Attempt at perfecting quantum mechanics based on interaction
Tian-Hai Zeng
Many wave phenomena are related to interactions. When preparing a particle in a superposed state starting from an eigenstate, we must let it interact with an apparatus or environment. Considering neglected interactions, states of apparatus and Newton's idea about measurement, we attempt to modify some concepts of quantum mechanics. Our modifications include wave-particle duality, the principle of superposition of states, the meaning of the Schr\"{o}dinger equation, the postulate of quantum measurement, and the uncertainty relations, without changing past correct results. These may help us to understand some typical quantum phenomena with near classical and intuitive ideas, to mediate some contradictions logically, and then to go a step further toward perfecting quantum mechanics. We also review the argument between Einstien and Bohr about quantum mechanics, and conclude that the result is a draw.
1307.1851.tex

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\title{Attempt at perfecting quantum mechanics based on interaction}% Force line breaks with \\
\author{Tian-Hai Zeng}
 \email{zengtianhai@bit.edu.cn}

 \affiliation{School of Physics, Beijing Institute of Technology, Beijing 100081, People's Republic of China}%

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\begin{abstract}
  Many wave phenomena are related to interactions. When preparing a particle in a superposed state starting from an eigenstate, we must let it interact with an apparatus or environment. Considering neglected interactions, states of apparatus and Newton's idea about measurement, we attempt to modify some concepts of quantum mechanics. Our modifications include wave-particle duality, the principle of superposition of states, the meaning of the Schr\"{o}dinger equation, the postulate of quantum measurement, and the uncertainty relations, without changing past correct results. These may help us to understand some typical quantum phenomena with near classical and intuitive ideas, to mediate some contradictions logically, and then to go a step further toward perfecting quantum mechanics. We also review the argument between Einstien and Bohr about quantum mechanics, and conclude that the result is a draw.
\end{abstract}

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\section{Introduction}
Through the application of quantum mechanics (QM), scientists have made many great achievements. These show the validity of the theory of QM, such that its initial form is still accepted. But some things such as the famous argument between Einstein's camp and Bohr's camp about the complete properties of QM, and Feynman's assertion that no one can understand it, motivate some people to try to complete and understand it. As far as I know, no such attempt has yet succeeded.

Interaction (INT) energies (or potential energies) in any composite system (CS) are singles or various composites of the four fundamental INT energies: gravitational, electromagnetic, strong and weak. In a large number of composite quantum systems composed of two or more subsystems, the formulae of INT energies between subsystems in Hamiltonian terms are unknown. If we know the formulae in a CS, we can solve the Schr\"{o}dinger equation in principle and obtain the knowledge of the state evolution of the system. So looking for unknown formulae of INT energies is the key for us to understand QM, rather than looking for some hidden variable \cite{D. Bohm} that it was guessed might allow us to complete and understand QM. However, looking for unknown formulae of INT energies and solving Schr\"{o}dinger equations are not the task of this article. We assume that the INTs in all CSs here are known.

Observations of some wave phenomena, for example, sound waves, water or liquid waves, or elastic waves in solids, let us find the common point that any one of these waves has at least one INT between particles. But the wave equation, for example, a sound wave equation \cite{R. P. Feynman} ${\partial^2\xi}/{\partial^2 x=(1/\nu^2)\partial^2\xi}/{\partial^2 t}$ (here $\nu$ is the wave velocity) may easily lead one to forget the existence of INT, for it does not appear clearly in the equation. The essence of superposition of any one of these waves is actually the superposition of INTs with the same particle, which induces interference phenomena. If there is no INT, these wave phenomena will not appear. When a large number of single particles go through a double-slit one by one, an interference phenomenon appears. Since there is no INT among single particles, single particles are considered to have an inherent probability wave property, i.e., every particle is in some superposed state after it goes through the double-slit. But there certainly exists some INT between a particle and the double-slit, in which case the superposed state of a particle and the interference phenomenon may be related with the INT. We will explain this in Sec. IV below entitled \textquotedblleft Understanding of typical quantum phenomena". In Sec. II, we offer a way of preparation of a superposed state starting from an eigenstate.  In Sec. III, we give out modifications of QM and some explanations. In Sec. V, we discuss some contradictions and mediations. In Sec. VI, we review the argument between Einstein and Bohr about physical reality and locality. The conclusion is in Sec. VII.

\section{PREPARE A SUPERPOSED STATE STARTING FORM AN EIGEINSTATE}

The starting point here is the consideration of how a particle state returns to the superposed state (or a wave packet state, expressed by a wave function or state vector) after its initial superposed state has collapsed to an eigenstate (eigenfunction). An energy eigenstate of a free particle never automatically evolves into a superposed state of different energy eigenstates according to the Schr\"{o}dinger equation. A general way of preparing a particle, which is initially in an eigenstate of some physical quantity, in a superposed state with different eigenstates of the same quantity, should let it interact with an apparatus or environment.

