-section?", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$m_{\\mathrm{p}}^{*}$": "effective mass of holes in germanium", + "$m_{0}$": "free electron mass", + "$\\gamma_{\\mathrm{p}}$": "hole capture coefficient", + "$N_{\\mathrm{t}}$": "trap concentration", + "$\\tau_{\\mathrm{p}}$": "hole capture time", + "$\\sigma_{\\mathrm{p}}$": "hole capture cross-section", + "$v_{\\mathrm{t}}$": "thermal velocity of holes", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 406, + "topic": "Semiconductors", + "question": "In a piece of p-type semiconductor, there exists a recombination-generation center. When slightly doped, the electrons captured by these centers are reemitted to the conduction band with the same probability as their recombination with holes. Try to find the energy level position of this recombination-generation center. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$n_{1}$": "number of conduction band electrons from recombination centers", + "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", + "$E_{\\mathrm{c}}$": "energy level of the conduction band", + "$E_{\\mathrm{t}}$": "energy level position of the recombination-generation center", + "$k_{0}$": "Boltzmann constant", + "$T$": "absolute temperature", + "$s_{-}$": "electron excitation probability", + "$n_{\\mathrm{t}}$": "number of electrons at the recombination-generation center", + "$r_{\\mathrm{n}}$": "electron capture probability", + "$r_{\\mathrm{p}}$": "hole capture probability", + "$p$": "hole concentration", + "$p_{0}$": "equilibrium hole concentration", + "$\\Delta p$": "excess hole concentration", + "$N_{\\mathrm{v}}$": "effective density of states in the valence band", + "$E_{\\mathrm{v}}$": "energy level of the valence band", + "$E_{\\mathrm{F}}$": "Fermi energy level", + "$E_{\\mathrm{i}}$": "intrinsic energy level" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 407, + "topic": "Semiconductors", + "question": "Illuminating an n-type silicon sample with a resistivity of $1 \\Omega \\cdot \\mathrm{~cm}$, non-equilibrium carriers are uniformly generated, with the generation rate of electron-hole pairs being $10^{17} \\mathrm{~cm}^{-3} \\cdot \\mathrm{~s}^{-1}$. Assume the lifetime of the sample is $10 \\mu \\mathrm{~s}$, and the surface recombination velocity is $100 \\mathrm{~cm} / \\mathrm{s}$. Calculate: The number of holes recombined at the surface per unit time per unit surface area;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$u_{\\mathrm{s}}$": "recombination rate", + "$\\nu_{\\mathrm{s}}$": "recombination rate at the surface", + "$s_{\\mathrm{p}}$": "surface recombination velocity", + "$p(x)$": "hole concentration as a function of x", + "$p_{0}$": "equilibrium hole concentration", + "$\\tau_{\\mathrm{p}}$": "hole lifetime", + "$g_{\\mathrm{p}}$": "generation rate of holes", + "$L_{\\mathrm{p}}$": "diffusion length of holes", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$N_{\\mathrm{D}}$": "donor concentration" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 408, + "topic": "Semiconductors", + "question": "Illuminate a $1 \\Omega \\cdot \\mathrm{~cm}$ n-type silicon sample, uniformly generating non-equilibrium carriers, with an electron-hole pair generation rate of $10^{17} \\mathrm{~cm}^{-3} \\cdot \\mathrm{~s}^{-1}$. Assume the sample's lifetime is $10 \\mu \\mathrm{~s}$, and the surface recombination velocity is $100 \\mathrm{~cm} / \\mathrm{s}$. Calculate: The number of holes recombined within three diffusion lengths from the surface, per unit time and per unit surface area.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\Delta p$": "change in hole concentration", + "$L_{\\mathrm{p}}$": "diffusion length of holes", + "$p(x)$": "hole concentration as a function of position", + "$p_{0}$": "equilibrium hole concentration", + "$\\tau_{\\mathrm{p}}$": "hole lifetime", + "$g_{\\mathrm{p}}$": "hole generation rate", + "$s_{\\mathrm{p}}$": "surface recombination velocity of holes" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 409, + "topic": "Semiconductors", + "question": "A silicon wafer with a donor concentration of $2 \\times 10^{16} \\mathrm{~cm}^{-3}$ is saturated with gold at $920^{\\circ} \\mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \\mathrm{~cm}^{-2}$. Calculate the bulk lifetime;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\tau$": "bulk lifetime", + "$r_{\\mathrm{p}}$": "hole capture rate", + "$N_{\\mathrm{t}}$": "trap concentration", + "$\\gamma_{\\mathrm{p}}$": "hole capture rate of gold", + "$\\mu_{\\mathrm{p}}$": "mobility", + "$N_{\\mathrm{i}}$": "total impurity concentration", + "$N_{\\mathrm{D}}$": "donor concentration", + "$D_{\\mathrm{p}}$": "diffusion coefficient", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$q$": "elementary charge", + "$L_{\\mathrm{p}}$": "diffusion length", + "$\\tau_{\\mathrm{p}}$": "carrier lifetime", + "$s_{\\mathrm{p}}$": "surface recombination velocity", + "$N_{\\mathrm{st}}$": "surface trap density" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 410, + "topic": "Semiconductors", + "question": "A silicon wafer with a donor concentration of $2 \\times 10^{16} \\mathrm{~cm}^{-3}$ is saturated with gold at $920^{\\circ} \\mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \\mathrm{~cm}^{-2}$. Calculate the surface recombination velocity;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\tau$": "bulk lifetime", + "$r_{\\mathrm{p}}$": "hole recombination rate", + "$N_{\\mathrm{t}}$": "trap density", + "$\\gamma_{\\mathrm{p}}$": "hole capture rate of gold", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$N_{\\mathrm{i}}$": "total impurity concentration", + "$N_{\\mathrm{D}}$": "donor concentration", + "$D_{\\mathrm{p}}$": "hole diffusion coefficient", + "$L_{\\mathrm{p}}$": "hole diffusion length", + "$s_{\\mathrm{p}}$": "surface recombination velocity", + "$N_{\\mathrm{st}}$": "surface trap density" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 411, + "topic": "Semiconductors", + "question": "A silicon wafer with a dopant concentration of $2 \\times 10^{16} \\mathrm{~cm}^{-3}$ is gold-doped to saturation concentration at $920^{\\circ} \\mathrm{C}$. After oxidation and other treatments, the surface recombination center of the silicon wafer is $10^{10} \\mathrm{~cm}^{-2}$. If the silicon wafer is uniformly illuminated, and the generation rate of electron-hole pairs is $10^{11} \\mathrm{~cm}^{-3} \\cdot \\mathrm{~s}^{-1}$, what is the hole concentration at the surface?", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$p(x)$": "hole concentration at position x", + "$p_{0}$": "equilibrium hole concentration", + "$\\tau_{\\mathrm{p}}$": "hole lifetime", + "$g_{\\mathrm{p}}$": "generation rate of holes", + "$s_{\\mathrm{p}}$": "surface recombination velocity for holes", + "$L_{\\mathrm{p}}$": "diffusion length for holes", + "$x$": "distance from the surface", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$n_{0}$": "equilibrium electron concentration", + "$N_{\\mathrm{D}}$": "donor concentration", + "$N_{\\mathrm{t}}$": "trap concentration", + "$J_{\\mathrm{p}}$": "hole current density" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 412, + "topic": "Semiconductors", + "question": "A silicon wafer doped with a donor concentration of $2 \\times 10^{16} \\mathrm{~cm}^{-3}$ is saturated with gold at $920^{\\circ} \\mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \\mathrm{~cm}^{-2}$. If the silicon wafer is illuminated and uniformly absorbed by the sample, the generation rate of electron-hole pairs is $10^{11} \\mathrm{~cm}^{-3} \\cdot \\mathrm{~s}^{-1}$. What is the hole current density flowing towards the surface?", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$p$": "hole density", + "$p_{0}$": "equilibrium hole density", + "$\\tau_{\\mathrm{p}}$": "hole lifetime", + "$g_{\\mathrm{p}}$": "generation rate of holes", + "$s_{\\mathrm{p}}$": "surface recombination velocity", + "$L_{\\mathrm{p}}$": "diffusion length of holes", + "$n_{0}$": "equilibrium electron density", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$N_{\\mathrm{D}}$": "donor concentration", + "$N_{\\mathrm{t}}$": "trap concentration", + "$J_{\\mathrm{p}}$": "hole current density" + }, + "chapter": "Non-equilibrium carriers", + "section": "Non-equilibrium carriers" + }, + { + "id": 413, + "topic": "Semiconductors", + "question": "A pn junction composed of p-type germanium with a resistivity of $1 \\Omega \\cdot \\mathrm{~cm}$ and n-type germanium with a resistivity of $0.1 \\Omega \\cdot \\mathrm{~cm}$, calculate the built-in potential difference $V_{\\mathrm{D}}$ at room temperature (300 K). Given that at the above resistivities, the hole mobility in the p-type region $\\mu_{\\mathrm{p}}=1650 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, electron mobility in the n-type region $\\mu_{\\mathrm{n}}=3000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the intrinsic carrier concentration of germanium $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$V_{\\mathrm{D}}$": "built-in potential difference", + "$\\mu_{\\mathrm{p}}$": "hole mobility in the p-type region", + "$\\mu_{\\mathrm{n}}$": "electron mobility in the n-type region", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration of germanium", + "$N_{\\mathrm{A}}$": "acceptor concentration in p-type region", + "$N_{\\mathrm{D}}$": "donor concentration in n-type region" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 414, + "topic": "Semiconductors", + "question": "A pn junction composed of p-type germanium with resistivity $1 \\Omega \\cdot \\mathrm{~cm}$ and n-type germanium with resistivity $0.1 \\Omega \\cdot \\mathrm{~cm}$, calculate the width of the depletion region at room temperature (300 K). Given that at these resistivities, the hole mobility in the p region is $\\mu_{\\mathrm{p}}=1650 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the electron mobility in the n region is $\\mu_{\\mathrm{n}}=3000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, the intrinsic carrier concentration of germanium $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3} $.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\mu_{\\mathrm{p}}$": "hole mobility in the p region", + "$\\mu_{\\mathrm{n}}$": "electron mobility in the n region", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration of germanium", + "$p_{\\mathrm{p} 0}$": "hole concentration in the p-type region at equilibrium", + "$N_{\\mathrm{A}}$": "acceptor concentration in the p-type region", + "$q$": "elementary charge", + "$\\rho_{\\mathrm{p}}$": "resistivity of p-type germanium", + "$n_{n 0}$": "electron concentration in the n-type region at equilibrium", + "$N_{\\mathrm{D}}$": "donor concentration in the n-type region", + "$\\rho_{\\mathrm{n}}$": "resistivity of n-type germanium", + "$V_{\\mathrm{D}}$": "built-in potential difference", + "$k_{0}$": "Boltzmann constant", + "$T$": "absolute temperature", + "$x_{\\mathrm{p}}$": "depletion width in the p-type region", + "$\\varepsilon$": "permittivity", + "$\\varepsilon_{0}$": "vacuum permittivity", + "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity of germanium", + "$x_{\\mathrm{n}}$": "depletion width in the n-type region", + "$x$": "total depletion width" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 415, + "topic": "Semiconductors", + "question": "Given a silicon abrupt junction, with resistivities on both sides being $\\rho_{\\mathrm{n}}=10 \\Omega \\cdot \\mathrm{~cm}$ for $\\mathrm{n}-\\mathrm{Si}$ and $\\rho_{\\mathrm{p}}=0.01 \\Omega \\cdot \\mathrm{~cm}$ for $\\mathrm{p}-\\mathrm{Si}$, and mobilities $\\mu_{\\mathrm{n}}=100 \\mathrm{~cm}^{2}/(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{p}}=300 \\mathrm{~cm}^{2}/(\\mathrm{V} \\cdot \\mathrm{s})$, find the barrier width at room temperature.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\rho_{\\mathrm{n}}$": "resistivity of n-type silicon", + "$\\rho_{\\mathrm{p}}$": "resistivity of p-type silicon", + "$\\mu_{\\mathrm{n}}$": "mobility of electrons in n-type silicon", + "$\\mu_{\\mathrm{p}}$": "mobility of holes in p-type silicon", + "$q$": "charge of an electron", + "$N_{\\mathrm{D}}$": "donor concentration", + "$N_{\\mathrm{A}}$": "acceptor concentration", + "$n_{\\mathrm{n} 0}$": "minority electron concentration in p-region", + "$p_{\\mathrm{p} 0}$": "minority hole concentration in n-region", + "$V_{\\mathrm{D}}$": "built-in potential", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$x_{\\mathrm{n}}$": "depletion width in n-region", + "$x_{\\mathrm{D}}$": "total barrier width", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of silicon" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 416, + "topic": "Semiconductors", + "question": "Assume a silicon abrupt junction with the impurity concentrations on both sides as $N_{\\mathrm{A}}=10^{17} / \\mathrm{cm}^{3}, N_{\\mathrm{D}}=4.5 \\times 10^{15} / \\mathrm{cm}^{3}$. The intrinsic carrier concentration of silicon at room temperature is known as $n_{\\mathrm{i}} = 1.5 \\times 10^{10} / \\mathrm{cm}^{3}$, vacuum permittivity $\\varepsilon_{0} = 8.85 \\times 10^{-14} \\mathrm{F/cm}$, the relative permittivity of silicon $\\varepsilon_{\\mathrm{rs}} = 11.9$, elementary charge $q = 1.6 \\times 10^{-19} \\mathrm{C}$, thermal voltage $kT/q \\approx 0.026 \\mathrm{V}$. Determine the value of total depletion width $x_{\\mathrm{D}}$ at zero applied voltage.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$N_{\\mathrm{A}}$": "acceptor impurity concentration", + "$N_{\\mathrm{D}}$": "donor impurity concentration", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration of silicon", + "$\\varepsilon_{0}$": "vacuum permittivity", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of silicon", + "$q$": "elementary charge", + "$kT/q$": "thermal voltage", + "$x_{\\mathrm{D}}$": "total depletion width", + "$V_{\\mathrm{D}}$": "built-in potential", + "$x_{\\mathrm{n}}$": "depletion width in the n-region", + "$x_{\\mathrm{p}}$": "depletion width in the p-region", + "$V_{\\mathrm{R}}$": "reverse bias voltage", + "$V$": "applied voltage across the junction" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 417, + "topic": "Semiconductors", + "question": "Suppose a silicon abrupt junction, with impurity concentrations on either side as $N_{\\mathrm{A}}=10^{17} / \\mathrm{cm}^{3}, N_{\\mathrm{D}}=4.5 \\times 10^{15} / \\mathrm{cm}^{3}$, when a reverse bias voltage of 10 V is applied, find the value of $x_{\\mathrm{D}}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$N_{\\mathrm{A}}$": "acceptor impurity concentration", + "$N_{\\mathrm{D}}$": "donor impurity concentration", + "$x_{\\mathrm{D}}$": "depletion layer width", + "$x_{\\mathrm{n}}$": "width of the depletion region on the n-side", + "$x_{\\mathrm{p}}$": "width of the depletion region on the p-side", + "$V_{\\mathrm{D}}$": "built-in potential of the diode", + "$V_{\\mathrm{R}}$": "reverse bias voltage", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", + "$q$": "elementary charge", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 418, + "topic": "Semiconductors", + "question": "Given that the impurity concentration gradient $\\alpha_{\\mathrm{j}}$ of the linear graded junction of silicon is $10^{22} / \\mathrm{cm}^{4}, V_{\\mathrm{D}}=0.68 \\mathrm{~V}$, Find the value of the barrier width $x_{\\mathrm{D}}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$x_{\\mathrm{D}}$": "barrier width", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity", + "$V_{\\mathrm{D}}$": "built-in potential", + "$\\alpha_{\\mathrm{j}}$": "doping gradient", + "$E_{\\max}$": "maximum electric field", + "$q$": "elementary charge", + "$V$": "external voltage" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 419, + "topic": "Semiconductors", + "question": "Assume the impurity concentration gradient $\\alpha_{\\mathrm{j}}$ of silicon's linearly graded junction is $10^{22} / \\mathrm{cm}^{4}, V_{\\mathrm{D}}=0.68 \\mathrm{~V}$, Find its maximum electric field,", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$x_{\\mathrm{D}}$": "barrier width of the junction", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", + "$V_{\\mathrm{D}}$": "built-in voltage of the diode", + "$q$": "elementary charge", + "$\\alpha_{\\mathrm{j}}$": "gradient of the doping concentration", + "$E_{\\max}$": "maximum electric field", + "$V$": "applied voltage across the junction" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 420, + "topic": "Semiconductors", + "question": "Assume the impurity concentration gradient $\\alpha_{\\mathrm{j}}$ of a silicon linear gradient junction is $10^{22} / \\mathrm{cm}^{4}, V_{\\mathrm{D}}=0.68 \\mathrm{~V}$, Find the value of $x_{\\mathrm{D}}$ when an external reverse bias of 10 V is applied.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$x_{\\mathrm{D}}$": "barrier width of the linear gradient junction", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", + "$V_{\\mathrm{D}}$": "built-in potential across the p-n junction", + "$q$": "elementary charge", + "$\\alpha_{\\mathrm{j}}$": "doping gradient coefficient", + "$E_{\\max }$": "maximum electric field", + "$V$": "external voltage" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 421, + "topic": "Semiconductors", + "question": "For a silicon $n^+p$ junction, given $x_p = 0.2\\ \\mu\\text{m}$, $L_n = 200\\ \\mu\\text{m}$, $N_A = 10^{15}\\ \\text{cm}^{-3}$, $n_i = 1.5 \\times 10^{10}\\ \\text{cm}^{-3}$, at room temperature $T = 300\\ \\text{K}$, find the value of voltage V corresponding to the condition where the barrier recombination current equals the diffusion current.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$x_p$": "width of the p-type region", + "$L_n$": "electron diffusion length", + "$N_A$": "acceptor concentration", + "$n_i$": "intrinsic carrier concentration", + "$T$": "temperature", + "$V$": "voltage", + "$I_{\\mathrm{r}}$": "barrier recombination current", + "$I_{\\mathrm{FD}}$": "diffusion current", + "$q$": "elementary charge", + "$k_{0}$": "Boltzmann constant" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 422, + "topic": "Semiconductors", + "question": "The relationship between the barrier capacitance $C_{\\mathrm{T}}$ of the $\\mathrm{p}^{+} \\mathrm{n}$ junction made from GaP material and the reverse voltage $V_{\\mathrm{R}}$ is measured as follows\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline$V_{\\mathrm{R}}(\\mathrm{V})$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\\\\n\\hline$C_{\\mathrm{T}}(\\mathrm{pF})$ & 20 & 17.3 & 15.6 & 14.3 & 13.3 & 12.4 & 11.6 \\\\\n\\hline\n\\end{tabular}\n\nThe $pn$ junction area $A=4 \\times 10^{-4} \\mathrm{~cm}^{2}$, try to find the built-in potential $V_{\\mathrm{D}}$ of the $\\mathrm{p}^{+} \\mathrm{n}$ junction.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$C_{\\mathrm{T}}$": "barrier capacitance", + "$V_{\\mathrm{R}}$": "reverse voltage", + "$A$": "$pn$ junction area", + "$V_{\\mathrm{D}}$": "built-in potential", + "$N_{\\mathrm{A}}$": "acceptor concentration", + "$N_{\\mathrm{D}}$": "donor concentration", + "$B$": "constant related to material properties", + "$q$": "elementary charge", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of GaP material" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 423, + "topic": "Semiconductors", + "question": "The relationship between the barrier capacitance $C_{\\mathrm{T}}$ and reverse voltage $V_{\\mathrm{R}}$ of a $\\mathrm{p}^{+} \\mathrm{n}$ junction made of GaP material is measured as follows\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline$V_{\\mathrm{R}}(\\mathrm{V})$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\\\\n\\hline$C_{\\mathrm{T}}(\\mathrm{pF})$ & 20 & 17.3 & 15.6 & 14.3 & 13.3 & 12.4 & 11.6 \\\\\n\\hline\n\\end{tabular}\n\nThe pn junction area $A=4 \\times 10^{-4} \\mathrm{~cm}^{2}$, try to find the built-in field $N_{\\mathrm{D}}$ of this $\\mathrm{p}^{+} \\mathrm{n}$ junction.