3D Quantum Trajectories. Quantum orbits of the Hydrogen's electron
\noindent In this paper, we introduce the 3D-Quantum Stationary Hamilton Jacobi Equation for a central potential, and establish the 3D quantum law of motion of an electron in the presence of such a potential. We establish a system of three differential equations from which, and as a numerical application, we plot the 3D quantum trajectories of Hydrogen's electron. We show also the existence of nodes for these trajectories.
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\begin {document}

\title {\bf 3D Quantum Trajectories. Quantum orbits of the Hydrogen's electron}
\author{ T.~Djama\footnote{Electronic address:
{\tt djam\_touf@yahoo.fr}}}

\date{April 1, 2004}

\maketitle
\begin{center}
Mail address:14, rue Si El Hou\`es, B{\'e}ja{\"\i}a, Algeria
\end{center}



\begin{abstract}
\noindent In this paper, we introduce the 3D-Quantum Stationary
Hamilton Jacobi Equation for a central potential, and establish
the 3D quantum law of motion of an electron in the presence of
such a potential. We establish a system of three differential
equations from which, and as a numerical application, we plot the
3D quantum trajectories of Hydrogen's electron. We show also the
existence of nodes for these trajectories.
\end{abstract}

\vskip\baselineskip

\noindent
PACS: 03.65.Bz; 03.65.Ca

\noindent Key words:  three dimension, quantum law of motion,
atomic potential, quantum trajectories, nodes.

\newpage


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip0.5\baselineskip
\noindent
{\bf 1.\ \ Introduction }
\vskip0.5\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Three years ago, we presented with Bouda a deterministic approach
of quantum mechanics by which we attempted to restore an objective
reality of quantum phenomena. This approach consist on the
introduction of a 1D quantum Lagrangian from which we derived the
1D quantum law of motion \cite {Dja1}. Then, we plotted the
quantum trajectories for several one dimensional systems \cite
{Dja2}. After, we generalized our approach to the relativistic
systems \cite {Dja3}, and obtained the relativistic quantum
trajectories and established the existence of nodes also for such
systems \cite {Dja4}. we also presented a study of the
relativistic spinning particle using the deterministic approach
that we developed in Refs. \cite {Dja1,Dja2,Dja3,Dja4}. Indeed, we
gave, first, the relativistic quantum stationary Hamilton-Jacobi
equation for a particle with a half (1/2) spin RQSHJES$_{{1 \over
2}}$ \cite {Dja5}, then, we established the Quantum law of motion
of such a system \cite {Dja6}.

\noindent  Our deterministic quantum approach is interesting,
however it was elaborated only for one dimensional systems. This
fact was a real handicap for studying the nature since the most
phenomena happens in real 3D space. For this reason, we presented
in Ref. \cite {Dja7}, a generalization of our approach to three
dimensional systems. First, we set the expression
%
%
\begin{equation}
S_0(\vec{r})=\hbar \arctan\left(a\ {\theta(\vec{r}) \over
\phi(\vec{r})}+b\right) \;
\end{equation}
%
%
of the quantum reduced action as a solution of the 3D-QSHJE
%
%
\begin{eqnarray}
{1 \over 2m_0}\left({\vec{\nabla}S_0(\vec r)}\right)^2-{\hbar^2
\over 2m_0}{\Delta R(\vec r) \over R(\vec r)}+(E-V(\vec r))=0\;
\; , \hskip15mm  (a)
\nonumber\\
\vec{\nabla} . \left(R^2(\vec r)\vec{\nabla}S_0\right)=0 \; \; .
\hskip35mm (b)
\end{eqnarray}
%
%
The expression of the function $R(\vec{r})$ is given as
%
%
\begin{equation}
R(\vec{r})=\sqrt{{k \over \hbar a}}\; \;
\left[(a\theta+b\phi)^2+\phi^2\right]^{1 \over 2}\; .
\end{equation}
%
%
Note that $\theta$ and $\phi$ are two real independent solutions
of the 3D Schr{\"o}dinger equation
%
%
\begin{equation}
-{\hbar^2 \over 2m_0}\Delta\Psi+\; V(\vec{r})\Psi=E\;\Psi .
\end{equation}
%
%
$a$, $b$ and $k$ are real constants. After the establishment of
the 3D quantum reduced action Eq. (1), we introduced a
coordinates transformation with which the 3D-QSHJE (Eq. (2)) takes
the form of the 3D Classical Hamilton-Jacobi equation (3D-CSHJE)
\cite {Dja7} exactly as what Faraggi and Matone done for the
1D-QSHJE \cite {FM1}. Then, we introduced the 3D quantum
Lagragian from which we derived the relation between the
conjugate momentum and the velocity of the particle expressed as
\cite {Dja7}
%
%
\begin{equation}
\vec{v}\; . \; \vec{\nabla S_0}=2\left[E-V(\vec{r})\right]\; .
\end{equation}
%
%
We consider this last equation as the Quantum law of 3D motions.
It is because of -as we will see later- the fact that we should
use this equation in order to plot the 3D quantum trajectories.
Also, it can be seen as a generalization into 3D of the 1D
quantum law of motion already established in Ref. \cite {Dja1}
%
%
\begin{equation}
\dot{x}{dS_0 \over dx}=2\; (E-V(x)) \; .
\end{equation}
%
%
The elaboration of the 3D quantum law of motion (Eq. (5)) for a
general form of the potential is an important step to investigate
the 3D motions. However, we should plot the 3D quantum
trajectories when it is possible in order to understand the
behaviour of the quantum particle in 3D spaces. With this aim, we
should act exactly as we have done for 1D quantum and
relativistic quantum trajectories \cite {Dja2,Dja4}. Indeed, we
must first resolve the Schr{\"o}dinger equation (Eq. (4)) to find
the functions $\theta$ and $\phi$, which we replace into the
expression of the reduced action (Eq. (1)). Then, we replace the
expression (2) into Eq. (5). After resolving Eq. (5), we can plot
the quantum trajectories. Note that for a general potential,
plotting the 3D quantum trajectories is not evident. This is why,
in this article, we study the case of the central potential, and
plot the trajectories of an electron moving under the atomic
potential.

The paper is organized as follows: In Sec. 2, we introduce the
3D-QSHJE and its solutions for the central potential. Then, in
Sec. 3, we establish the quantum law of motion for this potential
and elaborate a system of three differential equations which
describe the dynamical behaviour of the quantum particles. In
Sec. 4, we plot the 3D quantum trajectories of the Hydrogen's
electron, for several bound states. Finally, In sec. 5, we
present a conclusion in which we discuss our results.

\noindent Some calculation and demonstrations are given in the
and of the article as appendix.



% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip\baselineskip \noindent {\bf 2.\ \ The 3D-QSHJE for the
central potential} \vskip0.5\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The central potential is a potential for which only the radial
position determine its value
$$
V(\vec{r})=V(r)\; .
$$
In order to construct the 3D-QSHJE for such a potential, we write
the \hbox{Schr{\"o}-\hskip-2pt plus 2pt}\break dinger equation
(Eq. (4)) with the spherical coordinates $(r,\vartheta,\varphi)$
\cite {Coh1}
%
%
\begin{equation}
{\partial ^2 \Psi \over \partial r^2}+ {2 \over r}{\partial  \Psi
\over \partial r}+ {1 \over r^2}{\partial ^2 \Psi \over \partial
\vartheta^2} +{\cot \vartheta \over r^2}\,{\partial  \Psi \over
\partial \vartheta}+ {1 \over r^2 \sin^{2} \vartheta }\,{\partial
^2 \Psi \over \partial \varphi^2}+ {2m_0 \over \hbar^2}[E-V(r)]
\Psi=0 \ .
\end{equation}
%
%
$\Psi(r,\vartheta,\varphi)$ is the wave function of the quantum
system taken into consideration. To resolve Eq. (7), the variables
must be separated into $\Psi(r,\vartheta,\varphi)$ as \cite {Coh1}
%
%
$$
\Psi (r\ ,\vartheta\ ,\varphi)=R(r)\, T(\vartheta) F(\varphi)\ ,
$$
%
%
where $R(r)$, $T(\vartheta)$ and $F(\varphi)$ are real functions
of the variables $r$, $\vartheta$ and $\varphi$ respectively.
$R(r)$ is the radial wave function, $T(\vartheta )$ and
$F(\varphi )$ are the angular wave functions. It follows that Eq.
(7) will be separated into three equations, each corresponds to
one variable,
%
%
\begin{equation}
{\frac{r^{2}}{R}}{\frac{d^{2}R}{dr^{2}}}+{\frac{2r}{R}}{\frac{dR}{dr}}+{%
\frac{2m_0\, r^{2}}{\hbar ^{2}}}(E-V(r))=\lambda \;,
\end{equation}
%
%
\begin{equation}
{\frac{d^{2}T}{d\vartheta ^{2}}}+\cot \vartheta {\frac{dT}{d\vartheta }}%
+\left( \lambda -{\frac{m^{2}_{l}}{\sin ^{2}\vartheta
}}\right) T=0\;,
\end{equation}
%
%
\begin{equation}
{\frac{d^{2}F}{d\varphi ^{2}}}+m^2_{l}\ F=0\; .
\end{equation}
%
%
$\lambda $ is a constant such as
%
%
$$
\lambda =l\, (l+1),
$$
%
%
$l$ being a positive integer or vanish, and $m_{l}$ is
an integer satisfying to
%
%
$$
-l \leq m_{l}\leq {l}\;.
$$
%
%
$l(l +1)$ is the eingen value of $L^{2}$, $L$ representing
the angular momentum operator. $m_{l}$ represents the eingen
value of $L_{z}$, the operator projection of $L$ along the $z$
axis.