We consider preparing a particle in a superposed state with different momentum eigenstates. A particle is initially in one of its momentum eigenstates, expressed by state vector $|\vec{p}_1\rangle$,  and an apparatus is in $|\vec{P}_1\rangle$. After switching the INT between the two parts, the CS evolves into an entangled state, first named by Schr\"{o}dinger \cite{J. A. Wheeler}, which cannot be expressed in a factoring form, for example, $|\phi\rangle|\Phi\rangle$ of a new state $|\phi\rangle$ of the particle and a new state $|\Phi\rangle$ of the apparatus. According to the Schr\"{o}dinger equation, we may obtain in principle one entangled state for a specific INT in the form:
\begin{equation}\label{l}
|\Psi\rangle=|\vec{p}_1\rangle|\vec{P}_1\rangle+|\vec{p}_2\rangle|\vec{P}_2\rangle,
\end{equation}
where $|\vec{p}_2\rangle$ and $|\vec{P}_2\rangle$ are other momentum eigenstates for the particle and the apparatus, respectively. The normalizing factors are often neglected in this article. If the apparatus is much larger than the particle, the states $|\vec{P}_1\rangle$ and $|\vec{P}_2\rangle$ can be considered as approximately the same (the magnitudes and directions of the two momentum eigenvalues) in the considered evolution time, and then we can obtain an approximate product state:
\begin{equation}\label{2}
|\Psi\rangle\approx(|\vec{p}_1\rangle+|\vec{p}_2\rangle)|\vec{P}_1\rangle,
\end{equation}
The two parts are approximately separate, and so we prepare the particle in an approximately superposed state $|\vec{p}_1\rangle+|\vec{p}_2\rangle$ of different momentum eigenstates by an apparatus.

We fail to find any other way to obtain the superposed state of momentum of a particle. According to QM, if we measure the momentum of the apparatus in the base $\{|\vec{P}_1\rangle+|\vec{P}_2\rangle$, $|\vec{P}_1\rangle-|\vec{P}_2\rangle\}$, we seem to get a superposed state $|\vec{p}_1\rangle+|\vec{p}_2\rangle$ or $|\vec{p}_1\rangle-|\vec{p}_2\rangle$ from (1), but no base (two superposed states) of a macro-apparatus exists. If the apparatus is a micro-object, we seem to be able to measure the momentum of the apparatus in the base, or to use an eraser for a delayed-choice \cite{J. S. Tang}, and then to get a superposed state of the particle, but this is a pre-hypothesis that the two superposed states of an apparatus have existed, which contradicts the premise of preparing a particle or a micro-object in a superposed state. Tracing out the states of the apparatus in (1), or measuring the momentum of the apparatus but not reading the result, we only obtain a mixed state $|\vec{p}_1\rangle\langle\vec{p}_1|+|\vec{p}_2\rangle\langle\vec{p}_2|$   (without coherence) of the particle. According to QM, the entanglement still holds even after the INT ceases, so the approximation above may be the final step to obtain a superposed state of a particle with different momentum eigenstates, and we cannot prepare a free particle (without interacting and entangling with another object) in the pure superposed state $|\vec{p}_1\rangle+|\vec{p}_2\rangle$.

 In quantum optics, when an atom initially in ground level $|g\rangle$ interacts with a single mode field $|n+1\rangle$ with $n+1$ photons, the CS may evolve into an entangled state \cite{D. F. Walls}
\begin{equation}\label{3}
|atomphotons\rangle=|g\rangle|n+1\rangle+|e\rangle|n\rangle,
\end{equation}
where $|e\rangle$ is the excited level of the atom after it has absorbed a photon. When $n$ is large enough in (3), then $|n\rangle\approx|n+1\rangle$ and the atom is in the approximately superposed state $|g\rangle+|e\rangle$. This example is similar to Ref. \cite{P. Bertet}, Bertet \textit{et al}. think that the coherent state $|\beta_g\rangle\approx|\beta_e\rangle$ in the classical limit, then the equation (3) there $(|g\rangle|\beta_g\rangle+|e\rangle|\beta_e\rangle)/\sqrt{2}\approx(|g\rangle+|e\rangle)|\beta_g\rangle/\sqrt{2}$ and a superposed state of an atom with two levels is obtained.

Some physical quantities are exchanged between entangled parts by their INTs. If the INT ceases, then no momentum is exchanged between the particle and the apparatus in (1), so the entangled state is disentangled and no such superposed state $|\vec{p}_1\rangle+|\vec{p}_2\rangle$ of a free particle can be obtained. This example explains that the entangled states and the approximately superposed state are determined by the Schr\"{o}dinger equation including an INT. Can someone imagine any way to circumvent INTs to obtain superposed states of a free particle?

It is reasonable to consider the INT between a particle and a macro-apparatus similar to the case of decoherence \cite{W. H. Zurek}, in which the environment entangles with a quantum system, and then destroys the coherence among its states due to unavoidable INT between them. We suggest the mass ratio of proton/electron, 1836, or other constant in some specific case, being as a criterion of neglecting the change of state of one object in a CS, if the mass, or energy, magnitude of momentum,  \textit{etc.}, of the object is 1836 times that of a particle, like the proton in a hydrogen atom. We call such an object a  \textit{large object} (LO).

\section{MODIFICATIONS AND EXPLANATIONS}
 In the concept of wave-particle duality of QM, the wave and particle properties of a particle are equally intrinsic, and they are all independent of INT. Our modification \cite{T. H. Zeng}: Wave property is based on INT. An INT with an LO makes a particle behave with the property of superposition of probability waves or states, and then behave with the property of interference. If there is no INT, the probability wave phenomena of a particle will not appear. A de Broglie wave vector may be considered as only a parameter corresponding to a definite momentum and energy of a free particle.