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$C_{\\mathrm{T}}$": "barrier capacitance", + "$V_{\\mathrm{R}}$": "reverse voltage", + "$A$": "pn junction area", + "$N_{\\mathrm{D}}$": "built-in field of the junction", + "$N_{\\mathrm{A}}$": "acceptor concentration", + "$q$": "elementary charge", + "$\\varepsilon_{0}$": "vacuum permittivity", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity", + "$V_{\\mathrm{D}}$": "diffusion potential", + "$B$": "constant related to capacitance and voltage" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 424, + "topic": "Semiconductors", + "question": "A pn junction diode has the following parameters: $N_{\\mathrm{D}}=10^{16} \\mathrm{~cm}^{-3}, ~ N_{\\mathrm{A}}=5 \\times 10^{18} \\mathrm{~cm}^{-3}, ~ \\tau_{\\mathrm{n}}=\\tau_{p}=$ $1 \\mu \\mathrm{~s}, ~ A=0.01 \\mathrm{~cm}^{2}$. Assume that the widths on both sides of the junction are much larger than the diffusion lengths of minority carriers. Find the applied voltage at room temperature (300 K) when the forward current is 1 mA. Assume the electron mobility in the p-type region $\\mu_{\\mathrm{n}}=500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the hole mobility in the n-type region $\\mu_{\\mathrm{p}}=180 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$N_{\\mathrm{D}}$": "donor concentration", + "$N_{\\mathrm{A}}$": "acceptor concentration", + "$\\tau_{\\mathrm{n}}$": "electron lifetime", + "$\\tau_{p}$": "hole lifetime", + "$A$": "area of the diode", + "$\\mu_{\\mathrm{n}}$": "electron mobility in the p-type region", + "$\\mu_{\\mathrm{p}}$": "hole mobility in the n-type region", + "$D_{\\mathrm{p}}$": "hole diffusion coefficient", + "$D_{\\mathrm{n}}$": "electron diffusion coefficient", + "$L_{\\mathrm{p}}$": "hole diffusion length", + "$L_{\\mathrm{n}}$": "electron diffusion length", + "$p_{\\mathrm{n} 0}$": "minority hole concentration in n-type region", + "$n_{\\mathrm{p} 0}$": "minority electron concentration in p-type region", + "$I_{0}$": "saturation current", + "$V$": "applied voltage", + "$V_{\\mathrm{T}}$": "thermal voltage", + "$I$": "forward current" + }, + "chapter": "pn junction", + "section": "pn junction" + }, + { + "id": 425, + "topic": "Semiconductors", + "question": "An n-type single crystal silicon wafer with [100] crystal orientation forms a Schottky diode with a certain metal contact. The parameters are $W_{\\mathrm{m}}=4.7 \\mathrm{eV}, \\chi_{\\mathrm{s}}=4.0 \\mathrm{eV}, N_{\\mathrm{c}}=10^{19} \\mathrm{~cm}^{-3}, N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}$, and the relative permittivity of semiconductor silicon is $\\varepsilon_{\\mathrm{r}}=$ 12. Ignoring the effect of surface states, calculate at room temperature: Zero-bias depletion width;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$N_{\\mathrm{D}}$": "donor concentration", + "$n_{0}$": "initial electron concentration", + "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", + "$E_{\\mathrm{c}}$": "conduction band energy", + "$E_{\\mathrm{F}}$": "Fermi level energy", + "$k_{0}$": "Boltzmann constant", + "$T$": "absolute temperature", + "$W_{\\mathrm{s}}$": "barrier height with respect to semiconductor", + "$\\chi_{\\mathrm{s}}$": "electron affinity of the semiconductor", + "$q$": "elementary charge", + "$V_{\\mathrm{D}}$": "built-in potential difference", + "$W_{\\mathrm{m}}$": "work function of the material", + "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity", + "$\\varepsilon_{0}$": "permittivity of free space", + "$d$": "depletion width" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 426, + "topic": "Semiconductors", + "question": "An n-type monocrystalline silicon wafer with [100] crystal orientation forms a Schottky diode upon contact with a certain metal. Its parameters are $W_{\\mathrm{m}}=4.7 \\mathrm{eV}, \\chi_{\\mathrm{s}}=4.0 \\mathrm{eV}, N_{\\mathrm{c}}=10^{19} \\mathrm{~cm}^{-3}, N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}$, and the relative dielectric constant of semiconductor silicon is $\\varepsilon_{\\mathrm{r}}=$ 12. Ignoring the effect of surface states, calculate at room temperature: The thermionic emission current when forward biased at 0.2 V. Assume $\\frac{A^{*}}{A}=2.1, A=120 \\mathrm{~A} / \\mathrm{cm}^{2}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$A$": "Richardson constant for a material", + "$A^{*}$": "modified Richardson constant", + "$q$": "elementary charge", + "$\\varphi_{\\mathrm{ns}}$": "surface potential difference", + "$V_{\\mathrm{D}}$": "diffusion voltage", + "$E_{\\mathrm{n}}$": "energy level difference", + "$J$": "current density", + "$T$": "temperature", + "$\\varphi_{\\mathrm{n}}$": "potential barrier height", + "$k_{0}$": "Boltzmann constant", + "$V$": "applied voltage" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 427, + "topic": "Semiconductors", + "question": "Consider a metal forming a Schottky diode with (111) crystal plane $\\mathrm{n}-\\mathrm{Si}$. It is known that the barrier height on the semiconductor side after contact is $0.50 \\mathrm{eV}, N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}, N_{\\mathrm{c}}=2.8 \\times 10^{19} \\mathrm{~cm}^{-3}$, electron affinity $X=4.05 \\mathrm{eV}, I_{\\mathrm{p}}=10 \\mu \\mathrm{~m}$, $D_{\\mathrm{p}}=15 \\mathrm{~cm}^{2} / \\mathrm{s}, n_{\\mathrm{i}}=1.5 \\times 10^{10} \\mathrm{~cm}^{-3}, A^{*}=252 \\mathrm{~A} / \\mathrm{cm}^{2} \\mathrm{~K}^{2}$ (Richardson constant), find: Calculate the minority carrier injection ratio.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\gamma$": "minority carrier injection ratio", + "$J_{\\mathrm{p}}$": "hole current density", + "$J_{\\mathrm{n}}$": "electron current density" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 428, + "topic": "Semiconductors", + "question": "A metal contacts a uniformly doped $n-Si$ material, forming a Schottky barrier diode. The barrier height on the semiconductor side is known as $qV_{\\mathrm{D}}=0.6 \\mathrm{eV}, N_{\\mathrm{D}}=5 \\times 10^{16} \\mathrm{~cm}^{-3}$. Calculate the maximum electric field in the semiconductor at the interface under a reverse bias of 5 V.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n-Si$": "n-type silicon", + "$q$": "elementary charge", + "$V_{\\mathrm{D}}$": "Schottky barrier height on the semiconductor side", + "$N_{\\mathrm{D}}$": "doping concentration in silicon", + "$E_{\\mathrm{M}}$": "maximum electric field at the semiconductor interface", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of silicon", + "$d$": "depletion width", + "$C$": "unit-area barrier capacitance" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 429, + "topic": "Semiconductors", + "question": "A metal contacts a uniformly doped $n-Si$ material, forming a Schottky barrier diode. Given the barrier height on the semiconductor side $q V_{\\mathrm{D}}=0.6 \\mathrm{eV}, N_{\\mathrm{D}}=5 \\times 10^{16} \\mathrm{~cm}^{-3}$, determine the barrier capacitance per unit area under a reverse bias voltage of 5 V.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n$": "indicates n-type semiconductor, such as silicon (Si)", + "$q$": "elementary charge", + "$V_{\\mathrm{D}}$": "barrier height on the semiconductor side", + "$N_{\\mathrm{D}}$": "doping concentration of the n-type semiconductor", + "$d$": "space charge region width", + "$E_{\\mathrm{M}}$": "maximum electric field in the semiconductor at the interface", + "$N$": "doping concentration, replaced with $N_{\\mathrm{D}}$ for calculation", + "$\\varepsilon_{0}$": "permittivity of free space", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", + "$V$": "applied reverse bias voltage", + "$C$": "barrier capacitance per unit area" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 430, + "topic": "Semiconductors", + "question": "A metal plate and n-type silicon are separated by $0.4 \\mu \\mathrm{~m}$, forming a parallel plate capacitor. The relative permittivity of the dry air between them is $\\varepsilon_{\\mathrm{ra}}=1$. When a negative voltage is applied to the metal side, the semiconductor is in a depletion state. Ignoring the work function difference between the metal and the semiconductor, what is the voltage $V_{\\mathrm{G}}$ on the metal plate when the depletion layer width just reaches its maximum? ($N_{\\mathrm{D}}=10^{16} \\mathrm{~cm}^{-3}$)", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$V_{\\mathrm{G}}$": "gate voltage", + "$N_{\\mathrm{D}}$": "donor concentration", + "$X_{\\mathrm{dm}}$": "maximum depletion layer width", + "$Q_{\\mathrm{s}}$": "surface charge density on the semiconductor", + "$V_{0}$": "voltage across the air gap", + "$C_{0}$": "air capacitance", + "$\\varepsilon_{\\mathrm{r} 0}$": "relative permittivity of air", + "$\\varepsilon_{0}$": "vacuum permittivity", + "$\\mathrm{d}_{0}$": "air gap distance" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 431, + "topic": "Semiconductors", + "question": "The MOS structure capacitor formed by metal-$\\mathrm{SiO}_{2}-\\mathrm{Si}$ (p-type), with hole concentration $N_{\\mathrm{A}}=1.5 \\times 10^{15}$ $\\mathrm{cm}^{-3}, ~ \\mathrm{SiO}_{2}$ layer thickness $d_{0}=0.2 \\mu \\mathrm{~m}$, its relative permittivity $\\varepsilon_{\\mathrm{r}_{0}}=3.9$, the relative permittivity of silicon $\\varepsilon_{\\mathrm{rs}}=12, \\varepsilon_{0}=$ $8.85 \\times 10^{-14} \\mathrm{~F} / \\mathrm{cm}$, at room temperature $n_{\\mathrm{i}}=1.5 \\times 10^{10} \\mathrm{~cm}^{-3}$. If there is a fixed positive charge at the SiO$_2$-silicon interface, and the measured $V_{\\mathrm{T}}=2.6 \\mathrm{~V}$, find the amount of fixed positive charge per unit area (neglecting the influence of the work function difference);", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$V_{\\mathrm{T}}$": "threshold voltage", + "$Q_{\\mathrm{fc}}$": "amount of fixed positive charge", + "$C_{0}$": "capacitance per unit area", + "$\\Delta V_{\\mathrm{T}}$": "change in threshold voltage", + "$N_{\\mathrm{fc}}$": "number of fixed positive charges per unit area", + "$q$": "elementary charge" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 432, + "topic": "Semiconductors", + "question": "The MOS structure capacitor composed of metal-$\\mathrm{SiO}_{2}-\\mathrm{Si}$ (p-type), with hole concentration $N_{\\mathrm{A}}=1.5 \\times 10^{15}$ $\\mathrm{cm}^{-3}, ~ \\mathrm{SiO}_{2}$ layer thickness $d_{0}=0.2 \\mu \\mathrm{~m}$, relative permittivity of $\\varepsilon_{\\mathrm{r}_{0}}=3.9$, silicon's relative permittivity $\\varepsilon_{\\mathrm{rs}}=12, \\varepsilon_{0}=$ $8.85 \\times 10^{-14} \\mathrm{~F} / \\mathrm{cm}$, intrinsic carrier concentration at room temperature $n_{\\mathrm{i}}=1.5 \\times 10^{10} \\mathrm{~cm}^{-3}$. If the above positive charges are uniformly distributed in $\\mathrm{SiO}_{2}$, what is the measured $V_{\\mathrm{T}}$? (neglecting the effect of work function difference);", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$V_{\\mathrm{T}}$": "threshold voltage", + "$V_{\\mathrm{FB}}$": "flatband voltage", + "$C_{0}$": "capacitance", + "$d_{0}$": "thickness of SiO2 layer", + "$\\rho_{0}$": "charge density", + "$x$": "position variable in integration" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 433, + "topic": "Semiconductors", + "question": "The metallurgical junction area of a gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode is $10^{-3} \\mathrm{~cm}^{2}$, and the overlap area of the gate with the n region is $10^{-3} \\mathrm{cm}^{2}$. The substrate impurity concentration is $10^{16} \\mathrm{~cm}^{-3}$, the junction depth is $5 \\mu \\mathrm{~m}$, the oxide layer thickness is $0.2 \\mu \\mathrm{~m}$, the lifetime $\\tau=1 \\mu \\mathrm{~s}$, the surface recombination velocity $s_{0}=5 \\mathrm{~cm} / \\mathrm{s}$, the flat-band voltage is -2 V. Calculate the gate voltage when the substrate surface is intrinsic (when the junction voltage is zero at room temperature).", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$V_{\\mathrm{G}}$": "gate voltage", + "$V_{\\mathrm{s}}$": "substrate voltage", + "$V_{0}$": "voltage drop across the SiO2 layer", + "$V_{\\mathrm{FB}}$": "flat-band voltage", + "$V_{\\mathrm{B}}$": "voltage related to the band bending", + "$E_{\\mathrm{i}}$": "intrinsic energy level", + "$E_{\\mathrm{F}}$": "Fermi energy level", + "$q$": "elementary charge", + "$n_{0}$": "carrier concentration at equilibrium", + "$N_{\\mathrm{D}}$": "donor concentration", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$Q_{\\mathrm{s}}$": "surface charge", + "$C_{0}$": "oxide capacitance", + "$x_{\\mathrm{d}}$": "depletion layer thickness", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the substrate", + "$\\varepsilon_{0}$": "permittivity of free space", + "$d_{0}$": "oxide thickness" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 434, + "topic": "Semiconductors", + "question": "The metallurgical junction area of a gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode is $10^{-3} \\mathrm{~cm}^{2}$, the overlap area between the gate and the n region is $10^{-3} \\mathrm{~cm}^{2}$, the substrate impurity concentration is $10^{16} \\mathrm{~cm}^{-3}$, the junction depth is $5 \\mu \\mathrm{~m}$, the oxide thickness is $0.2 \\mu \\mathrm{~m}$, the lifetime $\\tau=1 \\mu \\mathrm{~s}$, the surface recombination velocity $s_{0}=5 \\mathrm{~cm} / \\mathrm{s}$, the flat-band voltage is -2 V. For the gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode described in the problem (metallurgical junction area $10^{-3} \\mathrm{~cm}^{2}$, overlap area between the gate and n region $10^{-3} \\mathrm{~cm}^{2}$, substrate impurity concentration $10^{16} \\mathrm{~cm}^{-3}$, oxide thickness $0.2 \\mu \\mathrm{~m}$, minority carrier lifetime $\\tau=1 \\mu \\mathrm{~s}$, flat-band voltage $V_{FB} = -2 \\mathrm{~V}$), when the diode is subjected to a reverse bias voltage of $V_{\\mathrm{R}}=1 \\mathrm{~V}$ at room temperature, calculate the change of the forward current $\\Delta I_p = I_p(V_G=-20V) - I_p(V_G=0V)$ induced by varying the gate voltage $V_G$ from $0 \\mathrm{~V}$ to $-20 \\mathrm{~V}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\tau$": "minority carrier lifetime", + "$V_{FB}$": "flat-band voltage", + "$V_{\\mathrm{R}}$": "reverse bias voltage", + "$V_G$": "gate voltage", + "$V_{\\mathrm{s}}$": "surface potential", + "$V_{\\mathrm{B}}$": "built-in potential", + "$V_{\\mathrm{F}}$": "forward voltage", + "$x_{\\mathrm{dm}}$": "maximum width of the depletion layer", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", + "$N_{\\mathrm{D}}$": "dopant concentration", + "$I_p$": "forward current", + "$I_{\\mathrm{rF}}$": "recombination current in the barrier region", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$S_{0}$": "interface state density", + "$A_{\\mathrm{s}}$": "interfacial surface area", + "$I_{\\mathrm{rs}}$": "interface state contribution to the forward recombination current", + "$I_{\\mathrm{D}}$": "drift current", + "$I_{\\mathrm{rm}}$": "metallurgical junction recombination current" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 435, + "topic": "Semiconductors", + "question": "The metallurgical junction area of a gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode is $10^{-3} \\mathrm{~cm}^{2}$, and the overlap area between the gate and n-region is $10^{-3} \\mathrm{~cm}^{2}$. The substrate impurity concentration is $10^{16} \\mathrm{~cm}^{-3}$, the junction depth is $5 \\mu \\mathrm{~m}$, the oxide layer thickness is $0.2 \\mu \\mathrm{~m}$, lifetime $\\tau=1 \\mu \\mathrm{~s}$, surface recombination velocity $s_{0}=5 \\mathrm{~cm} / \\mathrm{s}$, and the flat-band voltage is -2 V. Calculate: For the gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode (with relevant parameters: gate to n-region overlap area $A_s = 10^{-3} \\mathrm{~cm}^{2}$, used to calculate the diffusion current; n-region substrate impurity concentration $N_D = 10^{16} \\mathrm{~cm}^{-3}$; minority carrier hole lifetime $\\tau_p = 1 \\mu \\mathrm{s}$; hole diffusion coefficient $D_p = 13 \\mathrm{~cm}^{2} / \\mathrm{s}$; intrinsic carrier concentration $n_i = 1.5 \\times 10^{10} \\mathrm{~cm}^{-3}$; thermal voltage $k_0T/q = 0.026 \\mathrm{~V}$), when the diode is under forward bias voltage $V_F = 0.4 \\mathrm{~V}$ at room temperature, calculate the diffusion current component $I_D$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$A_s$": "gate to n-region overlap area", + "$N_D$": "n-region substrate impurity concentration", + "$\\tau_p$": "minority carrier hole lifetime", + "$D_p$": "hole diffusion coefficient", + "$n_i$": "intrinsic carrier concentration", + "$k_0T/q$": "thermal voltage", + "$V_F$": "forward bias voltage", + "$I_D$": "diffusion current component", + "$I_p$": "hole current", + "$q$": "elementary charge", + "$p_{n_0}$": "equilibrium hole concentration in the n-region", + "$L_{p}$": "hole diffusion length", + "$I_{\\mathrm{rm}}$": "barrier region recombination current", + "$x_{\\mathrm{D}}$": "width of the depletion region", + "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of silicon", + "$\\varepsilon_{0}$": "permittivity of free space", + "$V$": "voltage across the diode" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 436, + "topic": "Semiconductors", + "question": "Estimate the injection ratio between GaAs and $\\mathrm{Al}_{0.3} \\mathrm{Ga}_{0.7} \\mathrm{As}$ at 300 K.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$x$": "composition variable", + "$E_{\\mathrm{g} x}^{\\mathrm{Al}_{x} \\mathrm{Ga}_{1-x} x_{\\mathrm{s}}}$": "band gap energy of AlGaAs as a function of composition", + "$E_{\\mathrm{g}}^{\\mathrm{Al}_{0.3} \\mathrm{Ga}_{0.7} \\mathrm{As}^{\\mathrm{As}}}$": "band gap energy of Al0.3Ga0.7As", + "$E_{\\mathrm{g}}^{\\mathrm{GaAs}}$": "band gap energy of GaAs", + "$\\Delta E_{\\mathrm{g}}$": "difference in band gap energies", + "$j_{\\mathrm{n} 1}$": "electron injection current density", + "$j_{\\mathrm{p} 2}$": "hole injection current density", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature" + }, + "chapter": "Metal-Semiconductor Contact", + "section": "Metal-Semiconductor Contact" + }, + { + "id": 437, + "topic": "Semiconductors", + "question": "Try to derive the relationship between the absorption coefficient $\\alpha$ and the extinction coefficient $\\bar{k}$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\alpha$": "absorption coefficient", + "$\\bar{k}$": "extinction coefficient", + "$\\omega$": "angular frequency", + "$c$": "speed of light in vacuum", + "$I$": "intensity of light", + "$E_{x}$": "electric field component in the x-direction", + "$E_{0}$": "initial electric field amplitude", + "$t$": "time", + "$n$": "refractive index", + "$z$": "propagation distance", + "$\\lambda$": "wavelength in vacuum" + }, + "chapter": "Optical properties of semiconductors, photoelectric and luminescence phenomena", + "section": "Optical properties of semiconductors, photoelectric and luminescence phenomena" + }, + { + "id": 438, + "topic": "Semiconductors", + "question": "There is an n-type CdS cubic chip, with an edge length of 1 mm and a thickness of 0.1 mm, with a wavelength absorption limit of $5100 \\AA$. Now, a violet light of intensity $1 \\mathrm{~mW} / \\mathrm{cm}^{2}$ $(\\lambda=4096 \\AA)$ is used to illuminate the square surface, with a quantum yield of $\\beta=1$. Assuming all photo-generated holes are trapped and the lifetime of photo-generated electrons is $\\tau_{\\mathrm{n}}=10^{-5} \\mathrm{~s}$, the electron mobility is $\\mu_{\\mathrm{n}}=100 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$; and assuming the illuminative energy is completely absorbed by the chip. Calculate the number of electron-hole pairs generated per second in the sample;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\beta$": "quantum efficiency", + "$Q$": "number of electron-hole pairs generated per second", + "$I$": "light intensity represented by the number of photons", + "$E_{0}$": "energy of each photon", + "$h$": "Planck's constant", + "$\\nu$": "frequency of the photon", + "$c$": "speed of light", + "$\\lambda$": "wavelength of the photon", + "$S$": "sample area" + }, + "chapter": "Optical properties of semiconductors, photoelectric and luminescence phenomena", + "section": "Optical properties of semiconductors, photoelectric and luminescence phenomena" + }, + { + "id": 439, + "topic": "Semiconductors", + "question": "There is an n-type CdS cubic wafer with a side length of 1 mm and a thickness of 0.1 mm, with an absorption edge wavelength of $5100 \\AA$. Now, the square surface is irradiated with purple light $(\\lambda=4096 \\AA)$ at an intensity of $1 \\mathrm{~mW} / \\mathrm{cm}^{2}$, with a quantum yield of $\\beta=1$. Assume all photogenerated holes are trapped, the lifetime of photogenerated electrons is $\\tau_{\\mathrm{n}}=10^{-5} \\mathrm{~s}$, and the electron mobility is $\\mu_{\\mathrm{n}}=100 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$; further assume all illumination energy is absorbed by the wafer. calculate the increase in the number of electrons in the sample;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\Delta n^{\\prime}$": "increase in the number of electrons in the sample", + "$Q$": "total charge", + "$\\tau_{\\mathrm{n}}$": "lifetime of charge carriers", + "$\\Delta n$": "increase in the number of electrons per unit volume", + "$V$": "volume" + }, + "chapter": "Optical properties of semiconductors, photoelectric and luminescence phenomena", + "section": "Optical properties of semiconductors, photoelectric and luminescence phenomena" + }, + { + "id": 440, + "topic": "Semiconductors", + "question": "There is an n-type CdS square crystal with a side length of 1 mm and a thickness of 0.1 mm. Its wavelength absorption limit is $5100 \\AA$. Now, violet light with an intensity of $1 \\mathrm{~mW} / \\mathrm{cm}^{2}$ $(\\lambda=4096 \\AA)$ illuminates the square surface, and the quantum yield is $\\beta=1$. Assume all photogenerated holes are trapped, the lifetime of the photogenerated electrons is $\\tau_{\\mathrm{n}}=10^{-5} \\mathrm{~s}$, and the electron mobility is $\\mu_{\\mathrm{n}}=100 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$; also assume that the illumination energy is completely absorbed by the crystal. When the photocurrent when a 50 V voltage is applied to the sample, calculate photoconductive gain factor.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$G$": "photoconductive gain factor", + "$\\tau_{\\mathrm{n}}$": "carrier lifetime", + "$\\tau_{\\mathrm{t}}$": "transit time", + "$\\mu_{\\mathrm{n}}$": "mobility", + "$V$": "voltage", + "$l$": "characteristic length" + }, + "chapter": "Optical properties of semiconductors, photoelectric and luminescence phenomena", + "section": "Optical properties of semiconductors, photoelectric and luminescence phenomena" + }, + { + "id": 441, + "topic": "Semiconductors", + "question": "Given a piece of n-type semiconductor material with a room temperature dark conductivity of $100 \\mathrm{~S} / \\mathrm{cm}$, when illuminated with light at an intensity of $I=$ $10^{-6} \\mathrm{~W} / \\mathrm{cm}^{2}$, its absorption coefficient $\\alpha=10^{2} / \\mathrm{cm}$, the measured ratio of steady-state photoconductivity to dark conductivity $\\gamma$ $=10$, and the lifetime $\\tau$ is $10^{-4} \\mathrm{~s}, b=\\mu_{\\mathrm{n}} / \\mu_{\\mathrm{p}}=10, \\mu_{\\mathrm{n}}=10000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, determine the corresponding quantum yield.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$I$": "light intensity", + "$\\alpha$": "absorption coefficient", + "$\\gamma$": "ratio of steady-state photoconductivity to dark conductivity", + "$b$": "mobility ratio of electrons to holes", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$\\tau$": "lifetime", + "$q$": "elementary charge", + "$\\sigma_{0}$": "dark conductivity", + "$\\sigma$": "photoconductivity", + "$\\Delta n$": "change in electron concentration", + "$\\beta$": "quantum yield", + "$\\tau_{\\mathrm{n}}$": "electron lifetime", + "$n_{0}$": "initial electron concentration", + "$p_{0}$": "initial hole concentration" + }, + "chapter": "Optical properties of semiconductors, photoelectric and luminescence phenomena", + "section": "Optical properties of semiconductors, photoelectric and luminescence phenomena" + }, + { + "id": 442, + "topic": "Semiconductors", + "question": "In a p-type silicon with a hole concentration of $10^{16} \\mathrm{~cm}^{-3}$, a cold end temperature of $0^{\\circ} \\mathrm{C}$, and a hot end temperature of $50^{\\circ} \\mathrm{C}$, assuming long wavelength acoustic wave scattering, calculate the thermoelectric power.