Now, let us write Eqs. (8), (9) and (10) in a form analogous to
the one dimensional Schr\"{o}dinger equation. First, we take up
Eq. (8) and write
%
%
\begin{equation}
R(r)={\frac{\mathcal{X}(r)}{r}}\;,
\end{equation}
%
%
where $\mathcal{X}(r)$ is a function of the radius. Then by
taking twice the derivative of Eq. (11) with respect to $r$, we
obtain
%
%
\begin{equation}
{\frac{dR}{dr}}={\frac{1}{r}}{\frac{d\mathcal{X}}{dr}}-{\frac{\mathcal{X}}{%
r^{2}}}\ ,
\end{equation}
%
%
\begin{equation}
{\frac{d^{2}R}{dr^{2}}}={\frac{1}{r}}{\frac{d^{2}\mathcal{X}}{dr^{2}}}-2{%
\frac{d\mathcal{X}/dr}{r^{2}}}+2{\frac{\mathcal{X}}{r^{3}}}\ .
\end{equation}
%
%
substituting Eqs. (12) and (13) into Eq. (8), we get
%
%
\begin{equation}
{\frac{-\hbar ^{2}}{2m_0}}{\frac{d^{2}\mathcal{X}}{dr^{2}}}+\left[ V(r)+{\frac{%
\lambda \hbar ^{2}}{2m_0r^{2}}}\right] \mathcal{X}=E\mathcal{X}\ .
\end{equation}
%
%
This last equation has the same form as the one dimensional
Schr\"{o}dinger equation for $\mathcal{X}(r)$ with an energy $E$
and a fictitious potential
%
%
\begin{equation}
V^{^{\prime }}(r)=V(r)+{\frac{\lambda \hbar ^{2}}{2m_0r^{2}}}\ .
\end{equation}
%
%
Likewise, Eq. (9) can be reduced to the form of the 1D
Schr\"{o}dinger equation. This can be done by introducing the
function $\mathcal{T}(\vartheta )$ defined as
%
%
\begin{equation}
\mathcal{T}(\vartheta )=\sin ^{\frac{1}{2}}\vartheta T(\vartheta
)\ .
\end{equation}
%
%
Taking twice the derivative of this last equation with respect to
$\vartheta $, we deduce
%
%
\begin{equation}
{\frac{dT}{d\vartheta }}={\frac{1}{\sin ^{\frac{1}{2}}\vartheta }}{\frac{d%
\mathcal{T}}{d\vartheta }}-{\frac{1}{2}}{\frac{\cos \vartheta }{\sin ^{\frac{%
3}{2}}\vartheta }}\mathcal{T}\ ,
\end{equation}
%
%
\begin{equation}
{\frac{d^{2}T}{d\vartheta ^{2}}}={\frac{1}{\sin ^{\frac{1}{2}}\vartheta }}{%
\frac{d^{2}\mathcal{T}}{d\vartheta ^{2}}}-{\frac{\cos \vartheta }{\sin ^{%
\frac{3}{2}}\vartheta }}{\frac{d\mathcal{T}}{d\vartheta }}+{\frac{3}{4}}{%
\frac{\cos ^{2}\vartheta }{\sin ^{\frac{3}{2}}\vartheta }}+{\frac{1}{2}}{%
\frac{\mathcal{T}}{\sin ^{\frac{1}{2}}\vartheta }}\ .
\end{equation}
%
%
If we replace Eqs. (17) and (18) into Eq. (19) we obtain
%
%
\begin{equation}
{\frac{d^{2}\mathcal{T}}{d\vartheta ^{2}}}+(\lambda +{\frac{1}{4}})\mathcal{T%
}+{\frac{(1/4-m_{l}^{2})}{\sin ^{2}\vartheta }}\mathcal{T}=0\
.
\end{equation}
%
%
Remark that Eq. (19) has the form of the one dimensional
Schr\"{o}dinger equation with a fictitious potential
%
%
\begin{equation}
V(\vartheta )={\frac{\hbar ^{2}}{2m_0}}{\frac{(m_{l}^{2}-1/4)}{\sin ^{2}\vartheta }}\ ,
\end{equation}
%
%
and an energy
%
%
\begin{equation}
E_{\vartheta }=(\lambda +{\frac{1}{4}}){\frac{\hbar ^{2}}{2m_0}}\ .
\end{equation}
%
%
Remark also that Eq. (10) has the same form as the 1D
Schr\"{o}dinger equation with a vanishing potential and an energy
equal to $m_{l}^{2}\hbar ^{2}/2m$.

\noindent Because Eqs.(8), (9) and (10) come to the form of the 1D
Schr\"{o}dinger equation. So, in order to obtain the 3D-QSHJE, let
us write the functions $\mathcal{X}(r)$, $\mathcal{T}(\vartheta)$
and $F(\varphi )$ as
%
%
\begin{equation}
\mathcal{X}(r)=A(r)\left[ \alpha e^{{\frac{i}{\hbar }}Z(r)}+\beta
e^{-{\frac{i}{\hbar }}Z(r)}\right] \ ,
\end{equation}
%
%
\begin{equation}
\mathcal{T}(\vartheta )=\xi (\vartheta )\left[ \gamma
e^{{\frac{i}{\hbar} }L(\vartheta )}+\varepsilon
e^{-{\frac{i}{\hbar }}L(\vartheta )}\right] \ ,
\end{equation}
%
%
\begin{equation}
F(\varphi )=\eta (\varphi )\left[ \sigma e^{{\frac{i}{\hbar
}}M(\varphi )}+\omega e^{-{\frac{i}{\hbar }}M(\varphi )}\right] \
.
\end{equation}
%
%
By replacing Eqs. (22), (23) and (24) respectively in Eqs. (8),
(9) and (10), we get
%
%
\begin{equation}
{\frac{1}{2m_0}}\left( {\frac{dZ}{dr}}\right) ^{2}-{\frac{\hbar ^{2}}{2m_0}}{%
\frac{1}{A}}{\frac{d^{2}A}{dr^{2}}}-+V(r)+{\frac{\lambda \hbar ^{2}}{%
2m_0\;r^{2}}}=E\ ,
\end{equation}
%
%
\begin{equation}
{\frac{d}{dr}}\left( A^{2}\ {\frac{dZ}{dr}}\right)=0 \ ,
\end{equation}
%
%
\begin{equation}
\left( {\frac{dL}{d\vartheta }}\right) ^{2}-{\frac{1}{\xi
}}{\frac{d^{2}\xi }{d\vartheta ^{2}}}-(\lambda
+{\frac{1}{4}})\hbar ^{2}-{\frac{(1/4-m_{l}^{2})}{\sin
^{2}\vartheta }}\hbar ^{2}=(\lambda +{\frac{1}{4}})\hbar ^{2}\ ,
\end{equation}
%
%
\begin{equation}
{\frac{d}{d\vartheta }}\left( \xi ^{2}\ {\frac{dL}{d\vartheta
}}\right)=0 \;,
\end{equation}
%
%
\begin{equation}
\left( {\frac{dM}{d\varphi }}\right) ^{2}-{\frac{\hbar ^{2}}{\eta (\varphi )}%
}{\frac{d^{2}\eta }{d\varphi ^{2}}}-m_{l}^{2}\hbar ^{2}=0\;,
\end{equation}
%
%
\begin{equation}
{\frac{d}{d\varphi }}\left( \eta ^{2}\ {\frac{dM}{d\varphi
}}\right)=0 \;.
\end{equation}
%
%
Replacing Eq. (26) into Eq. (25), Eq. (28) into Eq. (27) and Eq.
(30) into Eq. (29), we find
%
%
\begin{equation}
{\frac{1}{2m_0}}\left( {\frac{dZ}{dr}}\right) ^{2}-{\frac{\hbar ^{2}}{4m_0}}%
\left\{ Z,r\right\} +V(r)+{\frac{\lambda \hbar ^{2}}{2m_0r^{2}}}=E\
,
\end{equation}
%
%
\begin{equation}
\left( {\frac{dL}{d\vartheta }}\right) ^{2}-{\frac{\hbar
^{2}}{2}}\left\{ L,\vartheta \right\} +{\frac{(m_{l}^{2}-1/4)}{\sin ^{2}\vartheta }}\hbar ^{2}=(\lambda
+{\frac{1}{4}})\hbar ^{2}\;,
\end{equation}
%
%
\begin{equation}
\left( {\frac{dM}{d\varphi }}\right) ^{2}-{\frac{\hbar
^{2}}{2}}\left\{ M,\varphi \right\} =m_{l}^{2}\hbar ^{2}\ .
\end{equation}
%
%
Eqs. (31), (32) and (33) represent the components of the QSHJE in
3-D for a central potential. These equations contain the
Schwarzian derivatives of the functions $Z$, $L$ and $M$ with
respect to $r$, $\vartheta$ and $\varphi$ respectively. Note
that, by deducing the value of $\lambda $ from (31) and the value
of $m_{l}$ from (33), then substituting in Eq. (32), one
obtain
%
%
\begin{eqnarray}
{\frac{1}{2m_0}}\left[ \left( {\frac{dZ}{dr}}\right) ^{2}+{\frac{1}{r^{2}}}%
\left( {\frac{dL}{d\vartheta }}\right) ^{2}+{\frac{1}{r^{2}\sin
^{2}\vartheta }}\left( {\frac{dM}{d\varphi }}\right) ^{2}\right]
-\hskip5mm
&&  \nonumber \\
{\frac{\hbar ^{2}}{4m_0}}\left[ \left\{ Z,r\right\}
+{\frac{1}{r^{2}}}\left\{ L,\vartheta \right\}
+{\frac{1}{r^{2}\sin ^{2}\vartheta }}\left\{ M,\varphi
\right\} \right] +\hskip5mm &&  \nonumber \\
V(r)-{\frac{\hbar ^{2}}{8m_0\ r^{2}}}-{\frac{\hbar ^{2}}{8m_0\
r^{2}\sin ^{2}\vartheta }} &=&E\ .
\end{eqnarray}
%
%
Now, considering the functions $Z$, $L$ and $M$ as the reduced
actions corresponding to the directions of the variables $r$,
$\vartheta$ and $\varphi$, we can define a total reduced action
of our system by summing the three reduced actions \footnote{
Which is also valid for the classical case.}
%
%
\begin{eqnarray}
S_{0}(r,\vartheta ,\varphi )=Z(r)+L(\vartheta )+M(\vartheta)\ .
\end{eqnarray}
%
%
Taking up this equation in Eq. (34), we find
%
%
\begin{eqnarray}
{\frac{1}{2m_0}}\left( \vec{\nabla}_{r,\vartheta ,\varphi }S_{0}\right) ^{2}-{%
\frac{\hbar ^{2}}{4m_0}}\left[ \left\{ S_{0},r\right\}
+{\frac{1}{r^{2}}}\left\{ S_{0},\vartheta \right\} +
\right.
\hskip5mm &&
\nonumber \\
\left. {\frac{1}{%
r^{2}\sin ^{2}\vartheta }}\left\{ S_{0},\varphi \right\} \right] +V(r)-{%
\frac{\hbar ^{2}}{8m_0\ r^{2}}}-{\frac{\hbar ^{2}}{8m_0\ r^{2}\sin
^{2}\vartheta }}=E\; .
\end{eqnarray}
%
%
Eq. (35) represents the QSHJE in 3-D for a central potential. At
the classical limit $(\hbar \to 0)$, Eq. (35) goes to
%
%
\begin{equation}
{\frac{1}{2m}}\left( \vec{\nabla}_{r,\vartheta ,\varphi
}S_{0}\right) ^{2}+V(r)=E\ .
\end{equation}
%
%
which is the Classical Stationary Hamilton-Jacobi Equation
(CSHJE). Remark that taking the classical limit in Eqs. (31), (32)
and (33) make $dL/d\vartheta $ and $dM/d\varphi $ vanish, then,
one cannot obtain the CSHJE (Eq.(37)). We deduce that the
classical limit must be taken into Eq. (36).