 The principle of superposition of states of QM is that a superposed state of two or more different states of a particle (or system) is still its state, and that it is also independent of INT. Our modification  \cite{Zeng}: Superposed states exist only in CSs with INTs between subsystems, they are entangled states, and INTs and the conservation laws (below) are restrictive conditions. If the state of a particle or a system can be expanded into a superposed state of different eigenstates, such as energy, momentum and angular momentum eigenstates, etc., it must be an approximate one due to the fact that an INT with an LO lets them evolve into an entangled state and the state of the LO is approximately unchanged in considered evolution time.

 The Schr\"{o}dinger equation in QM is a dynamic equation describing a CS, or a particle in a field, even a free particle or an isolated system in a superposed state with different energy eigenstates. Our modification: The Schr\"{o}dinger equation is a dynamic equation that strictly describes a CS with INTs between subsystems or particles, and the wave function must include all subsystems, even including an LO or environment. For example, an equation of a CS of two subsystems is:

 \begin{equation}\label{4}
i\hbar\frac{\partial\Psi}{\partial t}=(\frac{\hat{\overrightarrow{p_{1}}} ^{2}}{2m_{1}}+V(r_{12})+\frac{\hat{\overrightarrow{p_{2}}} ^{2}}{2m_{2}})\Psi,
\end{equation}
where ${\hat{\overrightarrow{p_{1}}} ^{2}}/{2m_{1}}$ and ${\hat{\overrightarrow{p_{2}}} ^{2}}/{2m_{2}}$ are the kinetic energy operators of micro-subsystems, or the latter ${\hat{\overrightarrow{p_{2}}} ^{2}}/{2m_{2}}$ is formally that of an LO, and $V(r_{12})$ is the INT (potential) energy belonging to the CS. The equation of a particle or a micro-system in an external field must be approximately as in (4), in which ${\hat{\overrightarrow{p_{2}}} ^{2}}/{2m_{2}}$ and the state of an LO in $\Psi$ have been neglected, for the state is approximately unchanged in the considered evolution time. The INT has been approximately considered as an external field and the INT energy is considered as belonging to the particle.

The Schr\"{o}dinger equation is also formally a kinematic equation of a free particle or an isolated quantum system, with definite energy, momentum and angular momentum, for there are no exchanges of these quantities with an LO or environment.

Conservation laws in QM are only of the sense of statistical average. Our modification: In a CS, three physical quantities, energy (mass-energy, in a relativistic case), momentum and angular momentum, are exchanged among subsystems by INTs and strictly maintain respective conservations at any time (or in a single process)  and naturally in the sense of the statistical average.

Some experimental results negated the point of Bohr \textit{et al}. \cite{N. Bohr} that the conservation laws in the micro-world only hold in the sense of the statistical average. For example, in 1925, W. Bothe and H. Geiger \cite{W. Bothe,  Bothe} by single Compton collisions, and in 1932, J. Chadwick's discovery of the neutron \cite{J. Chadwick}, proved experimentally that the conservation laws of energy and momentum hold in a single process.

For the state $|\vec{p}_1\rangle|\vec{P}_1\rangle+|\vec{p}_2\rangle|\vec{P}_2\rangle$ in (1), the INT exchanges the momentum difference $\vec{p}_1-\vec{p}_2=\vec{P}_2-\vec{P}_1$ between the two parts, and then $\vec{p}_1+\vec{P}_1=\vec{p}_2+\vec{P}_2$. The total momentum is conserved at any time even if the state collapses into $|\vec{p}_1\rangle|\vec{P}_1\rangle$ or $|\vec{p}_2\rangle|\vec{P}_2\rangle$. These terms are degenerate for momentum. If $\vec{p}_1+\vec{P}_1\neq\vec{p}_2+\vec{P}_2$, there is no such entangled state. When the state is expanded by angular momentum eigenstates, it must be of the form $|\vec{l}_1\rangle|\vec{L}_1\rangle+|\vec{l}_2\rangle|\vec{L}_2\rangle+\cdot\cdot\cdot+|\vec{l}_N\rangle|\vec{L}_N\rangle$ with $\vec{l}_1+\vec{L}_1=\vec{l}_2+\vec{L}_2=\cdot\cdot\cdot=\vec{l}_N+\vec{L}_N$, \textit{i.e.}, the conservation law of angular momentum holds at any time.

The conservation law of energy is a bit more complex than the conservation laws of momentum and angular momentum, due to interaction (INT) energy belonging to the whole CS, \textit{i.e.}, not belonging to one subsystem. Given a particle (or a micro-system) initially in one of its kinetic energy eigenstates $|k_1\rangle$ with kinetic energy $k_1$, and an apparatus also initially in one of its kinetic energy eigenstates $|K_1\rangle$ with kinetic energy $K_1$, after switching the INT between them, the CS evolves into a state $|\varepsilon_1\rangle+|\varepsilon_2\rangle$ for some INT, and the total energy is
\begin{equation}\label{5}
\varepsilon_1=k_1+K_1+V_1=k_2+K_2+V_2=\varepsilon_2,
\end{equation}
which holds conservation, where $V_1$ and $V_2$ are INT energies, and $k_2$ and $K_2$ are  kinetic energies of the particle and the apparatus, respectively. If the apparatus is an LO, then $K_1\approx K_2$ and the INT energy can be considered approximately to belong to the particle. Let the particle energy $en_1=k_1+V_1$, then the state can be entangled $|en_1\rangle|K_1\rangle+|en_2\rangle|K_2\rangle\approx(|en_1\rangle+|en_2\rangle)|K_1\rangle$, where $|en_1\rangle$ and $|en_2\rangle$ are its energy eigenstates. We believe that, affected by uncontrollably weak noise from the environment, the larger the interchange quantities (energy, momentum, angular momentum) of two or more subsystems are, the more strictly the conservation laws for the CS hold experimentally.