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\alpha_{\\mathrm{p}}$": "thermopower of a p-type semiconductor", + "$k_{0}$": "Boltzmann constant", + "$q$": "elementary charge", + "$\\gamma$": "scattering parameter for acoustic wave scattering", + "$\\xi_{\\mathrm{p}}$": "reduced Fermi energy fraction for p-type semiconductor", + "$E_{\\mathrm{F}}$": "Fermi energy", + "$E_{\\mathrm{v}}$": "valence band edge energy", + "$p$": "hole concentration", + "$N_{\\mathrm{v}}$": "effective density of states in the valence band", + "$T$": "absolute temperature", + "$\\Delta T$": "temperature difference between hot and cold ends", + "$N_{\\mathrm{v}^{\\prime}}$": "adjusted effective density of states in the valence band at cold end" + }, + "chapter": "Thermoelectric properties of semiconductors", + "section": "Thermoelectric properties of semiconductors" + }, + { + "id": 443, + "topic": "Semiconductors", + "question": "For n-type PoTe with a conductivity of $2000 \\mathrm{~S} / \\mathrm{cm}$, an electron mobility of $6000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and an electron effective mass of $0.2 m_{0}$, determine the thermoelectric power factor at room temperature assuming long-wavelength acoustic phonon scattering.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n$": "charge carrier concentration", + "$g$": "degeneracy factor", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$q$": "elementary charge", + "$\\gamma$": "gamma parameter for scattering", + "$\\xi_{\\mathrm{n}}$": "reduced Fermi energy", + "$\\sigma$": "electrical conductivity", + "$m_{\\mathrm{n}}^{*}$": "electron effective mass", + "$m_{0}$": "electron rest mass", + "$N_{\\mathrm{c}}$": "effective density of states for conduction band", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$h$": "Planck constant", + "$\\alpha_{\\mathrm{n}}$": "Seebeck coefficient for n-type semiconductor", + "$\\pi_{\\mathrm{ab}}$": "Peltier coefficient" + }, + "chapter": "Thermoelectric properties of semiconductors", + "section": "Thermoelectric properties of semiconductors" + }, + { + "id": 444, + "topic": "Semiconductors", + "question": "For an n-type PoTe with a conductivity of $2000 \\mathrm{~S} / \\mathrm{cm}$, electron mobility of $6000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and an electron effective mass of $0.2 m_{0}$, assuming long-wavelength acoustic scattering, determine the Peltier coefficient at room temperature.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n$": "electron density", + "$\\sigma$": "conductivity", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$m_{n}^{*}$": "electron effective mass", + "$q$": "elementary charge", + "$N_{\\mathrm{c}}$": "effective density of states", + "$k_{0}$": "Boltzmann constant", + "$T$": "temperature", + "$\\alpha_{\\mathrm{n}}$": "thermoelectric power factor", + "$\\gamma$": "scattering parameter", + "$\\xi_{\\mathrm{n}}$": "dimensionless factor in thermoelectric power", + "$\\pi_{\\mathrm{ab}}$": "Peltier coefficient" + }, + "chapter": "Thermoelectric properties of semiconductors", + "section": "Thermoelectric properties of semiconductors" + }, + { + "id": 445, + "topic": "Semiconductors", + "question": "The thermal conductivity of bismuth telluride ($\\mathrm{Bi}_{2} \\mathrm{Te}_{3}$) is $2.4[\\mathrm{~W} /(\\mathrm{m} \\cdot \\mathrm{K})$]. Calculate the percentage contribution of carrier to the thermal conductivity for n-type $\\mathrm{Bi}_{2} \\mathrm{Te}_{3}$ at $10^{5} \\mathrm{~s} / \\mathrm{m}$ and $300 \\mathrm{~K}$. (Assume long-wavelength acoustic phonon scattering)", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$K_{\\mathrm{c}}$": "thermal conductivity contribution of carriers", + "$\\sigma$": "electrical conductivity", + "$T$": "temperature", + "$k_{0}$": "Boltzmann constant", + "$q$": "elementary charge", + "$K$": "total thermal conductivity" + }, + "chapter": "Thermoelectric properties of semiconductors", + "section": "Thermoelectric properties of semiconductors" + }, + { + "id": 446, + "topic": "Semiconductors", + "question": "Try to find the Seebeck coefficient of intrinsic silicon at room temperature.\nAssume the effective masses of electrons and holes are equal, the band gap of silicon is 1.12 eV, and the mobilities of electrons and holes are $0.135 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $0.048 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, respectively.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$k$": "Boltzmann constant", + "$q$": "elementary charge", + "$p$": "hole concentration", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$n$": "electron concentration", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\xi_{\\mathrm{p}}$": "reduced Fermi level for holes", + "$\\xi_{\\mathrm{n}}$": "reduced Fermi level for electrons", + "$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", + "$m_{\\mathrm{p}}^{*}$": "effective mass of holes", + "$E_{\\mathrm{i}}$": "intrinsic Fermi level energy", + "$E_{\\mathrm{g}}$": "band gap energy", + "$k_{0}$": "Boltzmann constant (alternate symbol)", + "$T$": "temperature" + }, + "chapter": "Thermoelectric properties of semiconductors", + "section": "Thermoelectric properties of semiconductors" + }, + { + "id": 447, + "topic": "Semiconductors", + "question": "For an indium antimonide sample, the hole concentration at room temperature is 9 times the electron concentration. Calculate the Hall coefficient $R$. Assume at room temperature $b=\\mu_{\\mathrm{n}} / \\mu_{\\mathrm{p}}=100, n_{\\mathrm{i}}=1.1 \\times 10^{16} \\mathrm{~cm}^{-3}$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$R$": "Hall coefficient", + "$b$": "mobility ratio", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$n$": "electron concentration", + "$p$": "hole concentration", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$q$": "elementary charge" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 448, + "topic": "Semiconductors", + "question": "InSb electron mobility is $7.8 \\mathrm{~m}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and hole mobility is $780 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, with an intrinsic carrier concentration of $1.6 \\times 10^{16} \\mathrm{~cm}^{-3}$, at 300 K Hall coefficient of intrinsic material;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$R_{\\mathrm{H}}$": "Hall coefficient", + "$p$": "hole concentration", + "$n$": "electron concentration", + "$b$": "ratio of mobilities of electrons to holes", + "$q$": "elementary charge", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 449, + "topic": "Semiconductors", + "question": "The electron mobility of InSb is $7.8 \\mathrm{~m}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the hole mobility is $780 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. The intrinsic carrier concentration is $1.6 \\times 10^{16} \\mathrm{~cm}^{-3}$, find at 300 K intrinsic resistivity;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\rho_{\\mathrm{i}}$": "intrinsic resistivity", + "$\\sigma_{\\mathrm{i}}$": "intrinsic conductivity", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 450, + "topic": "Semiconductors", + "question": "The electron mobility of InSb is $7.8 \\mathrm{~m}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the hole mobility is $780 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. The intrinsic carrier concentration is $1.6 \\times 10^{16} \\mathrm{~cm}^{-3}$ at 300 K. When $B_{z}=0.1 \\mathrm{~Wb} / \\mathrm{m}^{2}$, calculate the resistivity of the material considering the scattering of long acoustic waves.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$B_{z}$": "magnetic field component in the z direction", + "$\\theta$": "angle related to the magnetic field and material parameters", + "$\\mu_{\\mathrm{H}}$": "Hall mobility", + "$\\sigma$": "electrical conductivity", + "$R_{\\mathrm{H}}$": "Hall coefficient", + "$\\rho$": "resistivity", + "$\\rho_{0}$": "initial resistivity", + "$\\xi$": "transverse magnetoresistance coefficient", + "$n$": "electron concentration", + "$p$": "hole concentration", + "$b$": "parameter related to the carrier concentration ratios", + "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", + "$\\mu_{\\mathrm{n}}$": "mobility of electrons", + "$\\mu_{\\mathrm{p}}$": "mobility of holes" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 451, + "topic": "Semiconductors", + "question": "For n-type GaAs with a thickness of 0.08 cm, a current of 50 mA is passed along the $x$ direction, and a magnetic field of 0.5 T is applied along the $z$ direction, resulting in a Hall voltage of -0.4 mV. Hall coefficient;", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$R_{\\mathrm{H}}$": "Hall coefficient", + "$V_{\\mathrm{H}}$": "Hall voltage", + "$d$": "thickness of the material", + "$I_{x}$": "current in the material", + "$B_{z}$": "magnetic field in the z direction" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 452, + "topic": "Semiconductors", + "question": "A silicon sample with a conductivity of $0.001 /(\\Omega \\cdot \\mathrm{cm})$ has zero Hall voltage under a weak magnetic field. Assuming the electron mobility $\\mu_{\\mathrm{n}}=1300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and the hole mobility $\\mu_{\\mathrm{p}}=300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the Hall factors for electrons and holes are the same. Try to determine the carrier density of electrons $n$ in the sample.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$n$": "carrier density of electrons", + "$b$": "ratio of hole to electron mobility", + "$\\sigma$": "conductivity", + "$e$": "elementary charge", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$p$": "carrier density of holes", + "$R$": "Hall coefficient" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 453, + "topic": "Semiconductors", + "question": "Given the band gap of InSb $E_{\\mathrm{g}}=0.15 \\mathrm{eV}$, the effective mass of electrons $m_{\\mathrm{e}}=0.014 m_{0}$, and the effective mass of holes $m_{\\mathrm{h}}=0.18 m_{0}$ (with $m_{0}$ as the inertial mass of the electron). If only electrons are the effective carriers, calculate the Hall coefficient of intrinsic InSb at 300 K.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$E_{\\mathrm{g}}$": "band gap energy", + "$m_{\\mathrm{e}}$": "effective mass of electrons", + "$m_{\\mathrm{h}}$": "effective mass of holes", + "$m_{0}$": "inertial mass of the electron", + "$R_{\\mathrm{H}}$": "Hall coefficient", + "$n$": "electron concentration", + "$k_{B}$": "Boltzmann constant", + "$T$": "temperature", + "$h$": "Planck's constant", + "$b$": "width of the sample", + "$d$": "thickness of the sample", + "$V_{\\mathrm{H}}$": "Hall voltage", + "$B_{z}$": "magnetic field component in the z-direction", + "$J$": "current density", + "$I$": "current intensity" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 454, + "topic": "Semiconductors", + "question": "Try to demonstrate in the Hall effect under conditions of two types of carriers and a weak magnetic field, the Hall angle $\\theta$ and Hall coefficient $R$ can be expressed as\n\n\\begin{align}\n& \\theta=\\arctan \\frac{p \\mu_{\\mathrm{p}}^{2}-n \\mu_{\\mathrm{n}}^{2}}{p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}}} B_{z} \\\\\n& R=\\frac{1}{q} \\frac{p \\mu_{\\mathrm{p}}^{2}-n \\mu_{\\mathrm{n}}^{2}}{(p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}})^{2}}\n\\end{align} If a given germanium sample is placed in a magnetic field of $B=0.1$ T, what is $\\tan \\theta$ when its conductivity is at a minimum? Let $\\mu_{\\mathrm{n}}=3900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{p}}=1$ $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$B$": "magnetic field", + "$\\mu_{\\mathrm{n}}$": "electron mobility", + "$\\mu_{\\mathrm{p}}$": "hole mobility", + "$\\theta$": "angle related to conductivity" + }, + "chapter": "Semiconductor magneto- and piezoresistive effects", + "section": "Semiconductor magneto- and piezoresistive effects" + }, + { + "id": 455, + "topic": "Others", + "question": "Briefly answer the following questions: Classified by symmetry type, how many types of point groups are there for Bravais lattices? How many types of space groups are there? How many types of point groups are there for crystal structures? How many types of space groups are there? You should return your answer as a tuple format.", + "final_answer": [], + "answer_type": "Tuple", + "answer": "", + "symbol": {}, + "chapter": "Crystal structure", + "section": "Crystal structure" + }, + { + "id": 456, + "topic": "Others", + "question": "A one-dimensional atomic chain consisting of N identical atoms with mass m and spacing a. Each atom has only one valence electron. Using the tight-binding approximation, only nearest-neighbor interactions are considered, Derive the expression for the density of states of the s-band electrons.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$G(E)$": "density of states", + "$L$": "length of the system", + "$a$": "lattice constant", + "$J_{1}$": "hopping integral", + "$k$": "wave vector", + "$N$": "number of states", + "$E_{0}$": "edge of the energy band", + "$E$": "energy" + }, + "chapter": "Band theory", + "section": "Band theory" + }, + { + "id": 457, + "topic": "Semiconductors", + "question": "The valence band of a semiconductor material is almost filled with electrons (nearly full band), and the expression for the energy of valence band electrons is $E(k)=-1.016 \\times 10^{-34} k^{2}(J)$, where the energy zero point is taken at the top of the valence band. At this time, if the electron at $k=1 \\times 10^{6} \\mathrm{~cm}^{-1}$ is excited to a higher energy band (conduction band), a hole is generated at this location. Try to find energy of this hole.", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$E$": "energy of valence band electrons", + "$k$": "wave vector", + "$m_{h}^{*}$": "effective mass of the hole", + "$m_{e}^{*}$": "effective mass of the electron", + "$m_{0}$": "rest mass of electron", + "$k_{h}$": "wave vector of the hole", + "$k_{e}$": "wave vector of the electron", + "$p_{h}$": "quasi-momentum of the hole", + "$v_{h}$": "velocity of the hole", + "$E_{h}$": "energy of the hole", + "$E_{e}$": "energy of the electron" + }, + "chapter": "Movement of electrons in a crystal in electric and magnetic fields", + "section": "Movement of electrons in a crystal in electric and magnetic fields" + }, + { + "id": 458, + "topic": "Others", + "question": "Calculate the mean free path of electrons at room temperature (T=295K). (The density of silver is $10.5 \\mathrm{~g} / \\mathrm{cm}^{3}$, atomic weight is 107.87, and its resistivity at T=295K is $1.61 \\times 10^{-6} \\Omega \\cdot \\mathrm{~cm}$)", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$T$": "temperature", + "$\\lambda$": "mean free path", + "$\\tau$": "relaxation time", + "$v_{F}$": "Fermi velocity", + "$\\hbar$": "reduced Planck's constant", + "$k_{F}$": "Fermi wavevector", + "$m$": "mass of electron", + "$\\sigma$": "electrical conductivity", + "$n$": "electron density", + "$e$": "elementary charge", + "$\\rho$": "resistivity" + }, + "chapter": "Metal Electron Theory", + "section": "Metal Electron Theory" + }, + { + "id": 459, + "topic": "Superconductivity", + "question": "Phase transition in BCS superconducting systems\n Consider the Hamiltonian with parameter $\\lambda$, $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$, and the corresponding Gibbs free energy is $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$. When $T>0K$, Feynman's theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For a BCS superconducting system, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.\n\n \n\n Please:\n Use Feynman's theorem to derive the difference in Gibbs free energy between the normal state and the superconducting phase\n\n Hint: If the answer is an integral with respect to $\\lambda$, you can just output the integrand.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\lambda$": "coupling parameter", + "$\\bar{H}$": "Hamiltonian", + "$\\bar{H}_0$": "free Hamiltonian component", + "$\\bar{H}_{\\mathrm{int}}$": "interaction Hamiltonian component", + "$\\Gamma(\\lambda)$": "Gibbs free energy", + "$k_B$": "Boltzmann constant", + "$T$": "temperature", + "$G(\\lambda)$": "Gibbs free energy as a function of $\\lambda$", + "$\\Delta$": "superconducting gap parameter", + "$V$": "interaction constant", + "$g(0)$": "density of states at the Fermi surface", + "$C_k$": "annihilation operator", + "$C_k^\\dagger$": "creation operator", + "$G_S$": "Gibbs free energy in superconducting state", + "$G_N$": "Gibbs free energy in normal state", + "$\\epsilon$": "energy variable", + "$\\xi(\\lambda)$": "quasiparticle energy" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 460, + "topic": "Superconductivity", + "question": "Phase transition in BCS superconducting systems\n Consider a Hamiltonian with a parameter $\\lambda$ given by $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$, with the corresponding Gibbs free energy $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$, when $T>0K$, Feynman's theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For a BCS superconducting system, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states on the Fermi surface is $g(0)$.\n\n Please:\n Derive the approximate expression for the difference in free energy near the superconducting critical temperature", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\lambda$": "interaction parameter in the Hamiltonian", + "$\\bar{H}$": "total Hamiltonian", + "$\\bar{H}_0$": "non-interacting part of the Hamiltonian", + "$\\bar{H}_{\\mathrm{int}}$": "interaction part of the Hamiltonian", + "$\\Gamma$": "Gibbs free energy", + "$k_B$": "Boltzmann constant", + "$T$": "temperature", + "$\\beta$": "inverse temperature", + "$\\Delta$": "superconducting energy gap", + "$V$": "interaction constant", + "$g(0)$": "density of states on Fermi surface", + "$C_k$": "annihilation operator for state k", + "$\\omega_n$": "Matsubara frequency", + "$\\epsilon$": "energy variable", + "$T_c$": "critical temperature for superconductivity", + "$\\zeta$": "Riemann zeta function" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 461, + "topic": "Superconductivity", + "question": "Phase transition in BCS superconducting systems\n Consider a Hamiltonian with parameter $\\lambda$ given by $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$, and the corresponding Gibbs free energy is $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$. When $T>0K$, the Feynman theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For BCS superconducting systems, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.\n\n Please:\n Calculate the change in entropy $\\Delta S = S_\\mathrm{S}-S_\\mathrm{N}$, and analyze the physical significance of the results", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\lambda$": "parameter of the Hamiltonian", + "$\\bar{H}$": "Hamiltonian", + "$\\bar{H}_0$": "unperturbed Hamiltonian", + "$\\bar{H}_{\\mathrm{int}}$": "interaction Hamiltonian", + "$\\Gamma$": "Gibbs free energy", + "$k_B$": "Boltzmann constant", + "$T$": "temperature", + "$G$": "Gibbs energy", + "$\\Delta$": "superconducting gap parameter", + "$V$": "interaction constant", + "$C_k$": "creation operator for a particle with momentum $k$", + "$g(0)$": "density of states at the Fermi surface", + "$\\Delta S$": "change in entropy between superconducting and normal states", + "$S_\\mathrm{S}$": "entropy of the superconducting state", + "$S_\\mathrm{N}$": "entropy of the normal state", + "$T_c$": "critical temperature" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 462, + "topic": "Superconductivity", + "question": "Phase transition in BCS superconducting system\n Consider a Hamiltonian with parameter $\\lambda$ given by $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$, the corresponding Gibbs free energy is $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$, when $T>0K$, the Feynman theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For a BCS superconducting system, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.\n\n Please:\n Calculate the discontinuity in the electronic specific heat.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\lambda$": "parameter for interaction", + "$\\bar{H}$": "Hamiltonian", + "$\\bar{H}_0$": "non-interacting Hamiltonian component", + "$\\bar{H}_{\\mathrm{int}}$": "interaction Hamiltonian", + "$G$": "Gibbs free energy", + "$T$": "temperature", + "$k_B$": "Boltzmann constant", + "$\\Delta$": "energy gap parameter", + "$V$": "interaction constant", + "$C_k$": "annihilation operator for momentum k", + "$C_{-k}$": "annihilation operator for momentum -k", + "$g(0)$": "density of states at the Fermi surface", + "$T_c$": "critical temperature", + "$\\Delta S$": "change in entropy" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 463, + "topic": "Superconductivity", + "question": "Phase transition in BCS superconducting system\n Consider a Hamiltonian $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$ with parameter $\\lambda$, the corresponding Gibbs free energy is $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$, when $T>0K$, Feynman's theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For BCS superconducting systems, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.\n\n Please:\n Calculate the critical magnetic field $H_c(T)$ with Euler constant", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\bar{H}$": "Hamiltonian", + "$\\bar{H}_0$": "unperturbed Hamiltonian", + "$\\bar{H}_{\\mathrm{int}}$": "interaction Hamiltonian", + "$\\lambda$": "coupling parameter", + "$\\Gamma(\\lambda)$": "Gibbs free energy as a function of lambda", + "$k_B$": "Boltzmann constant", + "$T$": "temperature", + "$\\beta$": "inverse thermal energy (1/k_B T)", + "$G(\\lambda)$": "Gibbs free energy with respect to lambda", + "$< \\bar{H}_{\\text{int}} (\\lambda) >_T$": "thermal expectation value of interaction Hamiltonian", + "$\\Delta$": "superconducting gap parameter", + "$V$": "interaction constant", + "$C_k$": "annihilation operator for state k", + "$C_k^\\dagger$": "creation operator for state k", + "$g(0)$": "density of states at the Fermi surface", + "$H_c(T)$": "critical magnetic field as a function of temperature", + "$H_c(0)$": "critical magnetic field at zero temperature", + "$T_c$": "critical temperature", + "$e$": "Euler's number", + "$\\gamma$": "Euler-Mascheroni constant" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 464, + "topic": "Superconductivity", + "question": "London Theory\n\n Superconductors have two properties:\n \\begin{itemize}\n \\item[(i)] The DC resistance disappears when $T < T_c$, and a resistance-free supercurrent exists, which is the ideal conductivity of the superconductor.