\noindent Note also that considering the 3D QSHJE Eq. (36), and
separating variables with taking into account Eq. (35), we get to
Eqs. (31), (32) and (33) which lead to Eqs. (8), (9) and (10).
After separating variables in Eq. (36), three constants of motion
appear, they are the energy $E$, $\lambda $ and $m_{l}$.

\vskip0.5\baselineskip The solutions of Eqs. (31), (32) and (33)
are respectively
%
%
\begin{equation}
Z(r)=\hbar \arctan \left\{ \alpha_r {{\cal X}_2 \over {\cal
X}_1}+b_r\right\} \;,
\end{equation}
%
%
\begin{equation}
L(\vartheta )=\hbar \arctan \left\{ \alpha_\vartheta {{\cal T}_2
\over {\cal T}_1}+b_\vartheta\right\} \;,
\end{equation}
%
%
\begin{equation}
M(\varphi )=\hbar \arctan \left\{  \alpha_\varphi {F_2 \over
F_1}+b_\varphi \right\} \;,
\end{equation}
%
%
Where $\alpha_{r}$, $b _{r}$, $\alpha _{\vartheta }$, $b
_{\vartheta }$, $\alpha_{\varphi }$ and $a _{\varphi }$ are real
constants. ${\cal X}_{1}$ and ${\cal X}_{2}$ are two real
independent solutions of Eq. (17). ${\cal T}_{1}$ and ${\cal
T}_{2}$ are two real independent solutions of Eq. (18). $F_1$ and
$F_2$ are two real independent solutions of Eq. (19).

\vskip0.5\baselineskip Now, knowing that: " the wronskian
%
%
$$
\psi_1 {d\psi_2 \over dq}-\psi_2 {d\psi_1 \over dq}
$$
%
%
of two independent solutions of a second order differential
equation with respect to $q$ is a constant $k_q$", the expressions
(38), (39) and (40) reduce to
%
%
\begin{equation}
Z(r)=\hbar \arctan \left\{ a_r \int{dr \over {\cal
X}^2_1}+b_r\right\} \;,
\end{equation}
%
%
\begin{equation}
L(\vartheta)=\hbar \arctan \left\{ a_\vartheta \int{d\vartheta
\over {\cal T}^2_1}+b_\vartheta\right\} \;,
\end{equation}
%
%
\begin{equation}
M(\varphi)=\hbar \arctan \left\{ a_\varphi \int{d\varphi \over
F^2_1}+b_\varphi\right\} \; ,
\end{equation}
%
%
where $a_r=k_r\alpha_r$, $a_\vartheta=k_\vartheta
\alpha_\vartheta$ and $a_\varphi=k_\varphi \alpha_\varphi$ are
real constants. In Eqs. (41), (42) and (43), we need to determine
only one of the two independent solutions of Eqs. (8), (9) and
(10) respectively.

Before entering into the elaboration of the quantum law of
motion, we would like to exhibit the fact that for the central
potential, only the radial function ${\cal X}$ depend on the form
of the potential, otherwise the two angular functions ${\cal T}$
and $F$ can be established without considering of the form of the
central potential. Indeed, if we take a look on the table of
solutions of the Schr\"odinger equation for such a potential we
should write the angular function ${\cal T}_1$ as \cite{Nik}
%
%
\begin{equation}
{\cal T}^{(lm_l)}_1(\vartheta)=(1-\cos^2(\vartheta))^{{|m_l| \over
2}+{1 \over 4}}\; {d^{l+|m_l|}\ \ \ \ \ \ \ \ \ \over
d(\cos(\vartheta))^{l+|m_l|}}\left[(1-\cos^2(\vartheta))^l\right]\;
,
\end{equation}
%
%
and the angular function $F_1$ as
%
%
\begin{eqnarray}
F_1(\varphi)=\sin(|m_l|\varphi) \hskip15mm if \hskip5mmm_l\neq 0
&& \nonumber \\   = 1 \hskip27mm if \hskip5mmm_l= 0\; .
\end{eqnarray}
%
%
It is usual to remark that we do not include the normalization
constants in the two last expressions of the angular functions,
since we consider the two mathematical solutions and not the
physical's.

%\newpage
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip\baselineskip \noindent {\bf 3.\ \ The Quantum law of motion
for a particle moving under the central potential}
\vskip0.5\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now let us investigate the quantum law of motion. In Ref.
\cite{Dja7}, we have presented a 3D formalism from which we
described the quantum law of 3D motions for a general potential.
We established a relation connecting between the 3D velocity and
the 3D quantum momentum (Eq. (5)). Using the spherical coordinates
into Eq. (5) for the central potential case, we write
%
%
\begin{equation}
\dot{r}{dZ \over dr}+\dot{\vartheta}{dL \over
d\vartheta}+\dot{\varphi}{dM \over d\varphi}=2\left[ E-V(r)
\right]
\end{equation}
%
%
As in classic mechanics, we do not use directly Eq. (46) to plot
the trajectories. We must first, separate the variables $r$,
$\vartheta$ and $\varphi$. This seems not possible from Eq. (46),
and must be done in the 3D QSHJE (Eq. (36)). We noted in Sec. 2
that the separation of variables in Eq. (36), after taking into
account of Eq. (35), leads directly to Eqs. (31), (32) and (33).