From the conservation laws, we can deduce that when INTs in a CS cease, \emph{i.e.}, there is no exchange of the three physical quantities between the subsystems, the subsystems must be in their respective eigenstates with definite energy, momentum and angular momentum, and the CS is not in an entangled state.

Here quantum measurement is limited to the measurement of micro-objects, not that of macro-objects reaching accuracy of the order of the standard quantum limit  \cite{V. B. Braginsky}. When an apparatus interacts with a measured quantum system and their respective states are all changed, the result of the measurement can be obtained from the change of the apparatus state. If the system state is unchanged, the apparatus state should be unchanged also, and this case also corresponds to no measurement or no measured system existing, so that no result or information about the system can be read. This real measurement of micro-systems is different from the measurement of macro-systems, in which the result can be obtained without disturbing the measured macro-system in many cases where only \textit{micro-action} exists; for example, measuring the length of a macro-object may neglect some photon-action on the two ends of the object. This point is close to Newton's idea \cite{Newton} that \textquotedblleft Relative quantities are not the actual quantities whose names they bear but are those sensible measures of them (whether true or erroneous) that are commonly used instead of the quantities being measured". This is suitable for directly measured physical quantities of quantum systems. Quantum measurement in QM is simple. The result is an eigenvalue with some probability if the initial state of a measured system is not an eigenstate of the measured quantity, and the state left is an eigenstate corresponding to the eigenvalue.

Our modification is complex. (a). The eigenvalue in an eigenstate of spin of a particle or polarization of a photon can be indirectly measured, and the eigenvalue and the eigenstate are not changed after measurement, while the value of at least one other quantity must be changed. An example is that the direction of an electron's motion is changed in the Stern-Gerlach experiment, and then the eigenvalue corresponding to an eigenstate of spin of the electron is indirectly measured. (b).Directly measured quantities, such as energy, momentum and angular momentum, etc., only measure the difference between an eigenvalue after the measurement and one of the eigenvalues of a superposed state before the measurement. These eigenvalues themselves are not observables. On carrying out a second measurement on the same quantity immediately after the first, the two results will be different and the two states after two measurements will also be different. In the experiment with an absorbing spectrum, for example, an atom at ground level absorbs a photon and its level jumps into an excited one, so the result of the measurement of its energy is then the difference of the two levels, equal to the energy of a photon, and the atom is in the excited level after measurement, rather than its initial state. (c). The error of a measurement result comes from the process of amplifying in apparatus, not from the difference of two eigenvalues of the measured system.

We may obtain the expectation value of direct measurement from (b) above. For example, if a particle is initially in an energy eigenstate $|E_0\rangle$ (assuming $E_0=0$), the final state is some mixed state $\rho_{f}=\Sigma P_{n} |E_{n}\rangle \langle E_{n}|$ (assuming all $E_{n}>E_0$) after measurement, and then the expectation value of energy is

\begin{equation}\label{6}
\langle E \rangle=\Sigma P_{n} E_{n}
\end{equation}

For an initial state $\Sigma b_{m}|E_{m}\rangle$, the final state is $\rho_{f}'=\Sigma p_{j} |E_{j}\rangle \langle E_{j}|$ after measurement, where $E_{m}$ and $E_{j}$ are ordered in increasing size. In the measurement of an absorption spectrum (or an emission spectrum), the eigenvalue of a final eigenstate is greater (or less) than that of at least one eigenstate of the initial state. According to our modification, a measured energy is the difference between two eigenvalues,  \textit{i.e.}, $E_{j}-E_{m}$. In the case of the measurement of an absorption spectrum, for example, $m=1,2$ and $j=2,3,4$, and the expectation value of the measured energy is

\begin{equation}\label{7}
\begin{split}
\langle E' \rangle &=p_2(E_{2}-E_{1})+ |b_1|^{2}[p_3(E_{3}-E_{1})+p_4(E_{4}-E_{1})]\\
&+|b_2|^{2}[p_3(E_{3}-E_{2})+p_4(E_{4}-E_{2})],
\end{split}
\end{equation}
where $|b_1|^{2}+|b_2|^{2}=1$, and $p_2+p_3+p_4=1$.

Two uncertainty relations in QM are $\triangle p \triangle x\geq\hbar/2$ and $\triangle E \triangle t\geq\hbar/2$. The explanation of the former is that the greater the accuracy with which the coordinate \emph{x} of a particle is known (\textit{i.e.}, the less  $\triangle x$), the greater the momentum uncertainty  $\triangle p$ of the particle \cite{L. D. Landau}. The explanation of the latter is that the smaller the time interval $\triangle t$ (of two measurements) is, the greater the energy change that is observed. Our modification: The least change of angular momentum is $h$ (Planck constant), which is the least exchanging action between one particle and another one or an apparatus; and a change of the energy or the momentum of a measured particle is equal to the whole or part (Compton collisions) of the energy or momentum of one photon. For example, if a photon interacts with an atom and is absorbed by the atom, the increased energy of the atom has been measured, $\triangle E=h/T$, where \emph{T} is the inverse of the photon frequency and may be considered as a characteristic time of the photon absorbed by another system. Completing a measurement needs a time interval, $\triangle t\geq T$, then we obtain $\triangle E \triangle t\geq h$, where \emph{h} is different from $\hbar/2$ in the Heisenberg uncertainty relation  and the $\triangle t$ is not a time interval of two measurements. Similarly, in \emph{x} direction the change of the momentum of the atom is also equal to that of the photon, $\triangle p=h/cT$, where \emph{c} is the speed of light. Completing a measurement also needs an  extent, $\triangle x\geq cT$, then $\triangle p \triangle t\geq h$.