\n \\item[(ii)] The Meissner effect, a weak magnetic field cannot penetrate the interior of a bulk superconducting sample, exhibiting complete diamagnetism.\n\\end{itemize}\n Now consider the following current\n \\begin{equation}\\label{eq:6.7.1}\n \\mathbf{j}_s = - \\frac{c}{\\Lambda}\\mathbf{A} \\quad \\Lambda = \\frac{m}{n_se^2},\n \\end{equation}\n where at $T=0K$ $n_s=n$ is the density of conduction electrons, choosing the transverse field condition $\\nabla \\cdot \\mathbf{A}=0$, please explain using Maxwell's equations how the above expression encompasses the two main properties of the superconductor.\n\n1. Ideal Conductivity You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$T$": "temperature", + "$T_c$": "critical temperature", + "$\\mathbf{j}_s$": "supercurrent density", + "$c$": "speed of light", + "$\\Lambda$": "London parameter", + "$m$": "electron mass", + "$n_s$": "density of superconducting electrons", + "$e$": "elementary charge", + "$\\mathbf{A}$": "vector potential", + "$\\boldsymbol{v}_s$": "velocity of superconducting electrons", + "$\\boldsymbol{E}$": "electric field", + "$n$": "density of conduction electrons", + "$\\nabla$": "nabla operator (grad)" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 465, + "topic": "Superconductivity", + "question": "London theory\n\n\n Superconductors have two properties:\n \\begin{itemize}\n \\item[(i)] When $T < T_c$, the DC resistance disappears, and there exists a resistance-free supercurrent, which is the ideal conductivity of a superconductor.\n \\item[(ii)] Meissner effect, a weak magnetic field cannot penetrate inside a bulk superconducting sample, exhibiting perfect diamagnetism.\n\\end{itemize}\n Now consider the following current\n \\begin{equation}\\label{eq:6.7.1}\n \\mathbf{j}_s = - \\frac{c}{\\Lambda}\\mathbf{A} \\quad \\Lambda = \\frac{m}{n_se^2},\n \\end{equation}\n where when $T=0K$, $n_s=n$ is the density of conduction electrons. Choose the transversal gauge condition $\\nabla \\cdot \\mathbf{A}=0$, and using Maxwell's equations, explain how the above expression encompasses the two main properties of a superconductor.\n\n Consider a semi-infinite superconducting sample with $z > 0$, and explain the perfect diamagnetism when the external magnetic field is parallel to the $z=0$ plane.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$T$": "temperature", + "$T_c$": "critical temperature", + "$c$": "speed of light in vacuum", + "$\\Lambda$": "London parameter", + "$m$": "mass of an electron", + "$n_s$": "density of superconducting electrons", + "$e$": "elementary charge", + "$n$": "density of conduction electrons", + "$\\mathbf{j}_s$": "supercurrent density", + "$\\mathbf{A}$": "magnetic vector potential", + "$\\boldsymbol{B}$": "magnetic induction", + "$\\lambda_L$": "London penetration depth" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 466, + "topic": "Superconductivity", + "question": "Pippard Theory\n\n In superconductors, within the coherence length $\\xi_0 = \\frac{\\hbar v_F}{\\pi \\Delta(0)}$, there exists a correlation of electron motion, hence a perturbing potential acting at one point will inevitably affect the velocity of superconducting electrons and current density within the spatial scale of $\\xi_0$. Conversely, the supercurrent density at a certain point in space must also be influenced by the perturbing potentials within its neighboring scale of $\\xi_0$. Thus, $j_s (\\boldsymbol{r})$ depends not only on $\\boldsymbol{A}(\\boldsymbol{r})$ at the same point but should also include contributions from the vector potential $\\boldsymbol{A}(\\boldsymbol{r}')$ at all points $\\boldsymbol{r}'$ within the range $|\\boldsymbol{r} - \\boldsymbol{r}'| < \\xi_0$. For this reason, one can consider the nonlocal relationship between $j_s$ and $\\boldsymbol{A}$ as (Pippard Theory):\n \\begin{equation*}\n j_s (\\boldsymbol{r}) = - \\frac{3}{4\\pi \\xi_0 \\lambda_c} \\int \\frac{\\boldsymbol{R} [\\boldsymbol{R} \\cdot \\boldsymbol{A}(\\boldsymbol{r}')]}{R^4} e^{-R/\\xi_0} d\\boldsymbol{r}' \\quad (\\boldsymbol{R} = \\boldsymbol{r} - \\boldsymbol{r}')\n \\end{equation*}\n Complete the following question:\n Analyze the approximate value of $j_s$ under the condition $q\\xi_0 \\ll 1$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\xi_0$": "coherence length", + "$\\hbar$": "reduced Planck's constant", + "$v_F$": "Fermi velocity", + "$\\Delta(0)$": "energy gap at zero temperature", + "$j_s$": "supercurrent density", + "$\\boldsymbol{r}$": "position vector", + "$\\boldsymbol{A}$": "vector potential", + "$\\lambda_c$": "characteristic length scale", + "$\\boldsymbol{R}$": "relative position vector", + "$q$": "wave vector magnitude", + "$\\lambda_L(0)$": "London penetration depth at zero temperature" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 467, + "topic": "Superconductivity", + "question": "Pippard Theory\n In superconductors, there is a correlation of electron motion within the coherence length $\\xi_0 = \\frac{\\hbar v_F}{\\pi \\Delta(0)}$, thus a perturbative potential acting at one point will inevitably affect the velocity and current density of superconducting electrons within the spatial scale of $\\xi_0$. Conversely, the supercurrent density at a certain point in space will inevitably be influenced by the perturbative potential within the scale of $\\xi_0$ around it. Therefore, $j_s (\\boldsymbol{r})$ is not only determined by $\\boldsymbol{A}(\\boldsymbol{r})$ at the same point, but should also include the contribution of vector potentials $\\boldsymbol{A}(\\boldsymbol{r}')$ at various points within the range $|\\boldsymbol{r} - \\boldsymbol{r}'| < \\xi_0$. To this end, one can consider the nonlocal relationship between $j_s$ and $\\boldsymbol{A}$ as follows (Pippard theory):\n \\begin{equation*}\n j_s (\\boldsymbol{r}) = - \\frac{3}{4\\pi \\xi_0 \\lambda_c} \\int \\frac{\\boldsymbol{R} [\\boldsymbol{R} \\cdot \\boldsymbol{A}(\\boldsymbol{r}')]}{R^4} e^{-R/\\xi_0} d\\boldsymbol{r}' \\quad (\\boldsymbol{R} = \\boldsymbol{r} - \\boldsymbol{r}')\n \\end{equation*}\n Please complete the following question:\n Analyze the approximate value of $j_s$ in the case of $q\\xi_0 \\gg 1$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\xi_0$": "coherence length", + "$\\hbar$": "reduced Planck's constant", + "$v_F$": "Fermi velocity", + "$\\Delta(0)$": "energy gap at zero temperature", + "$j_s$": "supercurrent density", + "$\\boldsymbol{r}$": "position vector", + "$\\boldsymbol{A}$": "vector potential", + "$\\lambda_c$": "penetration depth", + "$\\boldsymbol{r}'$": "position vector (integration variable)", + "$R$": "distance between points in space", + "$q$": "wave vector magnitude", + "$\\lambda_L(0)$": "London penetration depth at zero temperature", + "$K(q)$": "kernel function", + "$j_1(x)$": "spherical Bessel function of the first kind" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 468, + "topic": "Superconductivity", + "question": "The Current in Superconductors\n According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation it is given by:\n \\begin{align}\n \\hat{\\mathbf{j}}(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] - \\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &\\equiv \\hat{\\mathbf{j}}_1(\\mathbf{r}) + \\hat{\\mathbf{j}}_2(\\mathbf{r}) \\label{eq:6.8.9}\n \\end{align}\n Where\n \\begin{align}\n \\hat{\\mathbf{j}}_1(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] \\\\\n &= \\frac{e\\hbar}{2m} \\sum_{\\mathbf{k},\\mathbf{q}} (2\\mathbf{k} + \\mathbf{q}) e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} - C_{-\\mathbf{k}\\downarrow}^{\\dagger} C_{-\\mathbf{k}\\downarrow}) \\label{eq:6.8.10}\n \\end{align}\n \\begin{align}\n \\hat{\\mathbf{j}}_2(\\mathbf{r}) &= -\\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &= -\\frac{e^2}{mc} \\mathbf{A}(\\mathbf{r}) \\sum_{\\mathbf{k},\\mathbf{q}} e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} + C_{-\\mathbf{k}-\\mathbf{q}}^{\\dagger} C_{-\\mathbf{k}-\\mathbf{q}}) \\label{eq:6.8.11}.\n \\end{align}\n In terms of notation, we set $(\\mathbf{k},\\uparrow) = k,(-\\mathbf{k},\\downarrow)=-k$.\n\n The Hamiltonian of the system interaction $H_1$ can be expressed in terms of quasiparticle operators $\\alpha,\\alpha^+$ as:\n \\begin{align}\n H_1 = &-\\frac{e\\hbar}{mc} \\sum_{\\mathbf{k},\\mathbf{q}} [\\mathbf{k} \\cdot \\mathbf{A}(\\mathbf{q})] [(u_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}} - u_{\\mathbf{k}}v_{\\mathbf{k}+\\mathbf{q}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger} + \\alpha_{-\\mathbf{k}\\downarrow}\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}) + \\nonumber \\\\\n & (u_{\\mathbf{k}+\\mathbf{q}}u_{\\mathbf{k}} + v_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{\\mathbf{k}\\uparrow} - \\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow})] \\label{eq:6:8.8},\n \\end{align}\n Where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:\n \\begin{equation}\n u_k^2 = \\frac{1}{2} \\left(1 + \\frac{\\epsilon_k}{\\xi_k}\\right), \\quad v_k^2 = \\frac{1}{2} \\left(1 - \\frac{\\epsilon_k}{\\xi_k}\\right),\n \\end{equation}\n Where $\\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2}$.\n\n Based on the above information, please complete the following calculation:\n1. Under the influence of a weak magnetic field, calculate the current $j_2(r)$ at zero temperature, and express the result using the London penetration depth $\\lambda_L = \\left( \\frac{mc^2}{4\\pi ne^2} \\right)^{1/2}$;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathbf{j}$": "current density operator", + "$e$": "elementary charge", + "$\\hbar$": "reduced Planck's constant", + "$m_1$": "effective mass of particle 1", + "$m$": "effective mass of electron", + "$\\Psi$": "field operator", + "$\\mathbf{A}$": "vector potential", + "$\\mathbf{k}$": "momentum vector", + "$\\mathbf{q}$": "momentum transfer", + "$C_{\\mathbf{k}}$": "annihilation operator for the state with momentum $\\mathbf{k}$", + "$C_{\\mathbf{k}}^{\\dagger}$": "creation operator for the state with momentum $\\mathbf{k}$", + "$H_1$": "interaction Hamiltonian", + "$\\alpha$": "quasiparticle annihilation operator", + "$\\alpha^+$": "quasiparticle creation operator", + "$u_k$": "Bogoliubov transformation coefficient (u)", + "$v_k$": "Bogoliubov transformation coefficient (v)", + "$\\epsilon_k$": "single particle energy", + "$\\xi_k$": "quasiparticle energy", + "$\\Delta$": "superconducting energy gap", + "$\\lambda_L$": "London penetration depth", + "$n$": "electron density" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 469, + "topic": "Superconductivity", + "question": "Current in Superconductors\n According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation it is:\n \\begin{align}\n \\hat{\\mathbf{j}}(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] - \\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &\\equiv \\hat{\\mathbf{j}}_1(\\mathbf{r}) + \\hat{\\mathbf{j}}_2(\\mathbf{r}) \\label{eq:6.8.9}\n \\end{align}\n Where\n \\begin{align}\n \\hat{\\mathbf{j}}_1(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] \\\\\n &= \\frac{e\\hbar}{2m} \\sum_{\\mathbf{k},\\mathbf{q}} (2\\mathbf{k} + \\mathbf{q}) e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} - C_{-\\mathbf{k}\\downarrow}^{\\dagger} C_{-\\mathbf{k}\\downarrow}) \\label{eq:6.8.10}\n \\end{align}\n \\begin{align}\n \\hat{\\mathbf{j}}_2(\\mathbf{r}) &= -\\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &= -\\frac{e^2}{mc} \\mathbf{A}(\\mathbf{r}) \\sum_{\\mathbf{k},\\mathbf{q}} e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} + C_{-\\mathbf{k}-\\mathbf{q}}^{\\dagger} C_{-\\mathbf{k}-\\mathbf{q}}) \\label{eq:6.8.11}.\n \\end{align}\n Symbolically, we note $(\\mathbf{k},\\uparrow) = k,(-\\mathbf{k},\\downarrow)=-k$.\n\n The Hamiltonian of the system interaction $H_1$ can be expressed using the quasiparticle operators $\\alpha,\\alpha^+$ as:\n \\begin{align}\n H_1 = &-\\frac{e\\hbar}{mc} \\sum_{\\mathbf{k},\\mathbf{q}} [\\mathbf{k} \\cdot \\mathbf{A}(\\mathbf{q})] [(u_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}} - u_{\\mathbf{k}}v_{\\mathbf{k}+\\mathbf{q}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger} + \\alpha_{-\\mathbf{k}\\downarrow}\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}) + \\nonumber \\\\\n & (u_{\\mathbf{k}+\\mathbf{q}}u_{\\mathbf{k}} + v_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{\\mathbf{k}\\uparrow} - \\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow})] \\label{eq:6:8.8},\n \\end{align}\n where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:\n \\begin{equation}\n u_k^2 = \\frac{1}{2} \\left(1 + \\frac{\\epsilon_k}{\\xi_k}\\right), \\quad v_k^2 = \\frac{1}{2} \\left(1 - \\frac{\\epsilon_k}{\\xi_k}\\right),\n \\end{equation}\n where $\\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2}$.\n\n Based on the above information, please complete the following calculation:\n Under a weak magnetic field, the system is in the superconducting ground state. Considering the first-order approximation, calculate the current $j_1$. Hint: if the answer exists in an integral, then find the integrand.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$e$": "elementary charge", + "$\\hbar$": "reduced Planck's constant", + "$m_1$": "mass associated with the first type of particle", + "$m$": "mass", + "$\\Psi$": "wave function", + "$\\Psi^{\\dagger}$": "complex conjugate of the wave function", + "$\\mathbf{A}$": "vector potential", + "$C_{\\mathbf{k}}$": "quasiparticle operator for momentum state $\\mathbf{k}$", + "$C_{\\mathbf{k}}^{\\dagger}$": "creation operator for quasiparticle in momentum state $\\mathbf{k}$", + "$H_1$": "Hamiltonian of the system interaction", + "$\\alpha$": "quasiparticle operator", + "$\\alpha^+$": "creation operator for quasiparticles", + "$u_k$": "Bogoliubov transformation coefficient", + "$v_k$": "Bogoliubov transformation coefficient", + "$\\epsilon_k$": "kinetic energy of quasiparticle with wavevector $k$", + "$\\xi_k$": "energy spectrum related to superconductivity", + "$\\Delta$": "superconducting gap", + "$k_F$": "Fermi wavevector", + "$\\lambda_L(0)$": "London penetration depth at zero temperature" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 470, + "topic": "Superconductivity", + "question": "Current in Superconductors\n According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation, it is expressed as:\n \\begin{align}\n \\hat{\\mathbf{j}}(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] - \\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &\\equiv \\hat{\\mathbf{j}}_1(\\mathbf{r}) + \\hat{\\mathbf{j}}_2(\\mathbf{r}) \\label{eq:6.8.9}\n \\end{align}\n Where\n \\begin{align}\n \\hat{\\mathbf{j}}_1(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] \\\\\n &= \\frac{e\\hbar}{2m} \\sum_{\\mathbf{k},\\mathbf{q}} (2\\mathbf{k} + \\mathbf{q}) e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} - C_{-\\mathbf{k}\\downarrow}^{\\dagger} C_{-\\mathbf{k}\\downarrow}) \\label{eq:6.8.10}\n \\end{align}\n \\begin{align}\n \\hat{\\mathbf{j}}_2(\\mathbf{r}) &= -\\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &= -\\frac{e^2}{mc} \\mathbf{A}(\\mathbf{r}) \\sum_{\\mathbf{k},\\mathbf{q}} e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} + C_{-\\mathbf{k}-\\mathbf{q}}^{\\dagger} C_{-\\mathbf{k}-\\mathbf{q}}) \\label{eq:6.8.11}.\n \\end{align}\n In terms of notation, we consider $(\\mathbf{k},\\uparrow) = k,(-\\mathbf{k},\\downarrow)=-k$.\n\n The Hamiltonian $H_1$ of the system interaction can be expressed using the quasiparticle operators $\\alpha,\\alpha^+$ as:\n \\begin{align}\n H_1 = &-\\frac{e\\hbar}{mc} \\sum_{\\mathbf{k},\\mathbf{q}} [\\mathbf{k} \\cdot \\mathbf{A}(\\mathbf{q})] [(u_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}} - u_{\\mathbf{k}}v_{\\mathbf{k}+\\mathbf{q}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger} + \\alpha_{-\\mathbf{k}\\downarrow}\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}) + \\nonumber \\\\\n & (u_{\\mathbf{k}+\\mathbf{q}}u_{\\mathbf{k}} + v_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{\\mathbf{k}\\uparrow} - \\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow})] \\label{eq:6:8.8},\n \\end{align}\n where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:\n \\begin{equation}\n u_k^2 = \\frac{1}{2} \\left(1 + \\frac{\\epsilon_k}{\\xi_k}\\right), \\quad v_k^2 = \\frac{1}{2} \\left(1 - \\frac{\\epsilon_k}{\\xi_k}\\right),\n \\end{equation}\n where $\\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2}$.\n\n Based on the above information, please complete the following calculation:\n The total current can be written as $\\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})$, find the expression for $K(q)$.\n If the answer exists in an integral, then find the integrand.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$e$": "elementary charge", + "$\\hbar$": "reduced Planck's constant", + "$m_1$": "effective mass (component 1)", + "$m$": "electron mass", + "$c$": "speed of light", + "$\\Psi$": "wave function", + "$\\Psi^{\\dagger}$": "conjugate of the wave function", + "$\\mathbf{A}$": "vector potential", + "$H_1$": "interaction Hamiltonian", + "$\\alpha$": "quasiparticle operator", + "$\\alpha^+$": "conjugate quasiparticle operator", + "$\\mathbf{k}$": "wave vector", + "$\\mathbf{q}$": "momentum transfer vector", + "$u_k$": "Bogoliubov transformation coefficient (u)", + "$v_k$": "Bogoliubov transformation coefficient (v)", + "$\\xi_k$": "quasiparticle energy", + "$\\epsilon_k$": "kinetic energy", + "$\\Delta$": "superconducting gap", + "$n$": "conduction electron density", + "$\\lambda_L(0)$": "London penetration depth at T=0" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 471, + "topic": "Superconductivity", + "question": "Meissner Effect in Superconductors\n Assume the current in a superconductor follows $\\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})$. If $K(0)=0$, it indicates the absence of the Meissner effect in the superconductor; otherwise, it exists.\n \n At the microscopic level, the current in a BCS superconductor can be written as\n \\begin{equation*}\n \\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})\n \\end{equation*}\n \\begin{equation}\n K(q) = \\frac{1}{\\lambda_L^2(0)} \\left\\{1 - \\frac{3}{4} \\int_{-1}^{+1} (1-Z^2) dZ \\times \\int_{-\\infty}^{\\infty} d\\epsilon \\frac{1}{2} \\frac{\\xi_+ \\xi_- - \\epsilon_+ \\epsilon_- - \\Delta^2}{\\xi_+ \\xi_- (\\xi_+ + \\xi_-)} \\right\\} \\label{eq:6.8.24}\n \\end{equation}\n Here, $K(q)$ is isotropic, and\n \\begin{align}\n \\epsilon_{\\pm} &= \\epsilon_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad \\xi_{\\pm} = \\xi_{\\mathbf{k}\\pm\\mathbf{q}/2} = \\sqrt{\\epsilon_{\\pm}^2 + \\Delta^2} \\\\\n u_{\\pm} &= u_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad v_{\\pm} = v_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad Z = \\cos\\theta \n \\end{align}\n The coherence length of the superconductor is $\\xi_0 = \\frac{\\hbar v_F}{\\pi \\Delta(0)}$\n According to the information, perform the calculation under the assumption $q\\ll k_F$:\n1. For a normal conductor, calculate $K(q)$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathbf{j}(\\mathbf{q})$": "current density as a function of momentum", + "$c$": "speed of light", + "$K(q)$": "kernel function as a function of momentum", + "$\\mathbf{A}(\\mathbf{q})$": "vector potential as a function of momentum", + "$q$": "momentum", + "$\\lambda_L$": "London penetration depth", + "$Z$": "variable related to angle", + "$\\epsilon$": "energy variable", + "$\\xi_+$": "quasiparticle energy for positive momentum shift", + "$\\xi_-$": "quasiparticle energy for negative momentum shift", + "$\\Delta$": "superconducting energy gap", + "$\\epsilon_+$": "energy for positive momentum shift", + "$\\epsilon_-$": "energy for negative momentum shift", + "$u_+$": "coherence factor for positive momentum shift", + "$u_-$": "coherence factor for negative momentum shift", + "$v_+$": "coherence factor for positive momentum shift", + "$v_-$": "coherence factor for negative momentum shift", + "$\\theta$": "angle variable", + "$\\xi_0$": "coherence length of the superconductor", + "$\\hbar$": "reduced Planck's constant", + "$v_F$": "Fermi velocity", + "$k_F$": "Fermi wave number", + "$K_n(q)$": "kernel function for a normal conductor" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 472, + "topic": "Superconductivity", + "question": "Meissner Effect in Superconductors\n Assume the current in a superconductor follows $\\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})$. If $K(0)=0$, it can be shown that there is no Meissner effect in the superconductor, otherwise it exists.