In Ref. \cite{Dja7}, we have considered a deformation of the
space like for general relativity. We treated the 3D quantum
systems with a general potential by using the mechanics of a
deformed space. This allowed us to introduce a quantum Lagrangian
and derive the quantum law of motion (Eq. (5)). For the central
potential, we proceed with the same manner. Thus, let us write
Eq. (36) as
%
%
\begin{eqnarray}
\left({dZ \over dr}\right)^{2}\; {1 \over a_{rr}(r)}\, +\; {1 \over r^2}\,
\left({dL \over d\vartheta}\right)^{2} {1 \over
a_{\vartheta\vartheta}(\vartheta)} \, \, +
\hskip40mm\nonumber\\
\, {1 \over r^2\, sin^2(\vartheta)}\, \left({dM \over
d\varphi}\right)^{2}\; {1 \over a_{\varphi\varphi}(\varphi)}\, =
2m_0[E-V(r)] \; ,
\end{eqnarray}
%
%
where
%
%
$$
a_{rr}(r)=\left[1-{\hbar^2 \over 2}\left({dZ \over
dr}\right)^{-2}\{Z,r\}\right]\; , \hskip15mm
$$
%
%
\begin{equation}
a_{\vartheta\vartheta}(\vartheta)=\left[1-{\hbar^2 \over
2}\left({dL \over d\vartheta}\right)^{-2}\left(\{L,\vartheta\}-{1
\over  2}\right)\right]\; , \hskip5mm
\end{equation}
%
and
%
$$
a_{\varphi\varphi}(\varphi)=\left[1-{\hbar^2 \over 2}\left({dM
\over d\varphi}\right)^{-2}\left(\{M,\varphi\}-{1 \over
2}\right)\right]\; ,
$$
%
%
are real functions depending on $r$, $\vartheta$ and $\varphi$
respectively. They represents the non vanishing components of the
metric tensor of the curved space created by the central
potential. The Lagrangian is given as
%
%
\begin{eqnarray}
L(\dot{r},\dot{\vartheta},\dot{\varphi},r,\vartheta,\varphi)={m_0{\dot{r}}^2
\over 2}\; a_{rr}(r)+\, {m_0\, r^2\, {\dot{\vartheta}}^2 \over 2}\;
a_{\vartheta\vartheta}(\vartheta)+\,
\hskip20mm\nonumber\\
{m_0\, r^2\, sin^2(\vartheta)\, {\dot{\varphi}}^2 \over 2}\;
a_{\varphi\varphi}(\varphi)-V(r)\; ,
\end{eqnarray}
%
%
and leads to the dynamical equation
%
%
\begin{equation}
{m_0{\dot{r}}^2 \over 2}\; a_{rr}(r)+{m_0\, r^2\, {\dot{\vartheta}}^2 \over
2}\; a_{\vartheta\vartheta}(\vartheta)+{m_0\, r^2\, sin^2(\vartheta)\,
{\dot{\varphi}}^2
\over 2}\; a_{\varphi\varphi}(\varphi)+V(r)=E \; .
\end{equation}
%
%
After comparison between Eqs. (47) and (50), we get
%
%
\begin{equation}
a_{rr}={dZ/dr \over m_0{\dot{r}}}\; ,
\end{equation}
%
%
\begin{equation}
a_{\vartheta\vartheta}={dL/d\vartheta \over
m_0r^2{\dot{\vartheta}}}\; ,
\end{equation}
%
%
\begin{equation}
a_{\varphi\varphi}={dM/d\varphi \over
m_0r^2\sin^2(\vartheta){\dot{\varphi}}}\; .
\end{equation}
%
%
Replacing these last equations into Eq. (47), we deduce directly
Eq. (46).

\noindent Now, replacing Eqs. (51), (52) and (53) into Eqs. (31),
(32) and (33) respectively after taking into account of the
relations (48), we find
%
%
\begin{equation}
{dZ \over dr}\; \dot{r}=2\left[E-V(r)-{l(l+1)\hbar^2\over
2m_0r^2}\right]\; ,
\end{equation}
%
%
\begin{equation}
{dL \over d\vartheta}\; \dot{\vartheta}={l(l+1)\hbar^2 \over
m_0r^2}-{(m^2_l-{1/4})\hbar^2 \over m_0r^2\sin^2(\vartheta)}\; ,
\end{equation}
%
%
\begin{equation}
{dM \over d\varphi}\; \dot{\varphi}={(m^2_l-{1/4})\hbar^2 \over
m_0r^2\sin^2(\vartheta)}\; .  \hskip18mm
\end{equation}
%
%
The three above relations represent the decomposition of Eq. (46)
with respect to the variables $r$, $\vartheta$ and $\varphi$.
Indeed, if one takes the summation  of Eqs. (54), (55) and (56) we
find Eq. (46). In fact, these equations describe the dynamical
behaviour of a particle moving under the central potential, each
one describes it in the corresponding coordinate ($r$,
$\vartheta$ or $\varphi$).

\noindent Note that only Eq. (54) depends on the nature of the
central potential $V(r)$ while Eqs. (55) and (56) do not depend
on it. Then, Eqs. (55) and (56) can be studied whatever the
nature of $V(r)$. This is not the case for Eq. (54). So, as an
application of the 3D quantum law of motion (Eqs. (46), (54),
(55) and (56)), we propose in the following sections to plot the
trajectories of an electron moving under the action of the
Hydrogen potential (Hydrogen atom). This will allow us to acquire
a more comprehension of the 3D quantum law of motion.

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip\baselineskip \noindent {\bf 4.\ \ Classical trajectories
of a particle moving under a central potential}
\vskip0.5\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Before entering to plot the quantum trajectories, it is useful to
present the classical Hamilton-Jacobi formulation of mechanics
and also to establish the classical trajectories of a particle
moving under the central potential.

The Hamilton-Jacobi formulation is based on the CHJE given as


%
%
\begin{equation}
{\frac{1}{2m_0}}\left( \vec{\nabla}S_{0}\right) ^{2}
 +V({\vec r})=E\; .
\end{equation}
%
%
$S_0$ represents the reduced action. Eq. (57) is written for the
case of a central potential, after considering of the relation
(35), as follows
%
%
\begin{equation}
{\frac{1}{2m_0}}\left( {dZ \over dr} \right) ^{2}+{1 \over
2m_0}{1 \over r^2}\left( {dL \over d\vartheta} \right) ^{2}+{1
\over 2m_0}{1 \over r^2\sin^2(\vartheta)}\left( {dM \over
d\varphi} \right) ^{2}
 +V({r})=E\; .
\end{equation}
%
%
The separation of the variables into Eq. (58) gives
%
%
\begin{equation}
\left({dZ \over dr}\right)^2=2m_0\left[E-V(r)-{\alpha \over
2m_0r^2}\right]\; ,
\end{equation}
%
%
\begin{equation}
\left({dL \over d\vartheta}\right)^2={\alpha }-{\beta^2 \over
\sin^2(\vartheta)}\; , \hskip20mm
\end{equation}
%
and
%
\begin{equation}
\left({dM \over d\varphi}\right)^2={\beta^2}\; . \hskip25mm
\end{equation}
%
%



\noindent $\alpha$ and $\beta$ are real constants. As we know,
the momenta $dZ/dr$, $dL/d\vartheta$ and $dM/d\varphi$ are
connected to the speeds $\dot{r}$, $\dot{\vartheta}$ and
$\dot{\varphi}$ as follows
%
%
\begin{equation}
{dZ \over dr}=m_0\dot{r}\; , \hskip16mm
\end{equation}
%
%
\begin{equation}
{dL \over d\vartheta}=m_0r^2\dot{\vartheta}\; ,\hskip12mm
\end{equation}
%
%
\begin{equation}
{dM \over d\varphi}=m_0r^2\sin^2(\vartheta)\dot{\varphi}\; .
\end{equation}
%
%
Taking these equations into Eqs. (59), (60) and (61), we find
%
%
\begin{equation}
{\dot{r}}=\pm\sqrt{2 \over m_0}\sqrt{E-V(r)-{\alpha \over
2m_0r^2}}\; ,\hskip0mm
\end{equation}
%
%
\begin{equation}
{\dot{\vartheta}}=\pm{1 \over m_0r^2}\sqrt{{\alpha }-{\beta^2
\over \sin^2(\vartheta)}}\; ,\hskip10mm
\end{equation}
%
%
\begin{equation}
{\dot{\varphi}}=\pm{\beta \over m_0r^2\sin^2(\vartheta)}\;
,\hskip20mm
\end{equation}
%
%
The $\pm$ signs in Eqs. (65), (66) and (67) indicate that the
motion of the particle may be in either direction on the $r$
axis, $\vartheta$ and $\varphi$ circles respectively. Now, we can
integrate Eqs. (65), (66) and (67) to give the three time
equation with respect to the coordinates $r$, $\vartheta$ and
$\varphi$, but before, we write these equations as
%
%
\begin{equation}
\sqrt{2 \over m_0}\; {dt}=\pm{ dr \over \sqrt{E-V(r)-{\alpha /
(2m_0r^2)}}}\; ,\hskip0mm
\end{equation}
%
%
\begin{equation}
 {dt \over m_0r^2}=\pm {d\vartheta \over \sqrt{{\alpha }-{\beta^2
 /\sin^2(\vartheta)}}}\; ,\hskip10mm
\end{equation}
%
%
\begin{equation}
 {dt \over m_0r^2\sin^2(\vartheta)}=\pm {d\varphi \over \beta}\; .\hskip20mm
\end{equation}
%
%