The principle of the indistinguishability of identical particles in QM is that the wave function of the system is either symmetrical or antisymmetrical, and the system is in a pure entangled state. Our modification: The system can be in a pure entangled state only if there is the same INT between any pair of identical particles; when all INTs cease, any particle is free and is in an eigenstate having definite energy, momentum and angular momentum, and the pure entangled state of the system is disentangled and is only described as a mixed state. The indistinguishability fits for the mixed state as well as the pure entangled state.

We deem that the following three laws in the micro-world correspond to Newton's three laws. The Schr\"{o}dinger equation corresponds to Newton's second law. When INTs cease, a free particle has a definite energy, momentum and angular momentum, and it is still described formally by the Schr\"{o}dinger equation; this corresponds to Newton's first law, in which a macro-particle is still described formally by Newton's second law under the sum of all forces being zero. The modified principle of the superposition of states corresponds to Newton's third law.

\section{Understanding of typical quantum phenomena}

The most typical quantum phenomenon is double-slit interference of material particles. We may explain that, after an electron passes through a double-slit, the INT between the two parts in principle drives the CS to evolve an entangled state according to the Schr\"{o}dinger equation. Assume that two evolutions are $|\psi_1\rangle|S_1\rangle\rightarrow\alpha_1|\psi_1\rangle|S_1\rangle+\beta_1|\psi_2\rangle|S_2\rangle$, $|\psi_2\rangle|S_2\rangle\rightarrow\alpha_2|\psi_1\rangle|S_1\rangle+\beta_2|\psi_2\rangle|S_2\rangle$, $|\psi_1\rangle$ ($|\psi_2\rangle$) is the path state of an electron just passing through the slit 1 (slit 2) and $|S_1\rangle$ ($|S_2\rangle$) is the state of the matter of the double-slit corresponding to $|\psi_1\rangle$ ($|\psi_2\rangle$), where the normalizing factors $\alpha_1$, $\beta_1$, $\alpha_2$, $\beta_2$ express the difference of the two entangled states.  Then we can explain that the interference pattern changes with the distance from the double-slit to the screen. The two entangled states can be approximately expressed in one form $|\psi_1\rangle|S_1\rangle+|\psi_2\rangle|S_2\rangle$ after a suitable evolution time, \textit{i.e.}, the time of an electron from the double-slit to a screen, which is placed such that interference patterns can be observed. Since the states $|S_1\rangle$ and $|S_2\rangle$  are approximately the same within the evolution time, we can obtain an approximately superposed state $|\psi_1\rangle+|\psi_2\rangle$ of the electron due to the INT. This may also be an understanding of their experiment results and calculations of the simplest double-slit  (two protons) \cite{D. Akoury}, in which photoionization by circularly polarized light launches a coherent photoelectron at each nucleus of the molecule $H_{2}$, the Coulomb interactions between an electron, two protons and another electron are considered.

The phenomenon of particles tunneling through a barrier region with some probability is considered as a pure quantum effect and no one can understand it. According to our modification, when a particle enters the barrier region, the particle must interact with some LO and the energy of the potential barrier belongs to the CS, and then the system evolves into an entangled state. We use $|a_2\rangle$ expressing the energy eigenstate of the particle in the barrier region, where the energy may be greater than that of the barrier due to INT (it may get some energy from the LO), and use $|a_1\rangle$ and $|a_3\rangle$ expressing two energy eigenstates before and behind the region, respectively. The corresponding states of the LO are approximately unchanged and so the entangled state may be approximately expressed as $(c_1|a_1\rangle+c_2|a_2\rangle+c_3|a_3\rangle)|LO\rangle$. Therefore, the particle may be found in the three regions with probabilities $|c_1|^{2}$, $|c_2|^{2}$ and $|c_3|^{2}$.

The considerations of INT potential energy belonging to a particle sometimes bring an absurd understanding. In the case of describing a particle by the Schr\"{o}dinger equation, the potential energy in Hamiltonian terms is often considered as belonging to the particle, so the energy of the particle includes kinetic and potential barrier energy. If a particle enters the barrier region, it seems that its energy can become greater than its initial kinetic energy due to the potential barrier energy belonging to it, and then it can naturally jump the barrier with a large probability.

One coherent state \cite{ R. J. Glauber} of optical field $|\alpha\rangle$ ($\alpha$ is an arbitrary complex number) may be expanded in terms of photon number states $|n\rangle$ as $|\alpha\rangle=\Sigma f(\alpha,n) |n\rangle$ ($n=0,1,2,\cdots,\infty$), where $f(\alpha,n)$ are coefficients. This state can be explained as an approximately superposed state of number states, since the states of the light source apparatus, interacting with the field, can be considered as approximately unchanged. If the INT ceases, the coherent state will become a mixed state of number states.