\n \n Microscopically, the BCS superconducting current can be written as\n \\begin{equation*}\n \\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})\n \\end{equation*}\n \\begin{equation}\n K(q) = \\frac{1}{\\lambda_L^2(0)} \\left\\{1 - \\frac{3}{4} \\int_{-1}^{+1} (1-Z^2) dZ \\times \\int_{-\\infty}^{\\infty} d\\epsilon \\frac{1}{2} \\frac{\\xi_+ \\xi_- - \\epsilon_+ \\epsilon_- - \\Delta^2}{\\xi_+ \\xi_- (\\xi_+ + \\xi_-)} \\right\\} \\label{eq:6.8.24}\n \\end{equation}\n where $K(q)$ is orientation-independent, and\n \\begin{align}\n \\epsilon_{\\pm} &= \\epsilon_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad \\xi_{\\pm} = \\xi_{\\mathbf{k}\\pm\\mathbf{q}/2} = \\sqrt{\\epsilon_{\\pm}^2 + \\Delta^2} \\\\\n u_{\\pm} &= u_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad v_{\\pm} = v_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad Z = \\cos\\theta \n \\end{align}\n The coherence length of the superconductor is $\\xi_0 =\\frac{\\hbar v_F}{\\pi \\Delta(0)}$\n Based on the information, please complete the calculation under the assumption $q\\ll k_F$:\n If $q\\xi_0 \\ll 1$, calculate $K(q)$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathbf{j}$": "current", + "$\\mathbf{q}$": "momentum vector", + "$c$": "speed of light", + "$K(q)$": "function representing the kernel at momentum q", + "$\\mathbf{A}$": "vector potential", + "$\\lambda_L$": "London penetration depth", + "$Z$": "cosine of the angle theta", + "$\\xi_+$": "energy term for positive momentum shift", + "$\\xi_-$": "energy term for negative momentum shift", + "$\\epsilon_+$": "kinetic energy term for positive momentum shift", + "$\\epsilon_-$": "kinetic energy term for negative momentum shift", + "$\\Delta$": "superconducting energy gap", + "$\\epsilon$": "energy", + "$\\xi$": "energy related to the superconducting gap", + "$u_+$": "coefficient for particle-like quasiparticles with positive momentum shift", + "$u_-$": "coefficient for particle-like quasiparticles with negative momentum shift", + "$v_+$": "coefficient for hole-like quasiparticles with positive momentum shift", + "$v_-$": "coefficient for hole-like quasiparticles with negative momentum shift", + "$\\xi_0$": "coherence length of the superconductor", + "$\\hbar$": "reduced Planck's constant", + "$v_F$": "Fermi velocity", + "$k_F$": "Fermi momentum" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 473, + "topic": "Superconductivity", + "question": "Meissner effect in superconductors\n Assuming the current in a superconductor follows $\\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})$, if $K(0)=0$, it can be stated that the superconductor does not exhibit the Meissner effect, otherwise it does.\n \n Microscopically, BCS superconductor current can be expressed as\n \\begin{equation*}\n \\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})\n \\end{equation*}\n \\begin{equation}\n K(q) = \\frac{1}{\\lambda_L^2(0)} \\left\\{1 - \\frac{3}{4} \\int_{-1}^{+1} (1-Z^2) dZ \\times \\int_{-\\infty}^{\\infty} d\\epsilon \\frac{1}{2} \\frac{\\xi_+ \\xi_- - \\epsilon_+ \\epsilon_- - \\Delta^2}{\\xi_+ \\xi_- (\\xi_+ + \\xi_-)} \\right\\} \\label{eq:6.8.24}\n \\end{equation}\n Where $K(q)$ is orientation-independent, and\n \\begin{align}\n \\epsilon_{\\pm} &= \\epsilon_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad \\xi_{\\pm} = \\xi_{\\mathbf{k}\\pm\\mathbf{q}/2} = \\sqrt{\\epsilon_{\\pm}^2 + \\Delta^2} \\\\\n u_{\\pm} &= u_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad v_{\\pm} = v_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad Z = \\cos\\theta \n \\end{align}\n Coherence length of the superconductor $\\xi_0 =\\frac{\\hbar v_F}{\\pi \\Delta(0)}$\n Based on the information, under the assumption $q\\ll k_F$, complete the calculations:\n If $q\\xi_0 \\gg 1$, calculate $K(q)$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathbf{j}(\\mathbf{q})$": "current density as a function of wave vector $\\mathbf{q}$", + "$c$": "speed of light", + "$K(q)$": "kernel function for wave vector $q$", + "$\\mathbf{A}(\\mathbf{q})$": "vector potential as a function of wave vector $\\mathbf{q}$", + "$\\lambda_L^2(0)$": "squared London penetration depth at zero temperature", + "$Z$": "cosine of the angle $\\theta$", + "$\\epsilon_{\\pm}$": "energy at $\\mathbf{k} \\pm \\mathbf{q}/2$", + "$\\xi_{\\pm}$": "quasiparticle excitation energy at $\\mathbf{k} \\pm \\mathbf{q}/2$", + "$\\Delta$": "superconducting gap parameter", + "$u_{\\pm}$": "coherence factor $u$ at $\\mathbf{k} \\pm \\mathbf{q}/2$", + "$v_{\\pm}$": "coherence factor $v$ at $\\mathbf{k} \\pm \\mathbf{q}/2$", + "$\\xi_0$": "coherence length of the superconductor", + "$\\hbar$": "reduced Planck constant", + "$v_F$": "Fermi velocity", + "$k_F$": "Fermi wave vector" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 474, + "topic": "Superconductivity", + "question": "London penetration depth at finite temperatures\n The London equation is a significant equation describing superconductors, reflecting the perfect diamagnetism of superconductors and can be written as\n \\begin{equation}\n \\mathbf{j}_s(\\mathbf{r}) = - \\frac{c}{4\\pi}\\frac{1}{\\lambda_L^2}\\mathbf{A}(\\mathbf{r}),\n \\end{equation}\n where $\\lambda_L$ is called the London penetration depth, depicting the depth to which a magnetic field penetrates into a superconducting sample, and at zero temperature can be expressed as\n \\begin{equation}\n \\lambda_L^2(0) = \\left( \\frac{mc^2}{4\\pi ne^2} \\right)^{1/2}.\n \\end{equation}\n Based on the context, complete the calculation:\n\n Calculate the linear response coefficient of the diamagnetic current under first order approximation. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\mathbf{j}_s(\\mathbf{r})$": "supercurrent density at position r", + "$\\lambda_L$": "London penetration depth", + "$c$": "speed of light", + "$m$": "mass", + "$n$": "number density of Cooper pairs", + "$e$": "elementary charge", + "$T$": "temperature", + "$\\mathbf{A}(\\mathbf{r})$": "vector potential at position r", + "$\\mathbf{r}$": "position vector", + "$\\mathbf{k}$": "wave vector", + "$\\mathbf{q}$": "wave vector", + "$\\Psi$": "superconducting wave function", + "$C_{\\mathbf{k}}$": "annihilation operator for wave vector k", + "$\\Delta$": "superconducting gap", + "$\\epsilon_{\\mathbf{k}}$": "energy dispersion at wave vector k", + "$\\xi_{\\mathbf{k}}$": "quasiparticle energy at wave vector k", + "$K_2(q, T)$": "linear response coefficient of the diamagnetic current" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 475, + "topic": "Superconductivity", + "question": "Ginzberg Landau Theory\n Ginzburg and Landau proposed using a complex quantity $\\psi(r)$ to describe the 'effective wave function' of superconducting electrons, with charge $e^*$ and mass $m^*$, and the corresponding system free energy density and free energy are:\n \\begin{equation}\n f_s = f_n + \\alpha(T) |\\psi(\\mathbf{r})|^2 + \\frac{1}{2} \\beta(T) |\\psi(\\mathbf{r})|^4 + \\frac{1}{8\\pi} (\\nabla \\times \\mathbf{A}) \\cdot (\\nabla \\times \\mathbf{A}) + \\frac{1}{2m^*} \\left| \\left(-i\\hbar \\nabla - \\frac{e^*}{c}\\mathbf{A}\\right) \\psi(\\mathbf{r}) \\right|^2 \\label{eq:6.10.6},\n \\end{equation}\n The system's free energy is\n \\begin{equation}\n F_s = \\int f_s dr \\label{eq:6.10.7}\n \\end{equation}\n Please calculate:\n\nThe condition for thermodynamic equilibrium (consider variational performance with respect to the order parameter and vector potential) to derive the equation satisfied by $\\psi$, namely the Ginzburg-Landau equation. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\psi$": "effective wave function", + "$r$": "position", + "$e^*$": "effective charge of superconducting electrons", + "$m^*$": "effective mass of superconducting electrons", + "$f_s$": "system free energy density", + "$f_n$": "normal state free energy density", + "$\\alpha(T)$": "temperature-dependent coefficient for the quadratic term", + "$\\beta(T)$": "temperature-dependent coefficient for the quartic term", + "$\\mathbf{A}$": "vector potential", + "$\\hbar$": "reduced Planck's constant", + "$c$": "speed of light", + "$F_s$": "system free energy", + "$\\delta F_s$": "variation of system free energy", + "$\\mathbf{j}$": "current density", + "$\\mathbf{B}$": "magnetic field" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 476, + "topic": "Superconductivity", + "question": "Ginzburg-Landau Theory\n Ginzburg and Landau proposed using a complex variable $\\psi(r)$ to describe the \"effective wave function\" of superconducting electrons, with charge $e^*$ and mass $m^*$. The corresponding system free energy density and free energy are:\n \\begin{equation}\n f_s = f_n + \\alpha(T) |\\psi(\\mathbf{r})|^2 + \\frac{1}{2} \\beta(T) |\\psi(\\mathbf{r})|^4 + \\frac{1}{8\\pi} (\\nabla \\times \\mathbf{A}) \\cdot (\\nabla \\times \\mathbf{A}) + \\frac{1}{2m^*} \\left| \\left(-i\\hbar \\nabla - \\frac{e^*}{c}\\mathbf{A}\\right) \\psi(\\mathbf{r}) \\right|^2 \\label{eq:6.10.6},\n \\end{equation}\n The free energy of the system is\n \\begin{equation}\n F_s = \\int f_s dr \\label{eq:6.10.7}\n \\end{equation}\n Please calculate:\n \n Solve the Ginzburg-Landau equation under $A=0$;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\psi(r)$": "effective wave function of superconducting electrons", + "$e^*$": "charge of superconducting electrons", + "$m^*$": "mass of superconducting electrons", + "$f_s$": "superconducting free energy density", + "$f_n$": "normal state free energy density", + "$\\alpha(T)$": "temperature-dependent parameter (alpha)", + "$\\beta(T)$": "temperature-dependent parameter (beta)", + "$\\mathbf{r}$": "position vector", + "$\\mathbf{A}$": "magnetic vector potential", + "$\\hbar$": "reduced Planck's constant", + "$c$": "speed of light", + "$F_s$": "superconducting free energy", + "$\\psi_0$": "uniform solution for the effective wave function", + "$T$": "temperature", + "$T_c$": "critical temperature" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 477, + "topic": "Superconductivity", + "question": "Ginzburg Landau Theory\n Ginzburg and Landau propose using a complex quantity $\\psi(r)$ to describe the 'effective wave function' of superconducting electrons, with charge $e^*$ and mass $m^*$. The corresponding system free energy density and free energy are:\n \\begin{equation}\n f_s = f_n + \\alpha(T) |\\psi(\\mathbf{r})|^2 + \\frac{1}{2} \\beta(T) |\\psi(\\mathbf{r})|^4 + \\frac{1}{8\\pi} (\\nabla \\times \\mathbf{A}) \\cdot (\\nabla \\times \\mathbf{A}) + \\frac{1}{2m^*} \\left| \\left(-i\\hbar \\nabla - \\frac{e^*}{c}\\mathbf{A}\\right) \\psi(\\mathbf{r}) \\right|^2 \\label{eq:6.10.6},\n \\end{equation}\n The system's free energy is\n \\begin{equation}\n F_s = \\int f_s dr \\label{eq:6.10.7}\n \\end{equation}\n Please calculate:\n\n The penetration depth $\\lambda(T)$ in a weak magnetic field (expressed in terms of the result from question two);", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\psi(r)$": "effective wave function of superconducting electrons", + "$e^*$": "effective charge of superconducting electrons", + "$m^*$": "effective mass of superconducting electrons", + "$f_s$": "system free energy density", + "$f_n$": "normal state free energy density", + "$\\alpha(T)$": "temperature-dependent coefficient", + "$\\beta(T)$": "temperature-dependent coefficient", + "$\\mathbf{A}$": "magnetic vector potential", + "$\\hbar$": "reduced Planck's constant", + "$c$": "speed of light in vacuum", + "$F_s$": "system's free energy", + "$\\lambda(T)$": "penetration depth", + "$\\psi_0$": "field-free order parameter" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 478, + "topic": "Superconductivity", + "question": "Ginzburg-Landau Theory\n Ginzburg and Landau proposed using a complex quantity $\\psi(r)$ to describe the \"effective wave function\" of superconducting electrons, with charge $e^*$ and mass $m^*$; the corresponding system free energy density and free energy are:\n \\begin{equation}\n f_s = f_n + \\alpha(T) |\\psi(\\mathbf{r})|^2 + \\frac{1}{2} \\beta(T) |\\psi(\\mathbf{r})|^4 + \\frac{1}{8\\pi} (\\nabla \\times \\mathbf{A}) \\cdot (\\nabla \\times \\mathbf{A}) + \\frac{1}{2m^*} \\left| \\left(-i\\hbar \\nabla - \\frac{e^*}{c}\\mathbf{A}\\right) \\psi(\\mathbf{r}) \\right|^2 \\label{eq:6.10.6},\n \\end{equation}\n The free energy of the system is\n \\begin{equation}\n F_s = \\int f_s dr \\label{eq:6.10.7}\n \\end{equation}\n Please calculate:\n\n Calculate the magnetic flux and thereby demonstrate the quantization of magnetic flux.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\psi$": "effective wave function of superconducting electrons", + "$r$": "position vector", + "$e^*$": "effective charge of superconducting electrons", + "$m^*$": "effective mass of superconducting electrons", + "$f_s$": "system free energy density", + "$f_n$": "normal state free energy density", + "$\\alpha(T)$": "temperature-dependent Ginzburg-Landau coefficient", + "$\\beta(T)$": "temperature-dependent Ginzburg-Landau coefficient", + "$\\mathbf{A}$": "vector potential", + "$\\hbar$": "reduced Planck constant", + "$c$": "speed of light", + "$\\mathbf{j}$": "current density", + "$\\varphi$": "phase of the wave function", + "$C$": "circuit path", + "$\\mathbf{B}$": "magnetic flux density", + "$S$": "surface enclosed by loop C", + "$\\Phi$": "magnetic flux", + "$\\Phi_0$": "magnetic flux quantum", + "$n$": "integer (quantization index)" + }, + "chapter": "Superconductivity: BCS theory and microscopic mechanism", + "section": "" + }, + { + "id": 479, + "topic": "Strongly Correlated Systems", + "question": "Hubbard Model in Narrow Bandwidth\n Consider the following single band Hubbard model,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c,c^\\dagger$ are the annihilation and creation operators for electrons, $n$ is the particle number operator\n\n Please complete the following calculation:\n\n Calculate the off-diagonal elements of the single-particle Green's function obtained in the Bloch representation You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$H$": "Hamiltonian", + "$T_{ij}$": "hopping parameter between sites i and j", + "$c_{i\\sigma}$": "annihilation operator for an electron with spin sigma at site i", + "$c_{i\\sigma}^{\\dagger}$": "creation operator for an electron with spin sigma at site i", + "$U$": "on-site interaction energy", + "$n_{i\\sigma}$": "particle number operator for spin sigma at site i", + "$n_{i\\bar{\\sigma}}$": "particle number operator for the opposite spin at site i", + "$G_{kk'}^\\sigma(\\omega)$": "off-diagonal Green's function in Bloch representation", + "$C_{k\\sigma}$": "annihilation operator for an electron with wave vector k and spin sigma", + "$C_{k\\sigma}^+$": "creation operator for an electron with wave vector k and spin sigma", + "$N$": "number of lattice sites", + "$\\mathbf{k}$": "wave vector", + "$\\mathbf{R}_i$": "position vector of site i", + "$G_k^\\sigma(\\omega)$": "diagonal Green's function in Bloch representation", + "$\\delta_{kk'}$": "Kronecker delta, equals 1 if k=k' and 0 otherwise" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 480, + "topic": "Strongly Correlated Systems", + "question": "Hubbard Model under Narrow Bandwidth — Green's Function Analysis\n Consider the following single-band Hubbard model,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c, c^\\dagger$ are electron annihilation and creation operators, $n$ is the particle number operator.\n\n In the case of bandwidth $\\Delta \\neq 0$, the single-particle green function approximately satisfies\n \\begin{equation}\n G_k^\\sigma(\\omega) = \\frac{\\omega - T_0 - U(1 - (n_{\\bar{\\sigma}}))}{(\\omega - E_k)(\\omega - T_0 - U) + (n_{\\bar{\\sigma}}) U(T_0 - E_k)}\n \\end{equation}\n\n Please solve the following problem:\n\n Calculate the energy spectrum of the system;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$H$": "Hamiltonian", + "$T_{ij}$": "hopping matrix element", + "$c_{i\\sigma}$": "annihilation operator for an electron with spin \\sigma at site i", + "$c_{i\\sigma}^{\\dagger}$": "creation operator for an electron with spin \\sigma at site i", + "$U$": "onsite Coulomb interaction", + "$n_{i\\sigma}$": "number operator for electrons with spin \\sigma at site i", + "$n_{i\\bar{\\sigma}}$": "number operator for electrons with opposite spin of \\sigma at site i", + "$\\Delta$": "bandwidth", + "$G_k^\\sigma(\\omega)$": "single-particle Green's function for a given wave vector k and spin \\sigma", + "$\\omega$": "frequency variable", + "$T_0$": "average hopping energy", + "$E_k$": "energy spectrum related to wave vector k", + "$E_{k\\sigma}^{(1)}$": "first pole of the Green's function for wave vector k and spin \\sigma", + "$E_{k\\sigma}^{(2)}$": "second pole of the Green's function for wave vector k and spin \\sigma", + "$A_{k\\sigma}^{(1)}$": "amplitude or spectral weight related to the first pole of the Green's function", + "$A_{k\\sigma}^{(2)}$": "amplitude or spectral weight related to the second pole of the Green's function" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 481, + "topic": "Strongly Correlated Systems", + "question": "Hubbard Model in Narrow Band - Green's Function Analysis\n Consider the following single-band Hubbard model,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c,c^\\dagger$ are the annihilation and creation operators of electrons, and $n$ is the particle number operator.\n\n For non-zero bandwidth $\\Delta \\neq 0$, the single-particle Green's function approximately satisfies\n \\begin{equation}\n G_k^\\sigma(\\omega) = \\frac{\\omega - T_0 - U(1 - < n_{\\bar{\\sigma}} >)}{(\\omega - E_k)(\\omega - T_0 - U) + < n_{\\bar{\\sigma}} > U(T_0 - E_k)}\n \\end{equation}\n\n Please complete the following problem:\n\n Calculate the Green's function for $U=0$ You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$H$": "Hamiltonian", + "$T_{ij}$": "hopping parameter between sites i and j", + "$c_{i\\sigma}$": "annihilation operator for an electron at site i with spin sigma", + "$c_{i\\sigma}^{\\dagger}$": "creation operator for an electron at site i with spin sigma", + "$U$": "on-site interaction energy", + "$n_{i\\sigma}$": "particle number operator for electrons at site i with spin sigma", + "$n_{i\\bar{\\sigma}}$": "particle number operator for electrons at site i with opposite spin to sigma", + "$\\Delta$": "bandwidth", + "$G_k^\\sigma(\\omega)$": "single-particle Green's function for momentum k and spin sigma", + "$\\omega$": "frequency (or energy variable) in the Green's function", + "$T_0$": "on-site energy", + "$< n_{\\bar{\\sigma}} >$": "average particle number for opposite spin to sigma", + "$E_k$": "energy of the band electron with momentum k" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 482, + "topic": "Strongly Correlated Systems", + "question": "Hubbard Model in the Narrow Band Limit—Green's Function Analysis\n Consider the single-band Hubbard model as follows,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c,c^\\dagger$ are the annihilation and creation operators for electrons, and $n$ is the particle number operator.\n\n In the case where bandwidth $\\Delta \\neq 0$, the single-particle Green's function approximately satisfies\n \\begin{equation}\n G_k^\\sigma(\\omega) = \\frac{\\omega - T_0 - U(1 - < n_{\\bar{\\sigma}} >)}{(\\omega - E_k)(\\omega - T_0 - U) + < n_{\\bar{\\sigma}} > U(T_0 - E_k)}\n \\end{equation}\n\n Please complete the following problem:\n3. Calculate the density of states for the case of $U=0$ You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$H$": "Hamiltonian", + "$T_{ij}$": "hopping matrix element between sites i and j", + "$c_{i\\sigma}^{\\dagger}$": "creation operator for electrons with spin σ at site i", + "$c_{j\\sigma}$": "annihilation operator for electrons with spin σ at site j", + "$U$": "Coulomb interaction energy", + "$n_{i\\sigma}$": "particle number operator for electrons with spin σ at site i", + "$n_{i\\bar{\\sigma}}$": "particle number operator for electrons with the opposite spin at site i", + "$\\Delta$": "bandwidth", + "$G_k^\\sigma(\\omega)$": "single-particle Green's function for momentum k and spin σ", + "$\\omega$": "frequency", + "$T_0$": "center of the energy band", + "$E_k$": "energy of electrons with momentum k", + "$< n_{\\bar{\\sigma}} >$": "expectation value of the particle number operator for electrons with the opposite spin", + "$\\rho_\\sigma (\\omega)$": "density of states for spin σ", + "$N$": "number of lattice sites", + "$D(\\omega)$": "density of states function of a single band" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 483, + "topic": "Strongly Correlated Systems", + "question": "Narrow-band Hubbard model—Green's function analysis\n Consider the following single-band Hubbard model,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c,c^\\dagger$ are the annihilation and creation operators for electrons, and $n$ is the particle number operator.\n\n When the band width $\\Delta \\neq 0$, the single-particle Green's function approximately satisfies\n \\begin{equation}\n G_k^\\sigma(\\omega) = \\frac{\\omega - T_0 - U(1 - (n_{\\bar{\\sigma}}))}{(\\omega - E_k)(\\omega - T_0 - U) + (n_{\\bar{\\sigma}}) U(T_0 - E_k)}\n \\end{equation}\n\n Please complete the following problem:\n\n Calculate the Green's function in the case $U\\geq \\Delta$ You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$H$": "Hamiltonian", + "$T_{ij}$": "hopping matrix element between sites i and j", + "$c_{i\\sigma}$": "annihilation operator for an electron with spin sigma at site i", + "$c_{i\\sigma}^{\\dagger}$": "creation operator for an electron with spin sigma at site i", + "$U$": "on-site Coulomb repulsion", + "$n_{i\\sigma}$": "number operator for electrons with spin sigma at site i", + "$n_{i\\bar{\\sigma}}$": "number operator for electrons with opposite spin to sigma at site i", + "$\\Delta$": "band width", + "$G_k^\\sigma$": "Green's function for wave vector k and spin sigma", + "$\\omega$": "frequency variable", + "$T_0$": "reference energy level", + "$E_k$": "energy dispersion relation for wave vector k", + "$E_{k\\sigma}^{(1)}$": "first energy eigenvalue for wave vector k and spin sigma", + "$E_{k\\sigma}^{(2)}$": "second energy eigenvalue for wave vector k and spin sigma", + "$A_{k\\sigma}^{(1)}$": "spectral weight factor for first energy level for wave vector k and spin sigma", + "$A_{k\\sigma}^{(2)}$": "spectral weight factor for second energy level for wave vector k and spin sigma" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 484, + "topic": "Strongly Correlated Systems", + "question": "Anderson s-d exchange model and Green's function equation of motion\n In 1961, Anderson proposed the s-d mixing model, suggesting that to discuss the formation of local magnetic moments by transition metal impurity atoms in a non-magnetic metal matrix, two factors must be considered: First, similar to the formation of intrinsic magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant impact on the formation of local magnetic moments in the crystal; second, since the free atomic d-orbital states $\\phi_d(\\mathbf{r})$ of impurities in crystals are no longer purely eigenstates, especially due to the tendency for electron delocalization in metal crystals into Bloch states (s orbitals), there exists an electron transfer between $\\phi_d$ and $\\phi_k$ states, which Anderson called s-d mixing. Therefore, he pointed out that the system's Hamiltonian $H$ should consist of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, and suppose the Lande factor of electrons and impurities $g_0 = g_i = 2$, which is the non-degenerate orbital Anderson s-d mixing model.