\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%  \begin{center}
%   {\includegraphics[height=3.2in,width=4.in]{Fig1.png}}
%       \end{center}
%       \centerline { \vbox { \hsize=10cm\noindent \small  Fig. 1: Classical 3D
%    trajectory of the Hydrogen's electron. In order to consider the
%     atomic scalars, we have chosen in Eqs (71-73), $E=-13.6/4$\, eV,
%    $\alpha=2\, \hbar^2$ and $\beta=(\sqrt{3}/2)\, \hbar^2$. It is an
%    ellipse, and we see clearly that the motion is done in a plan.
%     }}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
We remark that to integrate Eq. (68) can be integrated directly
after knowing the form of the central potential (either
analytically or numerically). However, to integrate Eq. (69), we
must first write $r$ -already integrated from Eq. (68)- in
function of the time $t$. Also, to integrate Eq. (70), we should
set the variables $r$ and $\vartheta$ in function of $t$. In
spite of the importance of the time equations, we will focus on
the spatial trajectory equations. These last's can be derived
directly from Eqs. (68-70) after parameterizing the time $t$.
Thus, we deduce
%
%
\begin{equation}
\sin^2(\vartheta)d\varphi=\pm{\beta dr \over
r^2\sqrt{2m_0(E-V(r))-\alpha/r^2}}\; ,
\end{equation}
%
%
\begin{equation}
{d\vartheta \over \sqrt{{\alpha }-{\beta^2
 /\sin^2(\vartheta)}}}=\pm{ dr \over r^2\sqrt{2m_0(E-V(r))-{\alpha / r^2}}}\; ,\hskip0mm
\end{equation}
%
%
\begin{equation}
d\varphi=\pm{ d\vartheta \over
\sin(\vartheta)\sqrt{(\alpha/\beta^2)\sin^2(\vartheta)-1}}\; .
\hskip10mm
\end{equation}
%
%
Integrating these last equations we find the trajectory equation
${\cal F}(r, \vartheta, \varphi)=0$.

\noindent In order to compare the classical trajectories with the
quantum trajectories, we plot in Fig. 1 the trajectory of an
electron under the atomic potential (Hydrogen atom)
%
%
\begin{equation}
V(r)=-{k\; e^2 \over r}\; ,
\end{equation}
%
%
where $k=9 . 10^9 S.I$ is the electric constant and $e$ the
charge of the electron. We give the energy $E$ the value
-(13.6/4) eV, which correspond in the quantum case to the energy
of the first excited state of the Hydrogen atom ($n=2$, ($n$ is
the principle integer)). We chose this value of energy in order to
compare the classical trajectory (Fig. 1) with the quantum
trajectories corresponding to this value of energy. For the same
reason, we have chosen the constants $\alpha=2\, \hbar^2$ and
$\beta=(\sqrt{3}/2)\, \hbar$ equal to the values of $l(l+1)\,
\hbar^2$ and $\sqrt{m^2_l-1/4}\; \hbar$ respectively, for $l=1$
and $m_l=1$. Then, the quantum trajectories corresponding to the
classical trajectory are those plotted for $n=2$, $l=1$ and
$m_l=1$ (Figs. (8) and (10)).

\noindent The classical trajectory showed in Fig. 1 is an
inclined ellipse. This trajectory is obviously famous since it is
the trajectory of a planet submitted to the gravitation force of
the sun, and this force looks like the electric force of the atom.

%\newpage

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip\baselineskip \noindent {\bf 5.\ \ Quantum orbits of the
Hydrogen's electron } \vskip0.5\baselineskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we will plot the 3D quantum trajectories. First,
let us write Eqs. (54), (55) and (56) as follows
%
%
\begin{equation}
2\; dt=\left[E-V(r)-{l(l+1)\hbar^2 \over 2m_0r^2}\right]^{-1}{dZ
\over dr}\; dr\; ,\hskip5mm
\end{equation}
%
%
\begin{equation}
dt={m_0r^2 \over \hbar^2}\left[{l(l+1)}-{(m^2_l-{1/4}) \over
\sin^2(\vartheta)}\right]^{-1}{dL \over d\vartheta}\; d\vartheta\;
,
\end{equation}
%
%
\begin{equation}
dt={m_0r^2\sin^2(\vartheta) \over (m^2_l-{1/4})\hbar^2}{dM \over
d\varphi}\; d\varphi\; . \hskip26mm
\end{equation}
%
%
Integrating Eqs. (75), (76) and (77), we find the time equations.
In order to plot the quantum trajectories, we need to find the
trajectories equations. In this order, we parameterize the time
$dt$ into Eqs. (75), (76) and (77) so as to obtain
%
%
\begin{equation}
{\sin^2(\vartheta) \over (m^2_l-{1/4})}{dM \over d\varphi}\;
d\varphi=\left[{2m_0 \over \hbar^2}\; r^2 (E-V(r)) -l(l+1)
\right]^{-1}{dZ \over dr}\; dr\; ,\hskip5mm
\end{equation}
%
%
\begin{eqnarray}
\left[{l(l+1)}-{(m^2_l-{1/4}) \over
\sin^2(\vartheta)}\right]^{-1}{dL \over d\vartheta}\;
d\vartheta=\hskip40mm\nonumber\\
\left[{2m_0 \over \hbar^2}\; r^2 (E-V(r)) -l(l+1) \right]^{-1}{dZ
\over dr}\; dr\; ,
\end{eqnarray}
%
%
\begin{equation}
{dM \over d\varphi}\; d\varphi=\left[{l(l+1)\over (m^2_l-{1/4})}\;
\sin^2(\vartheta)-1\right]^{-1}{dL \over d\vartheta}\;
d\vartheta\; .\hskip25mm
\end{equation}
%
%
In what follows Eqs. (78), (79) and (80) are using to plot the 3D
quantum spatial trajectories (orbits). Expressions of the quantum
momenta ${dZ / dr}$, ${dL / d\vartheta}$ and ${dM / d\varphi}$ are
deduced directly from the corresponding quantum reduced actions
$Z$, $L$ and $M$ (Eqs. (41), (42) and (43) respectively)
%
%
\begin{equation}
{dZ \over dr}=\pm{\hbar a_r \over (a_r{\cal X}_1\int {dr \over
{\cal X}^2_1} +b_r{\cal X}_1)^2+{\cal X}^2_1}\; ,
\end{equation}
%
%
\begin{equation}
{dL \over d\vartheta}=\pm{\hbar a_\vartheta \over
(a_\vartheta{\cal T}_1\int {d\vartheta \over {\cal T}^2_1}
+b_\vartheta{\cal T}_1)^2+{\cal T}^2_1}\; ,
\end{equation}
%
%
\begin{equation}
{dM \over d\varphi}=\pm{\hbar a_\varphi \over (a_\varphi F_1\int
{d\varphi \over F^2_1} +b_\varphi F_1)^2+F^2_1}\ .
\end{equation}
%
%
Knowing the expressions of the functions ${\cal X}_1$, ${\cal
T}_1$ and $F_1$, the quantum momenta will be known and the system
of the three differential equations (Eqs. (78-80)) can be
resolved numerically for the different values of the constants
$a_r$, $b_r$, $a_\vartheta$, $b_\vartheta$, $a_\varphi$ and
$b_\varphi$. The $\pm$ sign in the expressions of the quantum
momenta explicit the fact that the motion of the particle may be
in either direction of the $r$ axis, $\vartheta$ semicircle and
$\varphi$ circle.

\noindent {\bf Note:} In fact two of Eqs. (78-80) are sufficient
to integrate and determine the quantum spatial trajectories since
the third equation is automatically satisfied when the two others
are.

\vskip0.5\baselineskip Now, let us consider the quantum motion of
the Hydrogen's electron. We take up the expression of the atomic
potential given in Eq. (74). In this case, the resolution of Eq.
(14) is obvious and gives for the radial function ${\cal X}$ the
following expression \cite {Nik}
%
%
\begin{equation}
{\cal X}^{(nl)}_1(r)=\exp\left({r \over na_{o}}\right)\; \; {r
\over a_o }^{-l}\; {d^{n-l-1} \over dr^{n-l-1}}\left[{r \over a_o
} ^{n+l}\exp\left(-{2r \over na_{o}}\right)\right]
\end{equation}
%
%
$n$ being an integer number representing the principle quantum
number, and
%
%
\begin{equation}
a_{o}={\ \hbar^2 \over m_0 k e^2}
\end{equation}
%
%
the Bohr's radius. The function ${\cal X}^{(nl)}_1$ corresponds
to the energy
%
%
\begin{equation}
E_n=-{k\, e^2 \over 2\, n^2\, a_{o}}\; .
\end{equation}
%
%
Fixing the values of $n$, $l$ and $m$ into Eqs. (84), (44) and
(45) and  substituting these last equations into Eqs. (81), (82)
and (83) respectively, then replacing these lasts' and Eqs. (85)
and (86) into Eqs. (78-80), we get a system of three differential
equations which after integrating leads to the 3D quantum
trajectories of the electron for the corresponding bound
state$(n,l,m_l)$. Naturally, these trajectories depend on the
constants $a_r$, $b_r$, $a_\vartheta$, $b_\vartheta$, $a_\varphi$
and $b_\varphi$. So, for several sets of those constants it
correspond many trajectories (See Figs. (2-12)). In other words,
for each bound state many trajectories exist exactly as we have
found for the 1D quantum motions. We will show this by plotting
the trajectories of four bound states, the fundamental state
$(n=1,\; l=0,\; m_l=0)$, and the excited bound states $(n=2,\;
l=0,\; m_l=0)$, ($n=2,\; l=1,\; m_l=0)$ and ($n=2,\; l=1,\;
m_l=1)$\footnote{The considered excited states in this article do
not obeys to the selection law that we consider in spectroscopy of
the Hydrogen atom. We just use these states in order to make a
simple description of the quantum trajectories for different
values of the quantum numbers $n$, $l$ and $m_l$.}. With this
aim, for each state we proceed as follows:

\noindent First we explicit the functions ${\cal X}^{(nl)}_1$,
${\cal T}^{(lm)}_1$ and $F_1$ from Eqs. (84), (44) and (45).
Secondly, we integrate numerically Eq. (75) and plot the time
equation $r(t)$ for different values of the constants $a_r$ and
$b_\vartheta$. After, by numerical methods we integrate two of
Eqs. (78-79) for different values of the constants and  plot the
quantum spatial trajectories.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newpage
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip0.5\baselineskip \noindent {\bf a- the fundamental state}
\vskip0.3\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=2.85in,width=3.85in]{Fig2.png}}
%      \end{center}
% \centerline { \vbox { \hsize=10cm\noindent \small  Fig. 2:
%Quantum radial time trajectories for the Hydrogen's electron in
%the fundamental state ($E=-13.6$\, eV, $n=1$, $l=0$, $m_l=0$).
%For all the curves, we have chosen $r(t=0)= 2a_o$ ($a_o=0.52917\,
%.\, 10^{-10}$ m  is the Bohr radius). }}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For this state, we have $n=1$, $l=0$ and $m_l=0$. The functions
${\cal X}^{(10)}_1$, ${\cal T}^{(00)}_1$ and $F_1$ are deduced
from Eqs. (84), (44) and (45) as
%
%
\begin{equation}
{\cal X}^{(10)}_1(r)={r \over a_o }\; \exp\left(-{r \over
a_{o}}\right)\; ,
\end{equation}
%
%
\begin{equation}
{\cal T}^{(00)}_1(\vartheta)=\sin^{1 \over 2}(\vartheta)\;
,\hskip7mm
\end{equation}
%
%
\begin{equation}
F^{(0)}_1(\varphi)=1\; .\hskip18mm
\end{equation}
%
%
Replacing Eqs. (87), (88) and (89) into Eqs. (81), (82) and (83),
we find
%
%
\begin{equation}
{dZ \over dr}=\pm{\hbar a_r\; \left({r \over a_o
}\right)^{-2}e^{2r \over a_o } \over (a_r\int {r \over a_o
}^{-2}e^{2r \over a_o }dr +b_r)^2+1}\; ,
\end{equation}
%
%
\begin{equation}
{dL \over d\vartheta}=\pm{\hbar a_\vartheta \sin^{-1}(\vartheta)
\over [a_\vartheta \log(|\tan({\vartheta \over
2})|)+b_\vartheta]^2+1}\; ,
\end{equation}
%
%
\begin{equation}
{dM \over d\varphi}=\pm{\hbar a_\varphi \over (a_\varphi \varphi
+b_\varphi)^2+1}\ . \hskip13mm
\end{equation}
%
%
Replacing Eq. (90) into Eq. (75), we find
%
%
\begin{equation}
{\hbar \over m_0}\; dt=\pm\, \left(-{r^2 \over a^2_o}+{2\, r \over
a_o}\right)^{-1}\, {a_r\; a^2_oe^{2r \over a_o } \over
\left(a_r\int \left({r \over a_o }\right)^{-2}e^{2r \over a_o }dr
+b_r\right)^2+1}\; dr\; ,\hskip5mm
\end{equation}
%
%
Integrating numerically Eq. (93), we plot the radial time
trajectories given in Fig. 2. In this figure, the electron moves
between two radial extremities $r=0$ and $r=2\, a_o$, and
describes different trajectories for the same state. All these
trajectories pass throw some points constituting nodes with the
same manner as for the 1D motions \cite{Dja2,Dja4}. The radial
position $0$ of the nodes correspond to the zero of the radial
function ${\cal X}^{(10)}_1$. In particular, the nodes situated
at $r=2\, a_o$ correspond to the vanishing value of the radial
velocity given by Eq. (93). Remark that all the trajectories
oscillate about the trajectory for which the radial hidden
variables take the values $a_r=0.36$ and $b_r=0.52$.
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%  \begin{center}
%    {\includegraphics[height=3.7in,width=4.5in]{Fig3.png}}
%  \end{center}
%  \centerline { \vbox {
%\hsize=10cm\noindent \small Fig. 3: Quantum 3D spatial
%trajectories of the Hydrogen's electron in the fundamental state
%($E=-13.6$\, eV, $n=1$, $l=0$, $m_l=0$).} }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent Now, taking Eqs. (90), (91) and (92) into Eqs. (78),
(79) and (80), we get
%
%
\begin{equation}
{4\, a_\varphi\, \sin^2(\vartheta)\,  \over (a_\varphi\, \varphi
+b_\varphi )^2+1}\, d\varphi=\pm{a_r\, a^2_o\left(-{r^2 \over
a^2_o}+{2 r\over a_o\, }\right)^{-1}\, r^{-2}\, e^{2r \over a_o}  \over
\left(a_r \int \left({r \over a_o }\right)^{-2}e^{2r \over
a_o}dr+b_r\right)^2+1}\, dr\; ,
\end{equation}
%
%
\begin{equation}
{4\, a_\vartheta\, \sin(\vartheta) \over [a_\vartheta
\log(|\tan({\vartheta \over 2})|)+b_\vartheta]^2+1}\,
d\vartheta=\pm{a_r\, a^2_o\left(-{r^2 \over a^2_o}+{2 r\over a_o\,
}\right)^{-1}\, r^{-2}\, e^{2r \over a_o}  \over \left(a_r \int \left({r
\over a_o }\right)^{-2}e^{2r \over a_o}dr+b_r\right)^2+1}\, dr\; ,
\end{equation}
%
%
\begin{equation}
{a_\varphi \over (a_\varphi \varphi +b_\varphi)^2+1}\, d\varphi
=\pm{a_\vartheta \sin^{-1}(\vartheta) \over [a_\vartheta
\log(|\tan({\vartheta \over 2})|)+b_\vartheta]^2+1}\, d\vartheta
\; .
\end{equation}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=2.7in,width=3.75in]{Fig4.png}}
%  \end{center}
%  \centerline { \vbox { \hsize=10cm\noindent \small Fig. 4:
%Quantum radial time trajectories for the Hydrogen's electron in
%the first excited state ($E=-13.6/4$\, eV, $n=2$, $l=0$,
%$m_l=0$). For all the curves, we have chosen $r(t=0)= 0$.} }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Integrating the last equations, we plot the 3D quantum spatial
trajectories given in Fig. 3.
This figure shows the 3D behaviour of the Hydrogen's electron. The electron
describes different trajectories which pass from three points constituting
spatial nodes. One of these nodes corresponds to $r=0$
\footnote{This position can be seen as the position of the nucleus in a first approximation
and we may deduce that the electron fall on the nucleus. This is not the case because both
electron and nucleus make a relative motion compared to
their gravity center, so that the nucleus oscillate between the two radial
positions $r_{nuc}=0$ and $r_{nuc}={m_o \over m_o+M_{nuc}}\, r_{el}$,
(for the hydrogen $r_{nuc}\simeq 10^{-13}\times m$). Then the electron
has a real possibility to pass throw the position $r_{el}=0$ without collision with the nucleus.  }
and the two others to $r=2\, a_o$.