In Ref.\cite{C. Monroe} C. Monroe \textit{et al}. prepared the \emph{Schr\"{o}dinger's cat}\cite{J. A. Wheeler} superposition state of an atom, $|x_{1}\rangle|\uparrow\rangle+|x_{2}\rangle|\downarrow\rangle$, an entangled state of its external space states and its internal energy states. It is actually the entangled state, $(|x_{1}\rangle|\uparrow\rangle|LO_1\rangle+|x_{2}\rangle|\downarrow\rangle|LO_2\rangle\approx(|x_{1}\rangle|\uparrow\rangle+|x_{2}\rangle|\downarrow\rangle)|LO_1\rangle$, of the CS of the atom and an LO, which is Raman beam and its state is approximately unchanged.

\section{Contradictions and mediations}

The following two points of measurement contradict each other. Landau \textit{et al}. \cite{L. D. Landau} thought that: \textquotedblleft After the measurement, however, the electron is in a state different from its initial one, and in this state the quantity \emph{f} does not in general take any definite value. Hence, on carrying out a second measurement on the electron immediately after the first, we should obtain for \emph{f} a value which did not agree with that obtained from the first measurement." This contradicts the point made by Dirac \cite{P. A. M. Dirac} that: \textquotedblleft From physical continuity, if we make a second measurement of the same dynamical variable $\xi$ immediately after the first, the result of the second measurement must be the same as that of the first. Hence after the first measurement has been made, the system is in an eigenstate of the dynamical variable, the eigenvalue it belongs to being equal to the result of the first measurement."

According to our modification, the eigenvalue in an eigenstate of spin of a particle or polarization of a photon can be indirectly measured, and the eigenvalue and the eigenstate are not changed after measurement, so the result of the second measurement must be the same as that of the first, while the value of at least one other quantity must be changed. This is close to Dirac's point. On the other hand, the directly measured quantities of a particle, such as energy, momentum and angular momentum, \textit{etc}., only measure the difference between an eigenvalue after the measurement and one of the eigenvalues of a superposed state before the measurement. On carrying out a second measurement on the same quantity immediately after the first, the two results will be different and the two states after two measurements will also be different. This nears the point of Landau \textit{et al}., but these eigenvalues themselves are not observables, which is different from Landau's point. So we partly mediate the contradiction between them.

In QM, a free atom can be in a superposed state $|g\rangle+|e\rangle$ of its ground state $|g\rangle$ and an excited state $|e\rangle$. A free electron (or one photon) can be in a superposed spin state $|s\rangle=|0\rangle+|1\rangle$ of its spin up (vertical polarized or path 0) state $|0\rangle$ and spin down (horizontal polarized or path 1) state $|1\rangle$. If the above states are expressed in the form $|live cat\rangle+|dead cat\rangle$, the cat should be even stranger than Schr\"{o}dinger's cat. When we discover that Schr\"{o}dinger's cat is dead, the reason must be a toxicant killing it, but we cannot find any matter that might have killed the cat.

According to our modification, when we discover the cat is dead, the reason must be that an INT with an LO killed it. There only exists an atom being in an approximately superposed state $|g\rangle+|e\rangle$ due to INT with an LO or apparatus. When an electron in some state $|s\rangle$ interacts with the non-uniform magnetic field $|SG\rangle$ of a Stern-Gerlach apparatus, the CS evolves into an entangled state $|s\rangle|SG\rangle\rightarrow|0\rangle|SG_1\rangle+|1\rangle|SG_2\rangle\approx(|0\rangle+|1\rangle)|SG_1\rangle$, where  $|SG_1\rangle$ and $|SG_2\rangle$ are states of the magnetic field after switching their interaction and are approximately the same, therefore the electron is in the approximately superposed state $|0\rangle+|1\rangle)$. A free electron in some state $|s\rangle$ must interact with an LO or apparatus, then the state $|s\rangle$ may be dissolved into the superposed state $|0\rangle+|1\rangle)$ in physics, though it is easily dissolved into a superposed state mathematically. A photon polarized state or a photon path state is similar. When the photon interacts with a beam splitter, its state may be dissolved into a superposed state of different polarized states or path states.

In QM, measuring the energy of a free atom in state $|g\rangle+|e\rangle$ brings a collapsed state $|g\rangle$ or $|e\rangle$. Assume that some energy change introduced by an apparatus is much smaller than the difference $\Delta E$ of the two atomic levels; then the energy conservation does not strictly hold in the CS of the atom and the apparatus. A lifetime of the free atom state $|g\rangle+|e\rangle$ is $\Delta t$ due to $\Delta E\Delta t\sim\hbar/2$, and the state spontaneously collapses into state $|g\rangle$ or $|e\rangle$, so the atomic energy is also non-conservation. The non-conservation is reluctantly explained as only existing in the short time $\Delta t$, whereas if we explain this by selecting the relation $\Delta E\Delta t>\hbar/2$, then big energy non-conservation may exist for a long time, which is not acceptable. If the short lifetime $\Delta t$ is acceptable, it contradicts the principle that the evolution time of a free atom state   has no limit according to Schr\"{o}dinger's equation.

We explain that, a free atom is not in state $|g\rangle+|e\rangle$. Before and after a measurement of the atomic energy (initially being in the approximately superposed state $|g\rangle+|e\rangle$), the energy conservation strictly holds considering the interchanges of energies among the atom, apparatus and an LO. When we discuss the evolution time of an atomic state $|g\rangle+|e\rangle$, it is actually that of a CS of the atom and an LO. This evolution time has no limit according to the Schr\"{o}dinger equation or even according to the relation $\Delta E\Delta t\sim\hbar/2$ due to $\Delta E=0$, \textit{i.e}., the energy conservation still holds strictly. If an INT quantum (or photon) escapes, or (and) the state of an LO is evidently changed, the evolution of $|g\rangle+|e\rangle$ ends, and the system of the atom and a new LO start a new evolution.