\n\n In dealing with the $s-d$ exchange model, the Green's function equation of motion method is often used:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n By utilizing a technique, differentiate the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$, yielding the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Perform a Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n Obtaining two forms of the general equation of motion for Green’s function\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the problem\n\n Calculate the Green's function equation of motion for $\\ll C_{k\\sigma} | C_{k'\\sigma}^+ \\gg_\\omega$ in the s-d exchange model. Hint: Let $a_{kk'\\sigma}$ symbolize $\\ll C_{k\\sigma} | C_{k'\\sigma}^+ \\gg_\\omega$, and $b_{k'\\sigma}$ symbolize $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\phi_d$": "d-orbital state of impurity atom", + "$\\phi_k$": "Bloch state of metal crystal", + "$H$": "system Hamiltonian", + "$k$": "momentum index", + "$\\sigma$": "spin index", + "$E_{k\\sigma}$": "energy of state with momentum $k$ and spin $\\sigma$", + "$n_{k\\sigma}$": "number operator for state with momentum $k$ and spin $\\sigma$", + "$E_{d\\sigma}$": "energy of d-state with spin $\\sigma$", + "$n_{d\\sigma}$": "number operator for d-state with spin $\\sigma$", + "$U$": "Coulomb interaction parameter", + "$\\bar{\\sigma}$": "opposite spin of $\\sigma$", + "$V_{kd}$": "s-d mixing term", + "$C_{k\\sigma}^\\dagger$": "creation operator for Bloch state with momentum $k$ and spin $\\sigma$", + "$d_\\sigma$": "annihilation operator for d-state with spin $\\sigma$", + "$\\mu_B$": "Bohr magneton", + "$\\omega$": "frequency variable in Fourier transform", + "$a_{kk'\\sigma}$": "Green's function for states $k\\sigma$ and $k'\\sigma$", + "$b_{k'\\sigma}$": "Green's function for d-state and state $k'\\sigma$", + "$\\delta_{k, k'}$": "Kronecker delta for momentum indices" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 485, + "topic": "Strongly Correlated Systems", + "question": "Anderson s-d exchange model and Green's function equation of motion\n In 1961, Anderson proposed the s-d mixing model, suggesting that when discussing the formation of local magnetic moments by transition metal impurity atoms in a non-magnetic metal matrix, it is essential to consider two factors: Firstly, similar to the inherent magnetic moment formed by a free atom, the Coulomb interaction among d-shell electrons in impurity atoms significantly affects the formation of local magnetic moments in the crystal. Secondly, since the d-orbital state of impurity atoms, $\\phi_d(\\mathbf{r})$, in a crystal are no longer rigorous eigenstates, especially due to the tendency of electrons becoming delocalized into Bloch states (s orbitals) in the metal crystal, there exists an electron transfer between $\\phi_d$ and $\\phi_k$ states, a phenomenon that Anderson termed as s-d mixing. Therefore, he stated that the Hamiltonian $H$ of the system should be composed of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here, $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, and assuming the Lande's g-factor of electron and impurity $g_0 = g_i = 2$, this constitutes the non-orbitally degenerate Anderson s-d mixing model.\n\n When dealing with the $s-d$ exchange model, the following Green's function equation of motion is often used:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n a technique is used to take derivatives of the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$ separately, yielding the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n By performing a Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n two representations of the general form of the Green's function equation of motion are obtained:\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the question\n\n Calculate the Green's function equation of motion related to $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$ in the s-d exchange model. Hint: Let $a_{\\sigma}$ syimbolize $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$, $b_\\sigma$ symbolize $\\sum_k V_{kd} \\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$, and $c_{\\sigma}$ symbolize $\\ll n_{d\\bar{\\sigma}} d_\\sigma | d_\\sigma^+ \\gg_\\omega$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\phi_d$": "d-orbital state of impurity atoms", + "$\\phi_k$": "Bloch state (s orbitals) in the metal crystal", + "$H$": "Hamiltonian of the system", + "$E_{k\\sigma}$": "energy of the k-th Bloch state with spin \\sigma", + "$n_{k\\sigma}$": "number operator for k-th Bloch state with spin \\sigma", + "$E_{d\\sigma}$": "energy of the d-orbital state with spin \\sigma", + "$n_{d\\sigma}$": "number operator for d-orbital state with spin \\sigma", + "$U$": "Coulomb interaction energy", + "$n_{d\\bar{\\sigma}}$": "number operator for d-orbital state with opposite spin", + "$V_{kd}$": "s-d coupling constant", + "$C_{k\\sigma}$": "annihilation operator for k-th Bloch state with spin \\sigma", + "$d_\\sigma$": "annihilation operator for d-orbital state with spin \\sigma", + "$\\mu_B$": "Bohr magneton", + "$h$": "external magnetic field", + "$g_0$": "Lande's g-factor for electron", + "$g_i$": "Lande's g-factor for impurity", + "$\\omega$": "angular frequency in Fourier transform", + "$a_{\\sigma}$": "Green's function for d-orbital interaction with self", + "$b_{\\sigma}$": "Green's function for coupling between Bloch states and d-orbitals", + "$c_{\\sigma}$": "Green's function for interaction involving Coulomb interaction" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 486, + "topic": "Strongly Correlated Systems", + "question": "Anderson s-d exchange model and Green's function equations of motion\n In 1961, Anderson proposed the s-d mixing model, arguing that to discuss the formation of localized magnetic moments by transition metal impurity atoms in a non-magnetic metallic matrix, two factors must be considered simultaneously: First, similar to the intrinsic magnetic moment formation in free atoms, the Coulomb interaction of d shell electrons in impurity atoms significantly influences the formation of localized magnetic moments in the crystal; Second, because the d orbital states $\\phi_d(\\mathbf{r})$ of the impurity in the crystal are no longer eigenstates, particularly due to the tendency of electron delocalization into Bloch states in the metallic crystal (s orbital), there is mutual electron transfer between the states $\\phi_d$ and $\\phi_k$, which Anderson termed the s-d mixing interaction. Therefore, he pointed out that the Hamiltonian $H$ of the system should consist of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, and the Landé factor for the electron and impurity is set as $g_0 = g_i = 2$, which defines the non-degenerate orbital Anderson s-d mixing model.\n\n In dealing with the $s-d$ exchange model, the following Green's function equation of motion method is often employed:\n\n Starting from the two-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n By employing a technique, the differential of the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$ can be taken respectively, resulting in the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Performing Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n achieves the two representations of the Green's function equation of motion\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the problem\n1. Derive the equation of motion for the s-d exchange model concerning the mixed Green's function $\\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$. Hint: Let $a_{k\\sigma}$ symbolize $\\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$, and $b_{\\sigma}$ symbolize $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\phi_d$": "d orbital state of the impurity", + "$\\phi_k$": "k orbital state in the host crystal", + "$H$": "Hamiltonian of the system", + "$E_{k\\sigma}$": "energy of the k state with spin sigma", + "$n_{k\\sigma}$": "number operator for k state with spin sigma", + "$E_{d\\sigma}$": "energy of the d state with spin sigma", + "$n_{d\\sigma}$": "number operator for d state with spin sigma", + "$U$": "Coulomb interaction term", + "$n_{d\\bar{\\sigma}}$": "number operator for d state with opposite spin sigma", + "$V_{kd}$": "s-d mixing interaction", + "$C_{k\\sigma}$": "annihilation operator for k state with spin sigma", + "$d_\\sigma$": "annihilation operator for d state with spin sigma", + "$\\mu_B$": "Bohr magneton", + "$\\sigma$": "spin index", + "$a_{k\\sigma}$": "Green's function symbol for $\\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$", + "$b_{\\sigma}$": "Green's function symbol for $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$", + "$\\omega$": "frequency", + "$h$": "magnetic field" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 487, + "topic": "Strongly Correlated Systems", + "question": "Anderson s-d exchange model and Green's function equation of motion\n In 1961, Anderson proposed the s-d mixing model, suggesting that discussing the formation of local magnetic moments of transition-metal impurity atoms in a non-magnetic metal matrix must simultaneously consider two factors: First, similar to the formation of inherent magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant effect on the formation of local magnetic moments in crystals. Second, since the free atomic d-orbital state $\\phi_d(\\mathbf{r})$ of impurities in crystals is no longer strictly an eigenstate, especially due to the tendency of electron delocalization into Bloch orbital states (s-orbital) in metal crystals, there is electronic transfer between $\\phi_d$ and $\\phi_k$ states, which Anderson termed s-d mixing. To this end, he pointed out that the system's Hamiltonian $H$ should be composed of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n here $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, assuming the Lande factor of electrons and impurities $g_0 = g_i = 2$, this is the non-degenerate orbital Anderson s-d mixing model.\n\n In handling the $s-d$ exchange model, the Green's function equation of motion method is often used:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n Using a technique, derive the derivatives of the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$ separately to obtain the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Performing a Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n leads to two expressions for the Green's function equation of motion\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the problem\n\n Calculate the equation of motion for the mixed Green's function $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$ in the s-d exchange model. Hint: Let $a_{k'\\sigma}$ symbolize $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$, and $b_\\sigma$ symbolize $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\phi_d$": "d-orbital wave function state of impurity", + "$\\phi_k$": "Bloch orbital state in the metal", + "$H$": "Hamiltonian of the system", + "$E_{k\\sigma}$": "energy of k-state with spin σ", + "$n_{k\\sigma}$": "number operator for electrons in k-state with spin σ", + "$E_{d\\sigma}$": "energy of d-state with spin σ", + "$n_{d\\sigma}$": "number operator for d-electrons with spin σ", + "$U$": "Coulomb interaction energy", + "$V_{kd}$": "s-d mixing (hybridization) matrix element", + "$C_{k\\sigma}$": "annihilation operator for electrons in k-state with spin σ", + "$d_\\sigma$": "annihilation operator for electrons in d-state with spin σ", + "$\\mu_B$": "Bohr magneton", + "$\\omega$": "angular frequency in Fourier domain", + "$a_{k'\\sigma}$": "symbol for mixed Green's function $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$", + "$b_\\sigma$": "symbol for Green's function $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 488, + "topic": "Strongly Correlated Systems", + "question": "In 1961, Anderson proposed the s-d mixing model, where he considered that the formation of localized magnetic moments by transition metal impurity atoms in non-magnetic metal matrices must take into account two factors: first, similar to the formation of intrinsic magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant influence on forming localized magnetic moments in crystals; second, due to the fact that the d-orbital state $\\phi_d(\\mathbf{r})$ of free atoms in the crystal is no longer strictly an eigenstate, especially due to the tendency of electron delocalization into Bloch states (s orbitals) in metal crystals, there exists electron transfer between $\\phi_d$ and $\\phi_k$ states, which Anderson referred to as s-d hybridization. Consequently, he pointed out that the Hamiltonian $H$ of the system should consist of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here, $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, and the Landé factor for electrons and impurities is set as $g_0 = g_i = 2$, which is the non-degenerate s-d mixing model of Anderson.\n\n In dealing with the $s-d$ exchange model, the following method of the equation of motion for Green's functions is often employed:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n Utilizing a technique, the derivatives of the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$ respectively yield the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Performing the Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n leads to two representations of the general form of the equation of motion for Green's functions\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the question:\n\n Truncate the higher order terms in the equations to write out the approximate equations.\n Hint: Let $a_\\sigma$ symbolize $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$, $b_\\sigma$ symbolize $\\sum_k V_{kd} \\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$. Anderson s-d exchange model and the equation of motion for Green's functions You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\phi_d$": "d-orbital state", + "$\\phi_k$": "k-orbital state", + "$H$": "Hamiltonian of the system", + "$E_{k\\sigma}$": "energy of the k-state with spin \\sigma", + "$E_{d\\sigma}$": "energy of the d-state with spin \\sigma", + "$n_{k\\sigma}$": "number operator for k-state with spin \\sigma", + "$n_{d\\sigma}$": "number operator for d-state with spin \\sigma", + "$U$": "Coulomb interaction energy", + "$\\mu_B$": "Bohr magneton", + "$V_{kd}$": "hybridization term between k and d states", + "$C_{k\\sigma}$": "annihilation operator for k-state with spin \\sigma", + "$d_\\sigma$": "annihilation operator for d-state with spin \\sigma", + "$g_0$": "Landé factor for electrons", + "$g_i$": "Landé factor for impurities", + "$A$": "arbitrary operator A", + "$B$": "arbitrary operator B", + "$a_\\sigma$": "Green's function element for d-state with spin \\sigma", + "$b_\\sigma$": "summed Green's function element for hybrid states involving k and d states with spin \\sigma", + "$\\bar{\\sigma}$": "opposite spin to \\sigma", + "$\\ll A(t); B(t') \\gg$": "double-time Green's function", + "$\\omega$": "frequency variable in Fourier transform" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 489, + "topic": "Strongly Correlated Systems", + "question": "Anderson s-d exchange model and Green's function equation of motion\n In 1961, Anderson proposed the s-d mixing model. He considered that the discussion of the formation of a localized magnetic moment by transition metal impurity atoms in non-magnetic metallic matrices must account for two factors: Firstly, similar to the inherent magnetic moment formation in free atoms, the Coulomb interaction of d shell electrons in impurity atoms has a significant impact on the formation of localized magnetic moments in crystals. Secondly, due to the tendency of electron delocalization to Bloch orbital states (s orbitals) in metallic crystals, there is an exchange between states $\\phi_d(\\mathbf{r})$ and $\\phi_k$, which Anderson termed s-d mixing. Thus, he pointed out that the system's Hamiltonian $H$ should consist of the following four components:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here, $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, with a Landé factor of $g_0 = g_i = 2$ for both electrons and impurities. This is the non-degenerate orbital Anderson s-d mixing model.\n\n When handling the s-d exchange model, the following Green's function equation of motion is often used:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n by employing a technique to differentiate the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$, the following two equations of motion can be obtained:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Performing a Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n yields the two forms of the Green's function equation of motion\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the question\n Solve the Green's function using the truncation approximation", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E_{k\\sigma}$": "energy of the k-th state with spin sigma", + "$n_{k\\sigma}$": "number operator for the k-th state with spin sigma", + "$E_{d\\sigma}$": "energy of the d-state with spin sigma", + "$n_{d\\sigma}$": "number operator for the d-state with spin sigma", + "$U$": "Coulomb interaction energy", + "$V_{kd}$": "s-d exchange coupling strength", + "$C_{k\\sigma}$": "annihilation operator for the k-th state with spin sigma", + "$d_\\sigma$": "annihilation operator for the d-state with spin sigma", + "$\\mu_B$": "Bohr magneton", + "$h$": "external magnetic field", + "$\\omega$": "frequency in the Green's function", + "$\\sigma$": "spin index", + "$\\rho^{(0)}(E)$": "density of states at energy E", + "$\\rho_F^{(0)}$": "density of states at the Fermi surface", + "$\\Gamma$": "broadening due to s-d exchange coupling" + }, + "chapter": "Strongly correlated system", + "section": "" + }, + { + "id": 490, + "topic": "Superconductivity", + "question": "The known Hamiltonian for electron-phonon interaction:\n \\begin{equation}\n H_{ep} = -i \\sum_{\\mathbf{k}, \\mathbf{k}'} \\sum_{\\mathbf{q}, s} \\sum_{\\mathbf{p}} \n \\left( \\frac{N \\hbar}{2 M \\omega_{\\mathbf{q}s}} \\right)^{1/2} \n (\\mathbf{e}_{\\mathbf{q}s} \\cdot \\mathbf{p}) \n \\left\\{ \\frac{1}{N} \\sum_{\\mathbf{l}} e^{i(\\mathbf{k} + \\mathbf{q} - \\mathbf{k}')\\cdot \\mathbf{l}} \\right\\} \\times\n V_{\\mathbf{p}} \\langle \\mathbf{k}' | e^{i\\mathbf{p}\\cdot\\mathbf{r}} | \\mathbf{k} \\rangle \n (a_{\\mathbf{q}s} + a_{-\\mathbf{q}s}^\\dagger) \n C_{\\mathbf{k}}^\\dagger C_{\\mathbf{k}'}\n \\end{equation}\n Now use plane wave instead of Bloch wave function. Please analyze the conservation laws during the electron-phonon interaction process. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$H_{ep}$": "Hamiltonian for electron-phonon interaction", + "$\\hbar$": "reduced Planck's constant", + "$\\mathbf{k}'$": "final wave vector of electron", + "$\\mathbf{k}$": "initial wave vector of electron", + "$\\mathbf{q}$": "wave vector of the phonon", + "$\\mathbf{K}_n$": "reciprocal lattice vector", + "$\\mathbf{l}$": "lattice vector", + "$N$": "number of lattice points", + "$\\Omega$": "crystal cell volume" + }, + "chapter": "", + "section": "" + }, + { + "id": 491, + "topic": "Superconductivity", + "question": "The known Hamiltonian for electron-phonon interaction:\n \\begin{equation}\n H_{ep} = -i \\sum_{\\mathbf{k}, \\mathbf{k}'} \\sum_{\\mathbf{q}, s} \\sum_{\\mathbf{p}} \n \\left( \\frac{N \\hbar}{2 M \\omega_{\\mathbf{q}s}} \\right)^{1/2} \n (\\mathbf{e}_{\\mathbf{q}s} \\cdot \\mathbf{p}) \n \\left\\{ \\frac{1}{N} \\sum_{\\mathbf{l}} e^{i(\\mathbf{k} + \\mathbf{q} - \\mathbf{k}')\\cdot \\mathbf{l}} \\right\\} \\times\n V_{\\mathbf{p}} \\langle \\mathbf{k}' | e^{i\\mathbf{p}\\cdot\\mathbf{r}} | \\mathbf{k} \\rangle \n (a_{\\mathbf{q}s} + a_{-\\mathbf{q}s}^\\dagger) \n C_{\\mathbf{k}}^\\dagger C_{\\mathbf{k}'}\n \\end{equation}\n Now replace Bloch wave functions with plane waves Please provide the expression for the transition probability of the system from the initial to the final state during electron-phonon interaction (considering the long-time limit). You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$W$": "transition probability", + "$i$": "initial state", + "$f$": "final state", + "$\\hbar$": "reduced Planck's constant", + "$E_f$": "energy of final state", + "$E_i$": "energy of initial state", + "$t$": "interaction time", + "$H_{ep}$": "electron-phonon interaction Hamiltonian" + }, + "chapter": "", + "section": "" + }, + { + "id": 492, + "topic": "Superconductivity", + "question": "The known electron-phonon interaction Hamiltonian:\n \\begin{equation}\n H_{ep} = -i \\sum_{\\mathbf{k}, \\mathbf{k}'} \\sum_{\\mathbf{q}, s} \\sum_{\\mathbf{p}} \n \\left( \\frac{N \\hbar}{2 M \\omega_{\\mathbf{q}s}} \\right)^{1/2} \n (\\mathbf{e}_{\\mathbf{q}s} \\cdot \\mathbf{p}) \n \\left\\{ \\frac{1}{N} \\sum_{\\mathbf{l}} e^{i(\\mathbf{k} + \\mathbf{q} - \\mathbf{k}')\\cdot \\mathbf{l}} \\right\\} \\times\n V_{\\mathbf{p}} \\langle \\mathbf{k}' | e^{i\\mathbf{p}\\cdot\\mathbf{r}} | \\mathbf{k} \\rangle \n (a_{\\mathbf{q}s} + a_{-\\mathbf{q}s}^\\dagger) \n C_{\\mathbf{k}}^\\dagger C_{\\mathbf{k}'}\n \\end{equation}\n Now replace Bloch functions with plane waves Please provide the energy conservation relation in the electron-phonon interaction process and explain the specific relationship between electron energy and phonon energy before and after scattering. We only consider the case that the electron absorbs a phonon. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$E_f$": "final energy", + "$E_i$": "initial energy", + "$\\varepsilon_{\\mathbf{k}}$": "initial electron energy", + "$\\varepsilon_{\\mathbf{k}'}$": "final electron energy", + "$\\hbar$": "reduced Planck's constant", + "$\\omega_{\\mathbf{q}}$": "phonon frequency", + "$\\mathbf{k}$": "initial electron wave vector", + "$\\mathbf{k}'$": "final electron wave vector", + "$\\mathbf{q}$": "phonon wave vector", + "$H_{ep}$": "electron-phonon interaction Hamiltonian" + }, + "chapter": "", + "section": "" + }, + { + "id": 493, + "topic": "Superconductivity", + "question": "The known collective coordinate expression of the electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Consider a monovalent metal's simple lattice composed of $N$ ions immersed in a uniform electron gas. The frequency of perturbed LA phonons at this moment can be expressed using the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here the crystal occupies unit volume, $\\Omega$ is the primitive cell volume, $N=\\Omega^{-1}$, corresponding to the LA phonon's Hamiltonian:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n Among them, $H_e$ is the free electron approximation Hamiltonian, and $H_{ee}$ represents the Coulomb interaction. These two terms are independent of the electron normal coordinates $Q$.