 It is useful to indicate that the motion of the electron in
a state for which $m_l=0$ is a purely quantum motion and cannot have a
classical correspondent. The reason is because in Eqs. (60) and (61), $\beta^2$ never
takes a negative values while in Eq. (55) and (56), $(m_l^2-1/4)\hbar^2$ takes
a negative values for $m_l=0$, and this make impossible the construction of the
classical corresponding of Eqs. (55) and (56) by given $\beta^2$ a negative values since
$\beta$ is a real constant.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \vskip0.5\baselineskip
 \noindent {\bf b- excited state: { \mathversion{bold}${n=2}$, $l=0$, $m_l=0$}}
 \vskip0.2\baselineskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For this state, the radial function
${\cal X}^{(20)}_1$ is
%
%
\begin{equation}
{\cal X}^{(20)}_1(r)=\left({r \over a_o }\right)\, \left(2-{r \over a_o }\right)\; \exp\left(-{r \over
2\, a_{o}}\right)\; ,
\end{equation}
%
%

while the angular functions still given by Eqs. (88) and (89).
Replacing Eq. (97) into Eq. (81), we find
%
%
\begin{equation}
{ dZ \over dr}=\pm{\hbar a_r\; \left({r \over
a _o}\right)^{-2}\left(2-{r \over a_o}\right)^{-2}e^{r \over a_o }
\over \left(a_r {\int }\left({r \over a_o}\right)^{-2}\left(2-{r
\over a_o}\right)^{-2}e^{r \over a_o }dr +b_r\right)^2+1}\; ,
\end{equation}
%%
%
Taking Eq. (98) into Eq. (75), we find
%
%
\begin{equation}
{\hbar \over m_0}\; dt=\pm\, \left(-{r^2 \over 4\, a^2_o}+{2\, r
\over a_o}\right)^{-1}\,  {a_r\; a^2_o\left(2-{r \over
a_o}\right)^{-2}\, e^{r \over a_o } \over \left(a_r\int \left({r
\over a_o}\right)^{-2}\left(2-{r \over a_o}\right)^{-2}e^{r \over
a_o }dr +b_r\right)^2+1}\; dr\; ,\hskip5mm
\end{equation}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=3.7in,width=4.5in]{Fig5.png}}
%  \end{center}
%  \centerline { \vbox {
%\hsize=10cm\noindent \small Fig. 5: Quantum 3D spatial
%trajectories of the Hydrogen's electron in the first excited
%state ($E=-13.6/4$\, eV, $n=2$, $l=0$, $m_l=0$).} }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Integrating numerically Eq. (99), we plot the radial time
equation (Fig. 4.) for this bound state. In this figure, the electron moves between two radial
extremities $r=0$ and $r=8\, a_o$, and describes different trajectories for the
same state. All these trajectories pass throw some points constituting nodes.
The radial positions $0$ and $2\, a_o$ of the nodes correspond to
the zero of the radial function ${\cal X}^{(20)}_1$. In particular, the nodes
situated at $r=8\, a_o$ correspond to the vanishing value of the radial
velocity given by Eq. (99).


\noindent Taking Eqs. (98), (91) and (92) into Eqs. (78), (79) and
(80), we get
%
%
\begin{equation}
{4\, a_\varphi\, \sin^2(\vartheta)\,  \over (a_\varphi\, \varphi
+b_\varphi )^2+1}\, d\varphi=\pm\,  {a_r\; a^2_o\, \left(-{r^2
\over 4\, a^2_o}+{2\, r \over a_o }\right)^{-1}\left(2-{r \over
a_o}\right)^{-2}\, r^{-2}\, e^{r \over a_o } \over \left(a_r\int \left({r
\over a_o}\right)^{-2}\left(2-{r \over a_o}\right)^{-2}e^{r \over
a_o }dr +b_r\right)^2+1}\, dr\; ,
\end{equation}
%
%
\begin{equation}
{4\, a_\vartheta\, \sin(\vartheta) \over [a_\vartheta
\log(\tan({\vartheta \over 2}))+b_\vartheta]^2+1}\,
d\vartheta=\pm{a_r\; a^2_o\, \left(-{r^2 \over 4\, a^2_o}+{2\, r
\over a_o }\right)^{-1}\left(2-{r \over a_o}\right)^{-2}\, r^{-2}\, e^{r
\over a_o } \over \left(a_r\int \left({r \over
a_o}\right)^{-2}\left(2-{r \over a_o}\right)^{-2}e^{r \over a_o
}dr +b_r\right)^2+1}\, dr\; ,
\end{equation}
%
\hskip0.5\baselineskip
%
\begin{equation}
{a_\varphi \over (a_\varphi \varphi +b_\varphi)^2+1}\, d\varphi
=\pm{a_\vartheta \sin^{-1}(\vartheta) \over [a_\vartheta
\log(\tan({\vartheta \over 2}))+b_\vartheta]^2+1}\, d\vartheta \;
.
\end{equation}
%
%
Integrating Eqs. (100), (101) and (102) we plot the 3D quantum
spatial trajectories given in Fig. 5 for this state. This figure
shows clearly that for different values of the hidden variables
the electron describes different trajectories which pass from
three points constituting spatial nodes. One of these nodes
corresponds to $r=0$ and the two others to $r=8\, a_o$. Remark
that the positions $r=2\, a_o$ constituting nodes for the time
radial trajectories do not constitute nodes for the spatial
trajectories. We think that this is due to the angular behaviour
of the electron. In fact the angular trajectories in the
$(\varphi,\vartheta)$ plan do not pass from any nodes for this
state (see the Appendix).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip\baselineskip \noindent {\bf c- excited state: { \mathversion{bold}${n=2}$,
${l=1}$, ${m_l=0}$}} \vskip0.5\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For this state , the radial function is
%
%
\begin{equation}
{\cal X}^{(2\, 1)}_1(\vartheta)={r \over a_o}^2\, \exp\left(-{r
\over 2\, a_{o}}\right)\; ,
\end{equation}
%
%
and the angular function is deduced from Eq. (44) as
%
%
\begin{equation}
{\cal T}^{(1\, 0)}_1(r)=\sin^{1 \over 2}(\vartheta)\,
\cos(\vartheta)\; ,
\end{equation}
%
%

without taking into account of the multiplicative constant (-2)
obtained in the calculation of ${\cal T}^{(1\, 0)}_1$ since it
has no influence on our equations. The angular function $F^{0}_1$
still given by Eq. (89). Replacing Eqs. (103) and (104) into Eqs.
(81) and (82), we find
%
%
\begin{equation}
{dZ \over dr}=\pm{\hbar a_r\; \left({r \over a_o}\right)^{-4}\,
e^{r \over a_o } \over \left(a_r\int \left({r \over
a_o}\right)^{-4}\, e^{2r \over a_o }dr +b_r\right)^2+1}\; ,
\end{equation}
%
%
\begin{equation}
{dL \over d\vartheta}=\pm{\hbar a_\vartheta\;
\sin^{-1}(\vartheta)\, \cos^{-2}(\vartheta) \over
\left[a_\vartheta \left({1\, +\, \cos(\vartheta) \over
\cos(\vartheta)}+\, \log(|\tan({\theta \over 2})|)\right)
+b_\vartheta\right]^2+1}\; ,
\end{equation}
%
%
Taking Eqs. (105) into Eqs. (75), we find
%
%
\begin{equation}
{\hbar \over m_0}\; dt=\pm\, {a_r\, a^2_o \left(-{r^2 \over 4\,
a^2_o}+{2 r\over a_o\, }-2\right)^{-1}\, \left({r \over
a_o}\right)^{-2}\, e^{r \over a_o}  \over \left(a_r \int \left({r
\over a_o}\right)^{-4}e^{r \over a_o}dr+b_r\right)^2+1}\; dr\;
,\hskip5mm
\end{equation}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=2.7in,width=3.75in]{Fig6.png}}
%  \end{center}
%  \centerline { \vbox { \hsize=10cm\noindent \small Fig. 6:
%Quantum radial time trajectories for the Hydrogen's electron in
%the second excited state ($E=-13.6/4$\, eV, $n=2$, $l=1$,
%$m_l=0$). For all the curves, we have chosen $r(t=0)= 0$.} }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Integrating numerically Eq. (107), we plot the radial time
equation (Fig. 6) for this bound state. The electron moves between two radial
extremities $r=(4-2\, \sqrt{2})\, a_o$ and $r=(4+2\, \sqrt{2})\, a_o$,
and describes different trajectories for the
same state. All these trajectories pass throw some points constituting nodes.
These nodes are
situated at $r=(4-2\, \sqrt{2})\, a_o$ and $r=(4+2\, \sqrt{2})\, a_o$
corresponding to the vanishing values of the radial
velocity given by Eq. (107).