It will produce another contradiction when we combine Dirac's point and the uncertainty relation given by Landau \emph{et al.} \cite{L. D. Landau}, that $\Delta E$ is the difference of the two measured energy eigenvalues and $\Delta t$ is the interval of the two measurements. If the energy of an atom is measured twice in a finite interval $\Delta t$ and the same energy is obtained according to Dirac's point, then the difference  $\Delta E=0\rightarrow\Delta E\Delta t=0$, and this contradicts the Heisenberg uncertainty relation.

According to our modified uncertainty relation $\Delta E\Delta t\geq h$, $\Delta E$ is the difference of two different energy eigenvalues and $\Delta t$ is the time in one measurement (not the time interval of two measurements), and this then avoids the relation $\Delta E\Delta t=0$.

In QM, after two identical particles separate distantly and the INT between them ceases, the entanglement of the initial state $|e_1\rangle|e_2\rangle+|e_2\rangle|e_1\rangle$ of the energy eigenstate still persists. This contradicts the idea that the energies of two particles should be exchanged with each other by INT. We can understand the state $|g\rangle|1\rangle+|e\rangle|0\rangle$, when $n=0$  in  (1), but we cannot understand the state $|g\rangle|0\rangle+|e\rangle|1\rangle$ in QM, due to not knowing how to exchange energy between the atom and the field.

We explain that, if there exists an LO interacting with the two particles or the atom and the field above, and the states of that LO are approximately unchanged, then we can obtain $(|e_1\rangle|e_2\rangle+|e_2\rangle|e_1\rangle)|LO\rangle$ or $(|g\rangle|0\rangle+|e\rangle|1\rangle)|LO\rangle$. Otherwise, if there is no LO interacting with the two particles and the INT between them ceases, the entangled state disentangles into a mixed state $|e_1\rangle|e_2\rangle\langle e_1|\langle e_2|+|e_2\rangle|e_1\rangle\langle e_2|\langle e_1|$.

\section{Review of the argument between Einstein and Bohr}

There are two focal points of the argument between Einstein \cite{A. Einstein} and Bohr \cite{Bohr}. The first is the physical reality presented by Einstein \emph{et al}.: \textquotedblleft If, without in any way disturbing a system, we can predict with certainty the value (\emph{i.e.}, with probability equal to unity) of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."  \textquoteleft Without in any way disturbing a system' means no INT with apparatus. However, Bohr refuted the physical reality with INT between a measured particle and a measuring apparatus, and he considered the momentum exchanged between the particle and the separate parts of the apparatus. We consider that if a system state is unchanged, the measuring apparatus state should be unchanged too, and that this case also corresponds to no measurement or no measured system existing, so that no result or information about the system can be read. So we think that Bohr won the argument on this point based on INT.

The second is the physical locality presented also by Einstein \emph{et al}.\cite{A. Einstein}. They considered that two systems become an entangled state from an initial product state due to interact. They thought: \textquotedblleft At the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system." They emphasized that INT decides the coherence of the two systems, \emph{i.e}., entanglement. But Bohr refuted the physical locality with his famous complementarity \cite{J. A. Wheeler}, which is unrelated to any INT. When his complementarity was used for a particle, it seemed to explain why a particle has wave-particle duality, and the retention of the entanglement after INT ceased (non-local property) for a CS. However, we have explained that the wave property of a particle is determined by INT, not by any other principle, and some entangled states are also kept by INT that exchanges energy, momentum and angular momentum. As far as I know, whether an entangled state is local or non-local has not been completely proved experimentally, since the detection loophole \cite{E. Togan} and the lightcone loophole \cite{M. A. Rowe}, \emph{i.e}., the low detection efficiency of photons and the lack of a spacelike separation associated with measurement, have not been closed at the same time. The experiment of the ghost diffraction \cite{D. V. Strekalov} from entangled photons may be considered a proof of the existence of quantum non-locality, but the ghost diffraction can also be observed from un-entangled photons \cite{J. Hua}. So we think that the physical locality related to INT is right, and that Einstein therefore won the argument on the second point.

As for the famous words of Einstein that God does not play dice, we think that the probability issue is decided in principle by INT, the Schr\"{o}dinger equation, and boundary and initial conditions.