\n\n Under the long-wavelength approximation, complete the following calculation Find the equation of motion for $Q_q$, and discuss how to derive the phonon frequency correction You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$Q_{\\mathbf{q}}$": "normal coordinate (q-component)", + "$P_{-\\mathbf{q}}$": "momentum conjugate to Q_q with negative q-component", + "$H$": "Hamiltonian", + "$\\hbar$": "reduced Planck's constant", + "$\\delta_{\\mathbf{q}\\mathbf{q}'}$": "Kronecker delta for wave vectors q and q'", + "$\\Omega_{\\mathbf{q}}$": "frequency for normal mode (q-component)", + "$M_{-\\mathbf{q}}$": "coupling coefficient", + "$\\rho_{\\mathbf{q}}$": "charge density component (q-component)", + "$N$": "number of particles", + "$M$": "mass", + "$e$": "elementary charge", + "$\\mathbf{e}_{\\mathbf{q}}$": "polarization vector for q-component", + "$\\mathbf{q}$": "wave vector" + }, + "chapter": "", + "section": "" + }, + { + "id": 494, + "topic": "Superconductivity", + "question": "The known collective coordinate expression of the electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Considering a monovalent metal, where a simple lattice consisting of $N$ ions is immersed in a uniform electron gas, the perturbated LA phonon frequency can be expressed by the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here, the crystal is taken with unit volume, $\\Omega$ is the volume of the primitive cell, $N=\\Omega^{-1}$, corresponding to the LA phonon Hamiltonian:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n where $H_e$ is the Hamiltonian for free electrons approximation, $H_{ee}$ represents the Coulomb interaction, both of which are independent of the normal coordinate $Q$.\n\n Complete the following calculation under the long-wavelength approximation Calculate the ionic density fluctuations $\\rho^i_q$ produced by lattice vibrations", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\rho^i_q$": "ionic density fluctuation for wave vector q", + "$N$": "number of unit cells", + "$e$": "electric charge", + "$\\mathbf{u}(\\mathbf{r})$": "displacement field of the lattice", + "$M$": "mass of unit cell", + "$\\mathbf{q}$": "wave vector", + "$\\mathbf{e}_{\\mathbf{q}}$": "polarization vector for wave vector q", + "$Q_{\\mathbf{q}}$": "normal coordinate for wave vector q", + "$\\rho_{\\mathbf{q}}^i$": "ionic density Fourier component for wave vector q", + "$\\mathbf{r}$": "position vector" + }, + "chapter": "", + "section": "" + }, + { + "id": 495, + "topic": "Superconductivity", + "question": "The known collective coordinate expression for the electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Consider a monovalent metal where a simple lattice composed of $N$ ions is immersed in a uniform electron gas. In this case, the perturbed LA phonon frequency can be expressed by the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here, the crystal is taken to be of unit volume, $\\Omega$ is the unit cell volume of the primitive lattice, and $N=\\Omega^{-1}$. The Hamiltonian corresponding to the LA phonons is:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n where $H_e$ is the Hamiltonian for free electrons approximately, and $H_{ee}$ represents the Coulomb interaction. These two terms are independent of the electronic normal coordinate $Q$.\n\n Under the long-wavelength approximation, complete the following calculation Using linear response theory, calculate the relationship between $\\rho_q$ and $\\rho^i_q$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\rho_{\\mathbf{q}}$": "charge density at wavevector q", + "$\\rho_{\\mathbf{q}}^i$": "ionic charge density at wavevector q", + "$\\epsilon(\\mathbf{q})$": "static dielectric function at wavevector q" + }, + "chapter": "", + "section": "" + }, + { + "id": 496, + "topic": "Superconductivity", + "question": "The collective coordinate expression for the known electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Consider a monovalent metal immersed in a uniform electron gas, forming a simple lattice composed of $N$ ions. The perturbed LA phonon frequency can be expressed in terms of the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here the crystal is taken with unit volume, $\\Omega$ is the unit cell volume of the Bravais lattice, and $N=\\Omega^{-1}$. The Hamiltonian corresponding to the LA phonons is:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n where $H_e$ is the Hamiltonian of free electrons in approximation, and $H_{ee}$ represents Coulomb interaction, both of which are independent of the normal coordinate $Q$ of phonons.\n\n Under the long-wavelength approximation, complete the following calculations. Under the long-wavelength approximation, considering the electron screening effect, derive the dispersion relation between the LA phonon angular frequency $\\omega_{\\mathbf{q}}$ and the wave vector $q$, and specify its form $\\omega_{\\mathbf{q}}$ with respect to $q$. Hint: You can use the Thomas-Fermi dielectric function. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\omega_{\\mathbf{q}}$": "LA phonon angular frequency for wave vector $\\mathbf{q}$", + "$q$": "wave vector magnitude", + "$M_{\\mathbf{q}}$": "mass term related to wave vector $\\mathbf{q}$", + "$\\Omega_{\\mathbf{q}}$": "unperturbed phonon frequency for wave vector $\\mathbf{q}$", + "$\\epsilon(\\mathbf{q})$": "dielectric function at wave vector $\\mathbf{q}$", + "$\\lambda^2$": "Thomas-Fermi screening parameter squared", + "$N$": "electron density", + "$e$": "elementary charge", + "$E_F$": "Fermi energy", + "$M$": "ionic mass in phonon system", + "$m$": "electron mass", + "$v_F$": "Fermi velocity" + }, + "chapter": "", + "section": "" + }, + { + "id": 497, + "topic": "Superconductivity", + "question": "The known collective coordinate expression for the electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Consider a monovalent metal, a simple lattice consisting of $N$ ions submerged in a uniform electron gas. At this time, the perturbed LA phonon frequency can be expressed by the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here the crystal takes unit volume, $\\Omega$ is the positive lattice unit cell volume, $N=\\Omega^{-1}$, corresponding to the Hamiltonian of the LA phonon:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n where $H_e$ is the Hamiltonian for free electrons, and $H_{ee}$ represents the Coulomb interaction, both terms are independent of the electron normal coordinate $Q$.\n\n Under the long wavelength approximation, complete the following calculations With the known dispersion relation of LA phonons in the form $\\omega_{\\mathbf{q}} = c_L q$, please provide the specific expression for the speed of sound of LA phonons $c_L$ (i.e., the Bohm-Staver speed of sound formula), and explain each physical quantity. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\omega_{\\mathbf{q}}$": "angular frequency of the phonon with wavevector \\mathbf{q}", + "$c_L$": "speed of sound of longitudinal acoustic (LA) phonons", + "$q$": "magnitude of the phonon wavevector", + "$m$": "electron mass", + "$M$": "ion mass", + "$v_F$": "Fermi velocity", + "$k_F$": "Fermi wavevector" + }, + "chapter": "", + "section": "" + }, + { + "id": 498, + "topic": "Superconductivity", + "question": "The longitudinal optical (LO) vibration mode in ionic crystals generates a polarization field, which strongly couples with conduction electrons in ionic crystals. This interaction is much stronger than the effect of longitudinal acoustic (LA) phonons (which involves center of mass motion and does not generate a polarization field) on conduction electrons. Therefore, the interaction between LO phonons and conduction electrons affects the carrier properties in ionic crystals. When electrons move within an ionic crystal, they cause relative displacements between positive and negative ions, forming a local polarization field. This polarization, accompanying the electron motion, excites LO phonons, leading to the renormalization of the electron ground state energy and effective mass, forming a quasi-particle coupled with phonons—polaron. Please discuss the classification of polarons. Then use the perturbation method starting from the Hamiltonian to calculate the effective mass of the polaron in the case of slow phonons $(\\mathbf{k}\\rightarrow 0)$.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$a$": "lattice constant", + "$\\xi$": "range of polarization", + "$\\mathbf{k}$": "wave vector of the electron", + "$\\varepsilon_{\\mathbf{k}}$": "energy of the electron in state $|\\mathbf{k}\\rangle$", + "$\\hbar$": "reduced Planck's constant", + "$m$": "effective mass of the band electron", + "$\\omega_L$": "longitudinal optical phonon frequency", + "$\\mathbf{q}$": "phonon wave vector", + "$H_{ep}$": "electron-phonon interaction Hamiltonian", + "$e$": "elementary charge", + "$F$": "field strength", + "$\\alpha$": "electron-phonon coupling constant", + "$\\epsilon_\\infty$": "high-frequency dielectric constant", + "$\\epsilon_0$": "static dielectric constant", + "$m^*$": "effective mass of the polaron" + }, + "chapter": "", + "section": "" + }, + { + "id": 499, + "topic": "Superconductivity", + "question": "The longitudinal optical (LO) phonon mode in an ionic crystal generates a polarization field, which strongly couples with the conduction electrons in the ionic crystal. This coupling is much stronger than the effect of longitudinal acoustic (LA) phonons (which represent center of mass motion and do not generate a polarization field) on the conduction electrons. Therefore, the interaction between LO phonons and conduction electrons affects the carrier characteristics in ionic crystals. Please calculate the average number of virtual phonons excited around the electron.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\langle N_{\\text{ph}} \\rangle$": "average number of virtual phonons", + "$\\mathbf{q}$": "phonon wave vector", + "$\\mathbf{k}$": "electron wave vector", + "$H_{ep}$": "electron-phonon interaction Hamiltonian", + "$\\varepsilon_{\\mathbf{k}}$": "electron energy with wave vector k", + "$\\varepsilon_{\\mathbf{k}-\\mathbf{q}}$": "electron energy with wave vector k-q", + "$\\hbar$": "reduced Planck constant", + "$\\omega_L$": "longitudinal optical phonon frequency", + "$e$": "elementary charge", + "$F$": "electric field or force constant", + "$m$": "electron mass", + "$\\alpha$": "dimensionless coupling constant in the expression" + }, + "chapter": "", + "section": "" + }, + { + "id": 500, + "topic": "Superconductivity", + "question": "Effective interaction of current electronic exchange virtual phonons\n \\begin{equation*}\n H_{\\text{eff}} = \\frac{1}{2} \\sum_{\\substack{\\mathbf{k}_1, \\mathbf{k}_2 \\\\ q_1, q_2}} V_{\\mathbf{k}_1, \\mathbf{q}} C^\\dagger_{\\mathbf{k}_1 + \\mathbf{q}, q_1} C^\\dagger_{\\mathbf{k}_2 - \\mathbf{q}, q_2} C_{\\mathbf{k}_2, q_2} C_{\\mathbf{k}_1, q_1} c\n \\end{equation*}\n \n The interaction coefficient is:\n \\begin{equation*}\n V_{\\mathbf{k}_1, \\mathbf{q}} = |D_{\\mathbf{q}}|^2 \\frac{2\\hbar\\omega_{\\mathbf{q}}}{(E_{\\mathbf{k}_1 + \\mathbf{q}} - E_{\\mathbf{k}_1})^2 - (\\hbar\\omega_{\\mathbf{q}})^2} \n \\end{equation*} Analyze the situation near the Fermi surface and describe the interaction when the attractive potential is greater than the screened Coulomb potential, and elaborate on the approximation method of BCS theory.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E_{\\mathbf{k}_1+\\mathbf{q}}$": "energy of electron state with momentum $\\mathbf{k}_1 + \\mathbf{q}$", + "$E_{\\mathbf{k}_1}$": "energy of electron state with momentum $\\mathbf{k}_1$", + "$\\hbar \\omega_{\\mathbf{q}}$": "energy of phonon with momentum $\\mathbf{q}$", + "$\\hbar \\omega_D$": "energy corresponding to Debye frequency", + "$\\omega_D$": "Debye frequency of phonons", + "$V_{\\mathbf{k}_1,\\mathbf{q}}$": "effective potential for electron interaction", + "$q$": "momentum transfer", + "$\\lambda$": "screening length", + "$e$": "elementary charge", + "$V_{\\text{net}}$": "net potential after considering electron interaction", + "$\\sigma_1$": "spin of first electron", + "$\\sigma_2$": "spin of second electron", + "$V$": "constant attractive interaction potential in BCS approximation" + }, + "chapter": "", + "section": "" + }, + { + "id": 501, + "topic": "Superconductivity", + "question": "Please discuss the interaction problem of adding two electrons outside a filled Fermi sea at $T=0K$. Approximation: You can assume that the electrons inside the Fermi sea are free electrons. Please calculate the interaction between two electrons in the weak coupling case.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$C_{\\mathbf{k}}$": "creation operator for an electron with momentum $\\mathbf{k}$", + "$C_{-\\mathbf{k}}$": "creation operator for an electron with momentum $-\\mathbf{k}$", + "$|F\\rangle$": "ground state of the normal electrons with a filled Fermi sea", + "$\\overline{H}$": "reduced Hamiltonian", + "$\\varepsilon_{\\mathbf{k}}$": "energy of an electron with momentum $\\mathbf{k}$", + "$V$": "interaction potential", + "$k_F$": "Fermi wave vector", + "$a(\\mathbf{k})$": "coefficient for electron pair state with momentum $\\mathbf{k}$", + "$E$": "energy of the system state", + "$\\lambda$": "Lagrange multiplier in the variational method", + "$\\hbar\\omega_D$": "cutoff energy determined by Debye frequency $\\omega_D$", + "$g(0)$": "density of states at the Fermi surface", + "$A$": "sum of coefficients $a(\\mathbf{k})$ over allowed states" + }, + "chapter": "", + "section": "" + }, + { + "id": 502, + "topic": "Superconductivity", + "question": "It is known that the BCS superconducting Hamiltonian can be written as:\n \\begin{equation} \\label{eq:6.4.23_again}\n \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}),\n \\end{equation}\n where $\\alpha$ is the quasiparticle operator, $\\Delta$ represents the superconducting energy gap, and $E_s(0)$ denotes the ground state energy, given by the expression:\n \\begin{equation}\n E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V},\n \\end{equation}\n with:\n \\begin{equation*}\n \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F),\n \\end{equation*}\n\n Please solve the following problem based on the Hamiltonian: Calculate the superconducting gap $\\Delta$ at zero temperature;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\Delta$": "superconducting gap", + "$T$": "temperature", + "$V$": "interaction potential", + "$\\alpha_k$": "quasiparticle operator", + "$\\alpha_{-k}$": "quasiparticle operator", + "$u_k$": "Bogoliubov transformation coefficient", + "$v_k$": "Bogoliubov transformation coefficient", + "$\\epsilon_k$": "quasiparticle energy", + "$\\xi_k$": "quasiparticle excitation energy", + "$g(0)$": "density of states at the Fermi surface", + "$\\omega_D$": "Debye frequency" + }, + "chapter": "", + "section": "" + }, + { + "id": 503, + "topic": "Superconductivity", + "question": "The known Hamiltonian of BCS superconductors can be expressed as:\n \\begin{equation} \\label{eq:6.4.23_again}\n \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}),\n \\end{equation}\n Here, $\\alpha$ is the quasiparticle operator, $\\Delta$ represents the energy gap of the quasiparticles, and $E_s(0)$ is the ground state energy, expressed as:\n \\begin{equation}\n E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V},\n \\end{equation}\n where:\n \\begin{equation*}\n \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F),\n \\end{equation*}\n\n Please solve the following problem starting from the Hamiltonian: Please calculate the superconducting critical temperature $T_c$ based on the gap equation.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$T_c$": "critical temperature for superconductivity", + "$T$": "temperature", + "$\\Delta$": "superconducting energy gap", + "$V$": "interaction potential", + "$k$": "wave number or momentum index", + "$\\alpha_k$": "quasiparticle annihilation operator", + "$\\alpha_k^\\dagger$": "quasiparticle creation operator", + "$\\xi_k$": "quasiparticle excitation energy", + "$\\epsilon_k$": "single-particle energy level", + "$u_k$": "transformation coefficient for the quasiparticle state", + "$v_k$": "transformation coefficient for the quasiparticle state", + "$\\beta$": "inverse temperature factor", + "$k_B$": "Boltzmann constant", + "$g(0)$": "density of states at the Fermi level", + "$\\omega_D$": "Debye frequency", + "$M$": "ionic mass", + "$e$": "base of the natural logarithm", + "$\\gamma$": "Euler's constant" + }, + "chapter": "", + "section": "" + }, + { + "id": 504, + "topic": "Superconductivity", + "question": "It is known that BCS superconducting Hamiltonian can be written as: \\begin{equation} \\label{eq:6.4.23_context} \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}), \\end{equation} where $\\alpha$ is the quasiparticle operator, $\\Delta$ represents the energy gap of the quasiparticles, $E_s(0)$ denotes the ground state energy, given by: \\begin{equation} E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V}, \\end{equation} where: \\begin{equation*} \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F), \\end{equation*} Please solve the following problem starting from the Hamiltonian: At $T \\ll T_c$, calculate the approximate expression for $\\Delta(T)$ according to the previous results", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$T$": "temperature", + "$T_c$": "critical temperature", + "$\\Delta(T)$": "energy gap at temperature T", + "$\\Delta(0)$": "energy gap at absolute zero", + "$\\hbar$": "reduced Planck's constant", + "$\\omega_D$": "Debye frequency", + "$\\beta$": "inverse of thermal energy (1/kB*T)", + "$\\epsilon$": "energy variable", + "$k_B$": "Boltzmann constant" + }, + "chapter": "", + "section": "" + }, + { + "id": 505, + "topic": "Superconductivity", + "question": "The Hamiltonian of known BCS superconductivity can be written as:\n \\begin{equation} \\label{eq:6.4.23_context}\n \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}),\n \\end{equation}\n where $\\alpha$ is the quasiparticle operator, $\\Delta$ represents the energy gap of the quasiparticles, $E_s(0)$ represents the ground state energy, expressed as:\n \\begin{equation}\n E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V},\n \\end{equation}\n where:\n \\begin{equation*}\n \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F),\n \\end{equation*}\n\n Please complete the following problem starting from the Hamiltonian: Calculate the approximate expression for $\\Delta(T)$ as $T \\rightarrow T_c$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\Delta(T)$": "energy gap as function of temperature T", + "$T$": "temperature", + "$T_c$": "critical temperature", + "$g(0)$": "density of states at the Fermi surface", + "$V$": "interaction potential", + "$\\hbar \\omega_D$": "Debye energy", + "$\\epsilon$": "energy variable for integration", + "$\\beta$": "inverse temperature in units of energy", + "$k_B$": "Boltzmann constant", + "$\\omega_n$": "Matsubara frequency", + "$\\xi$": "generic energy variable", + "$\\gamma$": "Euler constant", + "$\\zeta(3)$": "Riemann zeta function at 3" + }, + "chapter": "", + "section": "" + }, + { + "id": 506, + "topic": "Superconductivity", + "question": "It is known that the Hamiltonian for BCS superconductors is \n\n \\begin{equation}\n \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}),\n \\end{equation}\n where $\\alpha$ is the quasiparticle operator, $\\Delta$ indicates the quasiparticle energy gap, $E_s(0)$ represents the ground state energy, and the expression is:\n \\begin{equation}\n E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V},\n \\end{equation}\n where:\n \\begin{equation*}\n \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F),\n \\end{equation*}\n\n Please calculate: In the case $T \\ll T_c$, the low-temperature approximation of the electronic specific heat", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$T$": "temperature", + "$T_c$": "critical temperature", + "$c_{es}$": "electronic specific heat capacity", + "$\\Delta (0)$": "energy gap at zero temperature", + "$\\Delta (T)$": "energy gap at temperature $T$", + "$\\beta$": "inverse temperature factor (1/kBT)", + "$f$": "distribution function", + "$\\xi$": "variable related to energy", + "$k_B$": "Boltzmann constant", + "$g(0)$": "density of states at the Fermi level", + "$\\epsilon$": "energy variable", + "$\\hbar$": "reduced Planck's constant" + }, + "chapter": "", + "section": "" + }, + { + "id": 507, + "topic": "Magnetism", + "question": "Given the ferromagnetic Heisenberg model $H = -J \\sum_{l,\\delta}S_l \\cdot S_{l+\\delta}$, where $\\delta$ represents the difference in position between neighboring lattice sites, discuss the case of $J>0$. Please perform the following calculations: Solve for the ground state energy of the Heisenberg model.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\hat{\\mathbf{S}}_l$": "vector spin operator at lattice site", + "$\\hat{S}_l^x$": "x-component of spin operator at lattice site", + "$\\hat{S}_l^y$": "y-component of spin operator at lattice site", + "$\\hat{S}_l^z$": "z-component of spin operator at lattice site", + "$H$": "Hamiltonian", + "$\\hbar$": "reduced Planck's constant", + "$S$": "ion spin quantum number", + "$m$": "eigenvalue of the z-component of spin", + "$\\hat{S}_l^+$": "spin raising operator at lattice site", + "$\\hat{S}_l^-$": "spin lowering operator at lattice site", + "$l$": "lattice site index", + "$\\hat{S}_{l+\\delta}$": "spin operator at neighboring lattice site", + "$J$": "exchange interaction constant", + "$\\delta$": "neighbor index", + "$Z$": "coordination number of the lattice", + "$E_0$": "ground state energy" + }, + "chapter": "Electromagnetic properties", + "section": "" + }, + { + "id": 508, + "topic": "Strongly Correlated Systems", + "question": "Consider the action of the Hubbard model: \\begin{align}\n S_{\\text{loc}}[\\Phi^\\dagger, \\Phi] &= \\int_0^\\beta d\\tau \\sum_{\\mathbf{r}} \\Phi_{\\mathbf{r}}^\\dagger(\\partial_\\tau - \\mu - i\\Delta_c - \\Delta\\sigma^z)\\Phi_{\\mathbf{r}}, \\nonumber \\\\\n S_1[\\Phi^\\dagger, \\Phi, \\Omega] &= \\int_0^\\beta d\\tau \\sum_{\\mathbf{r}} \\Phi_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}}^\\dagger \\dot{R}_{\\mathbf{r}} \\Phi_{\\mathbf{r}}, \\nonumber \\\\\n S_2[\\Phi^\\dagger, \\Phi, \\Omega] &= - \\int_0^\\beta d\\tau \\sum_{\\mathbf{r},\\mathbf{r}'} t_{\\mathbf{r}\\mathbf{r}'} \\Phi_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}'} \\Phi_{\\mathbf{r}'}.\n \\label{eq:6.141}\n \\end{align}\nWhere: \\begin{equation} R_{\\mathbf{r}} = e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}\\sigma^z} e^{-\\frac{i}{2}\\theta_{\\mathbf{r}}\\sigma^y} e^{-\\frac{i}{2}\\psi_{\\mathbf{r}}\\sigma^z} = \\begin{pmatrix} \\cos\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}} & -\\sin\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}} \\\\ \\sin\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{\\frac{i}{2}\\varphi_{\\mathbf{r}}} & \\cos\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{\\frac{i}{2}\\varphi_{\\mathbf{r}}} \\end{pmatrix}, \\end{equation} \nWe consider the fields $\\delta_c$ and $\\delta$ at the saddle-point approximation: $\\delta_c = i(U/2)\\langle\\phi_{\\mathbf{r}}^\\dagger\\phi_{\\mathbf{r}}\\rangle$ and $\\delta = (U/2)\\langle\\phi_{\\mathbf{r}}^\\dagger\\sigma^z\\phi_{\\mathbf{r}}\\rangle$, for the half-filled case, $\\mu + i\\delta_c = 0$. Please calculate First-order moment $\\langle S_2 \\rangle$", + "final_answer": [], + "answer_type": "Numeric", + "answer": "", + "symbol": { + "$\\langle S_2 \\rangle$": "first-order moment", + "$\\langle \\Phi^*_{\\mathbf{r\\sigma}}\\Phi_{\\mathbf{r\\sigma}}\\rangle$": "average value of the product of field functions at location and spin", + "$\\mathbf{r}$": "position vector", + "$\\mathbf{r'}$": "another position vector", + "$\\sigma$": "spin index" + }, + "chapter": "Strong correlation basis", + "section": "" + }, + { + "id": 509, + "topic": "Others", + "question": "The first-order perturbation calculation for the Hamiltonian of interacting electron systems \\begin{equation}H = H_0 + H' = \\sum_{\\mathbf{k},\\sigma} \\frac{\\hbar^2 k^2}{2m} C_{\\mathbf{k}\\sigma}^{\\dagger} C_{\\mathbf{k}\\sigma} + \\frac{1}{2V} \\sum_{\\mathbf{q}} v(q) (\\rho_{\\mathbf{q}}^{\\dagger} \\rho_{\\mathbf{q}} - N) \\label{eq:4.9.1}\\end{equation} is called the Hartree-Fock approximation. Therefore, within the Hartree-Fock approximation, one only needs to take the diagonal average of $H$ with the ground state of non-interacting electrons (Fermi surface state) \\begin{equation}|0\\rangle_0 = \\prod_{k \\le k_F, \\sigma} C_{\\mathbf{k}\\sigma}^{\\dagger} |Vac\\rangle \\label{eq:4.9.2}\\end{equation}. Here, $|Vac\\rangle$ represents the state where all $\\mathbf{k}$ spaces are unoccupied, i.e., the true vacuum state. The average ground state energy per electron under the Hartree-Fock approximation (expressed in terms of Fermi wave vector);", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E^{\\text{HF}}_0$": "ground state energy in the Hartree-Fock approximation", + "$E_k$": "total kinetic energy of the electron system", + "$E_{\\text{ex}}$": "exchange energy", + "$k$": "wave vector", + "$\\sigma$": "spin index", + "$\\hbar$": "reduced Planck's constant", + "$m$": "electron mass", + "$k_F$": "Fermi wave vector", + "$E_F$": "Fermi energy", + "$q$": "momentum transfer", + "$v(q)$": "interaction potential", + "$N$": "number of electrons", + "$e$": "elementary charge", + "$\\mathscr{E}_{\\text{ex}}$": "self-energy correction of a single electron", + "$x$": "ratio of wave vector $k_1$ to Fermi wave vector $k_F$", + "$S(x)$": "exchange integral function", + "$V$": "volume" + }, + "chapter": "Basics of Condensed Matter Theory", + "section": "" + }, + { + "id": 510, + "topic": "Others", + "question": "In ionic crystals, long-wavelength optical modes represent the opposite motion of positive and negative ions within a unit cell, accompanied by polarization and interacting strongly with electromagnetic waves, thus having an important impact on the electrical and optical properties of ionic crystals. For simplicity, assume each unit cell contains only two ions with equal charge magnitude but opposite sign, still confined to an isotropic continuum model. Since, in the long-wavelength limit, the relative displacement $(u_+ - u_-)$ of the positive and negative ions within each unit cell is almost the same, a vector $\\mathbf{W}$ can be used to describe the optical branch vibration \\begin{equation} \\mathbf{W} \\equiv \\rho^{1/2} (u_+ - u_-) \\label{eq:2.6.1} \\end{equation} here $\\rho$ represents the reduced mass density \\begin{equation} \\rho = \\frac{M}{\\omega}, \\quad M = \\frac{M_+ M_-}{M_+ + M_-} \\label{eq:2.6.2} \\end{equation} $M_\\pm$ are the masses of positive and negative ions, $M$ is the reduced mass. The vector $\\mathbf{W}$ can be called the reduced displacement. Assume the polarization intensity of the crystal is $P$, the macroscopic field is $E$, and satisfy $P = \\gamma_{12} W + \\gamma_{22}E$, with elastic energy as $\\frac{1}{2}\\gamma_{11}\\mathbf{W}\\cdot \\mathbf{W}$ Please complete the following question: Calculate the Hamiltonian density", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathcal{T}$": "kinetic energy density", + "$\\phi$": "total potential energy density", + "$\\phi_{\\text{elastic}}$": "elastic potential energy density", + "$\\phi_{\\text{polarization}}$": "polarization potential energy density", + "$\\gamma_{11}$": "elastic coefficient", + "$\\gamma_{12}$": "coupling coefficient between ionic displacement and electric field", + "$\\gamma_{22}$": "electric field coefficient", + "$\\mathbf{W}$": "displacement field", + "$\\mathbf{E}$": "macroscopic electric field", + "$\\mathbf{P}$": "polarization intensity", + "$\\mathcal{L}$": "Lagrangian density", + "$\\mathcal{H}$": "Hamiltonian density" + }, + "chapter": "Basics of Condensed Matter Theory", + "section": "" + }, + { + "id": 511, + "topic": "Others", + "question": "In an ionic crystal, long-wavelength optical modes represent the opposite movement of positive and negative ions within the unit cell, accompanied by polarization and strong interaction with electromagnetic waves, thus having an important impact on the electrical and optical properties of the ionic crystal. For simplicity, assume that each unit cell contains only two ions with equal and opposite charges, still restricted to the isotropic continuous model. Because the relative displacement of positive and negative ions $(u_+ - u_-)$ in each unit cell at the long-wavelength limit is nearly the same, a vector $\\mathbf{W}$ can describe the optical branch vibration \\begin{equation} \\mathbf{W} \\equiv \\rho^{1/2} (u_+ - u_-) \\label{eq:2.6.1} \\end{equation} where $\\rho$ represents the density of the reduced mass \\begin{equation} \\rho = \\frac{M}{\\Omega}, \\quad M = \\frac{M_+ M_-}{M_+ + M_-} \\label{eq:2.6.2} \\end{equation} $M_\\pm$ are the masses of the positive and negative ions, and $M$ is the reduced mass. The vector $\\mathbf{W}$ can be termed as the reduced displacement. Assume the polarization intensity of the crystal is $P$, the macroscopic field is $E$, satisfying $P = \\gamma_{12} W + \\gamma_{22}E$, and the elastic energy is $\\frac{1}{2}\\gamma_{11}\\mathbf{W}\\cdot \\mathbf{W}$. Please complete the following question: Calculate the squared transverse vibration frequency $\\omega^2_L$", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\omega_L^2$": "squared transverse vibration frequency", + "$\\mathbf{E}_L$": "longitudinal electric field", + "$\\mathbf{W}_L$": "longitudinal wave amplitude", + "$\\mathbf{D}$": "electric displacement field", + "$\\mathbf{P}_L$": "longitudinal polarization", + "$\\mathbf{P}$": "polarization", + "$\\gamma_{12}$": "coupling coefficient (12)", + "$\\gamma_{22}$": "coupling coefficient (22)", + "$\\gamma_{11}$": "coupling coefficient (11)" + }, + "chapter": "Basics of Condensed Matter Theory", + "section": "" + }, + { + "id": 512, + "topic": "Theoretical Foundations", + "question": "The problem of motion in the Coulomb field can be well handled in parabolic coordinates. The parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined by: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} Or conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ can range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. The surfaces $\\xi = $ constant and $\\eta = $ constant are rotational paraboloids around the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following question: Write the line element in parabolic coordinates", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\xi$": "parabolic coordinate", + "$\\eta$": "parabolic coordinate", + "$\\varphi$": "azimuthal angle in parabolic coordinates" + }, + "chapter": "Other physical foundations", + "section": "" + }, + { + "id": 513, + "topic": "Theoretical Foundations", + "question": "The problem of motion in a Coulomb field can be well handled in parabolic coordinates. The parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined by the following equations: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), \\ r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} Or conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. The surfaces $\\xi =$ constant and $\\eta =$ constant are rotational paraboloids about the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following problem: Write out the Laplacian in parabolic coordinates", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\xi$": "parabolic coordinate (xi)", + "$\\eta$": "parabolic coordinate (eta)", + "$\\varphi$": "angular coordinate" + }, + "chapter": "Other physical foundations", + "section": "" + }, + { + "id": 514, + "topic": "Theoretical Foundations", + "question": "The problem of motion in a Coulomb field can be well handled in parabolic coordinates, where the parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined by the following: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), \\ r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} or conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ can range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. Surfaces of constant $\\xi$ and constant $\\eta$ are rotational paraboloids about the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following problem: Write down the single-particle Schrödinger equation You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\xi$": "coordinate variable xi", + "$\\eta$": "coordinate variable eta", + "$\\psi$": "wave function", + "$\\varphi$": "angle variable phi", + "$E$": "energy" + }, + "chapter": "Other physical foundations", + "section": "" + }, + { + "id": 515, + "topic": "Theoretical Foundations", + "question": "The problem of motion in a Coulomb field can be well-handled in parabolic coordinates. The parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined as follows: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), \\ r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} or conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. The surfaces $\\xi =$ constant and $\\eta =$ constant are rotational paraboloids around the $z$-axis, with the origin as the focus. This is an orthogonal coordinate system. Please complete the following task: Use the method of separation of variables to solve the Schrödinger equation corresponding to the discrete spectrum;", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\psi$": "eigenfunction", + "$m$": "magnetic quantum number", + "$E$": "energy", + "$\\xi$": "parabolic coordinate (xi)", + "$\\eta$": "parabolic coordinate (eta)", + "$\\beta_1$": "separation parameter 1", + "$\\beta_2$": "separation parameter 2", + "$n$": "principal quantum number", + "$\\rho_1$": "scaled parabolic coordinate 1", + "$\\rho_2$": "scaled parabolic coordinate 2", + "$n_1$": "parabolic quantum number 1", + "$n_2$": "parabolic quantum number 2", + "$\\varphi$": "azimuthal angle", + "$\\psi_{n_1 n_2 m}$": "wave function for discrete spectrum", + "$L_{n_1}^{|m|}(\\xi)$": "associated Laguerre polynomial for n1", + "$L_{n_2}^{|m|}(\\eta)$": "associated Laguerre polynomial for n2" + }, + "chapter": "Other physical foundations", + "section": "" + }, + { + "id": 516, + "topic": "Theoretical Foundations", + "question": "The problem of motion in a Coulomb field can be well treated in parabolic coordinates. The parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined by the following: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), \\ r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} Conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ can range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. The surfaces $\\xi = $ constant and $\\eta = $ constant are rotational paraboloids about the $z$-axis centered at the origin. This forms an orthogonal coordinate system. Please complete the following problem: Assuming a hydrogen atom in a uniform electric field, solve for the energy level correction to second order approximation (Stark effect).", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$E$": "energy", + "$\\mathscr{E}$": "electric field strength", + "$\\xi$": "parabolic coordinate (xi)", + "$\\eta$": "parabolic coordinate (eta)", + "$m$": "magnetic quantum number", + "$n$": "principal quantum number", + "$n_1$": "quantum number in parabolic coordinates (1)", + "$n_2$": "quantum number in parabolic coordinates (2)", + "$\\beta_1$": "eigenvalue of the operator for xi coordinate", + "$\\beta_2$": "eigenvalue of the operator for eta coordinate", + "$\\epsilon$": "parameter related to energy" + }, + "chapter": "Other physical foundations", + "section": "" + }, + { + "id": 517, + "topic": "Theoretical Foundations", + "question": "We discuss the propagation of light in conductive media, considering a uniform isotropic medium whose dielectric constant is $\\varepsilon$, magnetic permeability is $\\mu$, and conductivity is $\\sigma_0$. Utilizing the material equations $\\mathbf{j} = \\sigma \\mathbf{E}$, $\\mathbf{D} = \\varepsilon \\mathbf{E}$, $\\mathbf{B} = \\mu \\mathbf{H}$, the Maxwell equations take the following form: \\begin{align}\n\\text{curl } \\mathbf{H} - \\frac{\\varepsilon}{c} \\dot{\\mathbf{E}} &= \\frac{4\\pi}{c} \\sigma \\mathbf{E}, \\label{eq:1} \\\\\n\\text{curl } \\mathbf{E} + \\frac{\\mu}{c} \\dot{\\mathbf{H}} &= 0, \\label{eq:2} \\\\\n\\text{div } \\mathbf{E} &= \\frac{4\\pi}{\\varepsilon} \\rho, \\label{eq:3} \\\\\n\\text{div } \\mathbf{H} &= 0. \\label{eq:4}\n\\end{align} Please calculate: Charge density within a metal", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\varepsilon$": "permittivity", + "$c$": "speed of light", + "$\\rho$": "charge density", + "$\\dot{\\mathbf{E}}$": "time derivative of electric field", + "$\\sigma$": "conductivity", + "$\\rho_0$": "initial charge density", + "$t$": "time", + "$\\tau$": "time constant" + }, + "chapter": "Other physical foundations", + "section": "" + }, + { + "id": 518, + "topic": "Theoretical Foundations", + "question": "We discuss the propagation of light in conductive media, considering a homogeneous isotropic medium with a dielectric constant $\\varepsilon$, magnetic permeability $\\mu$, and conductivity $\\sigma_0$. Using the material equations $\\mathbf{j} = \\sigma \\mathbf{E}$, $\\mathbf{D} = \\varepsilon \\mathbf{E}$, $\\mathbf{B} = \\mu \\mathbf{H}$, Maxwell's equations take the following form:\n\n\\begin{align}\n\\text{curl } \\mathbf{H} - \\frac{\\varepsilon}{c} \\dot{\\mathbf{E}} &= \\frac{4\\pi}{c} \\sigma \\mathbf{E}, \\label{eq:1} \\\\\n\\text{curl } \\mathbf{E} + \\frac{\\mu}{c} \\dot{\\mathbf{H}} &= 0, \\label{eq:2} \\\\\n\\text{div } \\mathbf{E} &= \\frac{4\\pi}{\\varepsilon} \\rho, \\label{eq:3} \\\\\n\\text{div } \\mathbf{H} &= 0. \\label{eq:4}\n\\end{align} Introduce the complex dielectric constant $\\hat{\\epsilon} = \\epsilon + i \\frac{4\\pi \\sigma}{\\omega}$, the complex phase velocity $\\hat{v} = \\frac{c}{\\sqrt{\\mu\\hat{\\epsilon}}}$, and the complex refractive index $\\hat{n} = \\frac{c}{\\hat{v}} = \\frac{c}{\\omega}k$, let $\\hat{n}=n(1+i\\kappa)$, where $\\kappa$ is the attenuation coefficient. Please express the refractive index $n$ using the material constants $\\epsilon,\\mu,\\sigma$. You should return your answer as an equation.", + "final_answer": [], + "answer_type": "Equation", + "answer": "", + "symbol": { + "$\\hat{\\epsilon}$": "complex dielectric constant", + "$\\epsilon$": "real part of the dielectric constant", + "$\\sigma$": "electrical conductivity", + "$\\omega$": "angular frequency", + "$\\hat{v}$": "complex phase velocity", + "$c$": "speed of light in vacuum", + "$\\mu$": "magnetic permeability", + "$\\hat{n}$": "complex refractive index", + "$n$": "real refractive index", + "$\\kappa$": "attenuation coefficient", + "$v$": "velocity" + }, + "chapter": "Other physical foundations", + "section": "" + }, + { + "id": 519, + "topic": "Theoretical Foundations", + "question": "We discuss the propagation of light in conducting media, considering a homogeneous isotropic medium with dielectric constant $\\varepsilon$, magnetic permeability $\\mu$, and conductivity $\\sigma_0$. Using the material equations $\\mathbf{j} = \\sigma \\mathbf{E}$, $\\mathbf{D} = \\varepsilon \\mathbf{E}$, $\\mathbf{B} = \\mu \\mathbf{H}$, the Maxwell equations take the following form:\n\\begin{align} \\text{curl } \\mathbf{H} - \\frac{\\varepsilon}{c} \\dot{\\mathbf{E}} &= \\frac{4\\pi}{c} \\sigma \\mathbf{E}, \\\\ \\text{curl } \\mathbf{E} + \\frac{\\mu}{c} \\dot{\\mathbf{H}} &= 0, \\\\ \\text{div } \\mathbf{E} &= \\frac{4\\pi}{\\varepsilon} \\rho, \\\\ \\text{div } \\mathbf{H} &= 0. \\end{align} \n\nPlease answer the following question. If the electric field is a plane wave, calculate the energy density of the wave.", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$\\mathbf{E}$": "electric field vector", + "$\\mathbf{E}_0$": "initial electric field vector amplitude", + "$\\mathbf{k}$": "wave vector", + "$\\mathbf{r}$": "position vector", + "$\\omega$": "angular frequency", + "$t$": "time", + "$\\hat{k}$": "unit wavevector", + "$\\hat{n}$": "unit vector in direction of wave propagation", + "$c$": "speed of light in vacuum", + "$n$": "refractive index", + "$\\kappa$": "extinction coefficient or absorption index", + "$\\lambda$": "wavelength in the medium", + "$\\mathbf{s}$": "unit vector in position direction", + "$w$": "energy density of the wave", + "$w_0$": "initial energy density of the wave", + "$\\chi$": "absorption coefficient", + "$\\lambda_0$": "wavelength in vacuum", + "$v$": "frequency" + }, + "chapter": "Other physical foundations", + "section": "" + }, + { + "id": 520, + "topic": "Strongly Correlated Systems", + "question": "Consider the action of the Hubbard model:\n\\begin{align}\n S_{\\text{loc}}[\\Phi^\\dagger, \\Phi] &= \\int_0^\\beta d\\tau \\sum_{\\mathbf{r}} \\Phi_{\\mathbf{r}}^\\dagger(\\partial_\\tau - \\mu - i\\Delta_c - \\Delta\\sigma^z)\\Phi_{\\mathbf{r}}, \\nonumber \\\\\n S_1[\\Phi^\\dagger, \\Phi, \\Omega] &= \\int_0^\\beta d\\tau \\sum_{\\mathbf{r}} \\Phi_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}}^\\dagger \\dot{R}_{\\mathbf{r}} \\Phi_{\\mathbf{r}}, \\nonumber \\\\\n S_2[\\Phi^\\dagger, \\Phi, \\Omega] &= - \\int_0^\\beta d\\tau \\sum_{\\mathbf{r},\\mathbf{r}'} t_{\\mathbf{r}\\mathbf{r}'} \\Phi_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}'} \\Phi_{\\mathbf{r}'}.\n \\label{eq:6.141}\n\\end{align}\nwhere:\n\\begin{equation}\n R_{\\mathbf{r}} = e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}\\sigma^z} e^{-\\frac{i}{2}\\theta_{\\mathbf{r}}\\sigma^y} e^{-\\frac{i}{2}\\psi_{\\mathbf{r}}\\sigma^z} = \\begin{pmatrix} \\cos\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}} & -\\sin\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}} \\\\ \\sin\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{\\frac{i}{2}\\varphi_{\\mathbf{r}}} & \\cos\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{\\frac{i}{2}\\varphi_{\\mathbf{r}}} \\end{pmatrix},\n\\end{equation}\nWe consider the fields $\\Delta_c$ and $\\Delta$ at the saddle-point approximation level: $\\Delta_c = i(U/2)\\langle\\Phi_{\\mathbf{r}}^\\dagger\\Phi_{\\mathbf{r}}\\rangle$ and $\\Delta = (U/2)\\langle\\Phi_{\\mathbf{r}}^\\dagger\\sigma^z\\Phi_{\\mathbf{r}}\\rangle$. At half-filling, we have $\\mu + i\\Delta_c = 0$.\n\nPlease answer the problem\n\\begin{enumerate}\n \\item the effective action $S[\\Omega]$ accurate to the first order in $\\partial_\\tau$ and the second order in $t$:\n \\begin{equation}\n S[\\Omega] = \\langle S_1 + S_2 \\rangle - \\frac{1}{2}\\langle S_2^2 \\rangle_c,\n \\label{eq:6.142}\n \\end{equation}\n where the expectation value $\\langle \\cdots \\rangle$ is taken with respect to the local action $S_{\\text{loc}}$.\n\n This coincides with the action of the spin- $1/2$ Heisenberg model, what is its exchange coupling $J$?\n\\end{enumerate}", + "final_answer": [], + "answer_type": "Expression", + "answer": "", + "symbol": { + "$S_{\\text{loc}}$": "local action", + "$\\Phi^\\dagger$": "creation field", + "$\\Phi$": "annihilation field", + "$\\beta$": "inverse temperature", + "$\\mathbf{r}$": "lattice site position", + "$\\partial_\\tau$": "imaginary time derivative", + "$\\mu$": "chemical potential", + "$i$": "imaginary unit", + "$\\Delta_c$": "complex field at saddle-point", + "$\\Delta$": "real field at saddle-point", + "$\\sigma^z$": "Pauli matrix in z-direction", + "$R_{\\mathbf{r}}$": "rotation matrix at site", + "$t_{\\mathbf{r}\\mathbf{r}'}$": "hopping amplitude between sites", + "$\\varphi_{\\mathbf{r}}$": "azimuthal angle at site", + "$\\theta_{\\mathbf{r}}$": "polar angle at site", + "$\\psi_{\\mathbf{r}}$": "phase angle at site", + "$U$": "interaction strength", + "$t$": "hopping parameter", + "$J$": "exchange coupling", + "$G_\\sigma$": "single-particle propagator" + }, + "chapter": "", + "section": "" + } +] \ No newline at end of file