\section{Conclusion}

In classical mechanics, the external force or the potential energy in a Lagrangian or Hamiltonian may cause one to forget that it comes from INT between a considered macro-particle and another object. In QM, we often concentrate our attention on a considered particle and its potential energy, and then sometimes forget other objects and the potential energy being the INT energy belonging to the CS. If we consider those once neglected INTs between micro-systems and some macro-objects or environments, especially the latter states, and consider Newton's idea on measurement, we may modify the theory of QM, understand some quantum phenomena, and mediate some contradictions logically and with internal consistency. These may partly satisfy our logic and intuition, and then may enable us to perfect the theory.  We also review the arguments between Einstein and Bohr on physical reality and physical locality, and think that the reality is wrong while the locality is right. In order to solve issues of QM, two important things may be to find unknown INTs in practical CSs and even to find the fifth fundamental INT, rather than finding hidden variables.
\vspace{-0.2cm}
\begin{acknowledgments}
\vspace{-0.2cm}
 I thank Shou-Yong Pei, Jian Zou, Bin Shao, Feng Wang, Xiu-San Xing, Jun-Gang Li, Xiang-Dong Zhang, Yu-Gui Yao, Jin-Fang Cai, Rui Wang, Chang-Hong Lu, Wen-Yong Su and Fan Yang for discussions and comments, Jin-Song Miao for translations of early German literature, Li-Fan Ying, Gui-Qin Li, Hao Wei and Yong-Jun Lu for help, and Pei-Zhu Ding and Shou-Fu Pan for encouragement. The work was partly supported by the fundamental research fund of Beijing Institute of Technology.
\end{acknowledgments}
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\end{thebibliography}
\end{document}
%
% ****** End of file apssamp.tex ******

1307.1851.extracted_bib
\begin{thebibliography}{99}
\bibitem{D. Bohm} D. Bohm, \textit{Quantum theory}, (Prentice-Hall Inc., New York, 1951).

\bibitem{R. P. Feynman} R. P. Feynman, \textit{The Feynman Lectures on Physics, Vol. I}, (Pearson Education Asia Limited and Beijing World Publishing Corporation, 2004).

\bibitem{J. A. Wheeler} J. A. Wheeler and W. H. Zurek, \textit{Quantum Theory and Measurement}, (Princeton University Press, Princeton, 1983)

\bibitem{J. S. Tang} J. S. Tang, Y. L. Li, X. Y. Xu, G. Y. Xiang, C. F. Li, and G. C. Guo, Nphoton \textbf{6}, 600 (2012).

\bibitem{D. F. Walls}D. F. Walls and G. J. Milburn, \textit{Quantum Optics}, (Springer-Verlag, Berlin Heidelberg, 1994).

\bibitem{P. Bertet} P. Bertet, S. Osnaghi, A. Rauschenbeutel, G. Nogues, A. Auffeves,
M. Brune, J. M. Raimond, and S. Haroche, Nature \textbf{411}, 166 (2001).

\bibitem{W. H. Zurek} W. H. Zurek, Rev. Mod. Phys. \textbf{75}, 715 (2003).

\bibitem{T. H. Zeng}T. H. Zeng, arXiv:1008.1691.

\bibitem{Zeng}T. H. Zeng, www.paper.edu.cn:2009.05.06.

\bibitem{N. Bohr} N. Bohr, H. A. Kramers, and J. C. Slater, Z. f. Physik A \textbf{32}, 69 (1924).

\bibitem{W. Bothe} W. Bothe and H. Geiger, Z. f. Physik A \textbf{32}, 639 (1925).

\bibitem{Bothe} W. Bothe and H. Geiger, Die Naturwissenschaften, \textbf{13}, 440 (1925).

\bibitem{J. Chadwick} J. Chadwick, Nature \textbf{29}, 312 (1932).

\bibitem{V. B. Braginsky}V. B. Braginsky and F. Y. Khalili, \textit{Quantum Measurement}, (Cambridge University Press, Cambridge, 1992).

\bibitem{Newton} I. Newton, \textit{The Principia: mathematical principles of natural philosophy}, (California University Press, California, 1999).

\bibitem{L. D. Landau}L. D. Landau and E. M. Lifshitz, \textit{Quantum Mechanics}, (Pergamon Press Ltd., Oxford, 1977).

\bibitem{D. Akoury} D. Akoury, K. Kreidi,  T. Jahnke,  Th. Weber,  A. Staudte,  M. Sch\"{o}ffler,
N. Neumann, J. Titze,  L. Ph. H. Schmidt,  A. Czasch,  O. Jagutzki,  R. A. Costa Fraga,
R. E. Grisenti,  R. D\'{\i}ez Mui\~{n}o,  N. A. Cherepkov,  S. K. Semenov,  P. Ranitovic,
C. L. Cocke,  T. Osipov,  H. Adaniya,  J. C. Thompson,  M. H. Prior,  A. Belkacem,
A. L. Landers,  H. Schmidt-B\"{o}cking, and R. D\"{o}rner, Science \textbf{318}, 949 (2007).

\bibitem{R. J. Glauber} R. J. Glauber, Phys. Rev. \textbf{131}, 2766 (1963).

\bibitem{C. Monroe} C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, Science \textbf{272}, 1131 (1996).

\bibitem{P. A. M. Dirac} P. A. M. Dirac, \textit{The Principle of Quantum Mechanics}, (Science Press, Beijing, 2008).

\bibitem{A. Einstein} A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. \textbf{47}, 777 (1935).

\bibitem{Bohr} N. Bohr, Phys. Rev. \textbf{48}, 696 (1935).

\bibitem{E. Togan} E. Togan,  Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. S{\o}rensen,
P. R. Hemmer, A. S. Zibrov, and M. D. Lukin
, Nature \textbf{466}, 730 (2010).

\bibitem{M. A. Rowe} M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano,
C. Monroe, and D. J. Wineland, Nature \textbf{409}, 791 (2001).

\bibitem{D. V. Strekalov} D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, , Phys. Rev. Lett. \textbf{74}, 3600 (1995).

\bibitem{J. Hua}  J. Hua and R. P. Wang, Acta Sci. Nat. Univ. Pekinensis \textbf{42}, 334 (2006).
\end{thebibliography}