\noindent Taking Eqs. (105), (106) and (92) into Eqs. (78), (79)
and (80), we get
%
%
\begin{eqnarray}
{4\, a_\varphi\, sin^2(\vartheta)\,  \over (a_\varphi\, \varphi
+b_\varphi )^2+1}\, d\varphi=
\nonumber\hskip60mm \\
\pm\,  {a_r\, a^2_o \left(-{r^2 \over
4\, a^2_o}+{2 r\over a_o\, }-2\right)^{-1}\, \left({r \over
a_o}\right)^{-2}\, r^{-2}\, e^{r \over a_o} \over \left(a_r \int \left({r
\over a_o}\right)^{-4}e^{r \over a_o}dr+b_r\right)^2+1}\, dr\; ,
\end{eqnarray}
%
%
\begin{eqnarray}
{4\, a_\vartheta\; (8\, \sin^2(\vartheta)+1)^{-1}\,
\sin(\vartheta)\, \cos^{-2}(\vartheta) \over \left[a_\vartheta
\left({1\, +\, \cos(\vartheta) \over \cos(\vartheta)}+\,
\log(|\tan(\vartheta/2)|)\right) +b_\vartheta\right]^2+1}\,
d\vartheta=
\nonumber\hskip30mm \\
 \pm\, {a_r\, a^2_o \left(-{r^2 \over
4\, a^2_o}+{2 r\over a_o\, }-2\right)^{-1}\, \left({r \over
a_o}\right)^{-2}\, r^{-2}\, e^{r \over a_o} \over \left(a_r \int \left({r
\over a_o}\right)^{-4}e^{r \over a_o}dr+b_r\right)^2+1}\, dr\; ,
\end{eqnarray}
%
\hskip0.5\baselineskip
%
\begin{equation}
{a_\varphi \over (a_\varphi \varphi +b_\varphi)^2+1}\, d\varphi
=\pm{a_\vartheta\; (8\, \sin^2(\vartheta)+1)^{-1}\,
\sin^{-1}(\vartheta)\, \cos^{-2}(\vartheta) \over
\left[a_\vartheta \left({1\, +\, \cos(\vartheta) \over
\cos(\vartheta)}+\, \log(|\tan(\vartheta/2)|)\right)
+b_\vartheta\right]^2+1}\, d\vartheta\; .
\end{equation}
%
%
Integrating Eqs. (108), (109) and (110) we plot the 3D quantum
spatial trajectories given in Fig. 7 for this state. This figure shows clearly
that for different values of the hidden variables
the electron describes different trajectories which pass from two points constituting
spatial nodes corresponding to $r=(4-2\, \sqrt{2})\, a_o$
and $r=(4+2\, \sqrt{2})\, a_o$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=3.4in,width=4.5in]{Fig7.png}}
%  \end{center}
%  \centerline { \vbox {
%\hsize=10cm\noindent \small Fig. 7: Quantum 3D spatial
%trajectories of the Hydrogen's electron in the second excited
%state ($E=-13.6/4$\, eV, $n=2$, $l=1$, $m_l=0$). } }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=3.in,width=3.8in]{Fig8.png}}
%  \end{center}
% \centerline{ \vbox { \hsize=10cm\noindent \small Fig. 8: This
%figure shows a comparison between the purely 3D classical
%trajectory to the one of the possible 3D quantum trajectories of
%the Hydrogen's electron for the state ($E=-13.6/4$\, eV, $n=2$,
%$l=1$, $m_l=1$). For the corresponding values of the hidden
%variables ($a_r=3.1$, $b_r=-0.4$, $a_\vartheta=0.6$,
%$b_\vartheta=-0.2$, $a_\varphi=1$, $b_\varphi=0$), we see easily
%from this picture, that the quantum trajectory can be considered
%as a deformation of the elliptic purely classical 3D trajectory.
%More comparison is done in Fig. 9, where projections of the two
%trajectories into $YZ$ and $XZ$ plans are done. } }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip\baselineskip \noindent {\bf d- excited state: { \mathversion{bold}${n=2}$,
${l=1}$, ${m_l=1}$} }\vskip0.5\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For this state , the radial function is given by Eq. (103), while
the angular functions are
%
%
\begin{equation}
{\cal T}^{(1\, 1)}_1(\vartheta)=\sin^{3 \over 2}(\vartheta)\, \; ,
\end{equation}
%
%
\begin{equation}
F^{(1)}_1=\sin(\varphi)\; ,\hskip9mm
\end{equation}
%
%
without taking into account of the multiplicative constant (-2)
obtained in the calculation of ${\cal T}^{(1\, 1)}_1$. Replacing
Eqs. (111) and (112) into Eqs. (82) and (83), we find
%
%
\begin{equation}
{dL \over d\vartheta}=\pm{\hbar a_\vartheta\;
\sin^{-3}(\vartheta)\,  \over \left[a_\vartheta
\left({\cos(\vartheta) \over \cos(2\, \vartheta)-1}+\, {1 \over
2} \log(|\tan({\theta \over 2})|)\right)
+b_\vartheta\right]^2+1}\; ,
\end{equation}
%
%
\begin{equation}
{dM \over d\varphi}=\pm\, {\hbar\, a_\varphi \over (a_\varphi\,
\sin(\varphi)+b_\varphi\, \cos(\varphi))^2+\cos(\varphi)^2}\; .
\end{equation}
%
%
The radial time equation is given by Fig. 6 since the radial
function is the same for the present and the previous bound
states.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=2.5in,width=4.5in]{Fig9.png}}
%  \end{center}
%  \centerline { \vbox {
%\hsize=10cm\noindent \small Fig. 9: Projection onto the $YZ$ and $XZ$ plans of the spatial
%trajectory for the state ($E=-13.6/4$\, eV, $n=2$, $l=1$, $m_l=1$)($a_r=3.1$, $b_r=-0.4$,
%$a_\vartheta=0.6$, $b_\vartheta=-0.2$, $a_\varphi=1$,
%$b_\varphi=0$), and it corresponding to the purely classical trajectory. the quantum trajectory seems
%to be a deformation of the classical one.
%} }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent Taking Eqs. (105), (113) and (114) into Eqs. (78), (79)
and (80), we get
%
%
\begin{eqnarray}
{{4 \over 3}\; a_\varphi\, sin^2(\vartheta)\,  \over (a_\varphi\,
\sin(\varphi) +b_\varphi \cos(\varphi))^2+\cos^2(\varphi)}\,
d\varphi=
\nonumber\hskip45mm \\
\pm\, {a_r\, a^2_o \left(-{r^2 \over 4\, a^2_o}+{2 r\over a_o\,
}-2\right)^{-1}\, \left({r \over a_o}\right)^{-2}\, r^{-2}\, e^{r \over
a_o} \over \left(a_r \int \left({r \over a_o}\right)^{-4}e^{r
\over a_o}dr+b_r\right)^2+1}\, dr\; ,
\end{eqnarray}
%
%
\begin{eqnarray}
{{4 \over 3}\; a_\vartheta\; \left({8 \over 3}\,
\sin^2(\vartheta)-1\right)^{-1}\, \sin^{-1}(\vartheta) \over
\left[a_\vartheta \left({\cos(\vartheta) \over \cos(2\,
\vartheta)-1}+{1 \over 2}\, \log(|\tan(\theta/2)|)\right)
+b_\vartheta\right]^2+1}\, d\vartheta=
\nonumber\hskip30mm \\
 \pm\, {a_r\, a^2_o \left(-{r^2 \over
4\, a^2_o}+{2 r\over a_o\, }-2\right)^{-1}\, \left({r \over
a_o}\right)^{-2}\, r^{-2}\, e^{r \over a_o} \over \left(a_r \int \left({r
\over a_o}\right)^{-4}e^{r \over a_o}dr+b_r\right)^2+1}\, dr\; ,
\end{eqnarray}
%
\hskip0.5\baselineskip
%
\begin{eqnarray}
{a_\varphi\, \over (a_\varphi\, \sin(\varphi) +b_\varphi
\cos(\varphi))^2+\cos^2(\varphi)}\, d\varphi =
\nonumber\hskip40mm \\
 \pm{a_\vartheta\; \left({8 \over 3}\,
\sin^2(\vartheta)-1\right)^{-1}\, \sin^{-3}(\vartheta) \over
\left[a_\vartheta \left({\cos(\vartheta) \over \cos(2\,
\vartheta)-1}+{1 \over 2}\, \log(|\tan({\theta \over 2})|)\right)
+b_\vartheta\right]^2+1}\, d\vartheta\; .
\end{eqnarray}
%
%


Integrating the last equations we plot the 3D quantum spatial
trajectories. In Fig. 8, we compare a quantum trajectory integrated from Eqs. (115),
(116) and (117) for the following values of the hidden variables
$a_r=3.1$, $b_r=-0.4$, $a_\vartheta=0.6$, $b_\vartheta=-0.2$,
$a_\varphi=1$ and $b_\varphi=0$ with the corresponding classical trajectory plotted from
the classical equations (71), (72) and (73). We remark that the quantum trajectory seems to
be, more or less, a deformation of the classical trajectory. To render this Note more
obvious, we plot in Fig. 9 the projection of the two trajectories into $YZ$ and $XZ$ plans.
The Fig. 9 shows that the quantum trajectory has a deformed elliptic form.

In Fig. 10, we plot different trajectories for different values of the hidden variables.
These trajectories pass from two points constituting
spatial nodes corresponding to $r=(4-2\, \sqrt{2})\, a_o$
and $r=(4+2\, \sqrt{2})\, a_o$.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=3.5in,width=4.3in]{Fig10.png}}
%  \end{center}
%  \centerline { \vbox {
%\hsize=10cm\noindent \small Fig. 10: Quantum 3D spatial trajectories of the Hydrogen's electron
%in the third excited
%state ($E=-13.6/4$\, eV, $n=2$, $l=1$, $m_l=1$).} }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip\baselineskip \noindent {\bf 6- Conclusion}
\vskip0.5\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

First, we would like to emphasize that this article is a
continuation of the work already presented in Refs. \cite
{Dja1,Dja2,Dja3,Dja4,Dja5,Dja6,Dja7}, especially in Ref. \cite
{Dja7}. By this work, we want to bring a clarification and a
checking of the results presented in Ref. \cite {Dja7}. Indeed,
we have introduced in Ref. \cite {Dja7} a generalization to 3D of
the deterministic approach of quantum mechanics presented in Refs.
\cite {Dja1,Dja2}. So, a 3D quantum law of motion is established
in Ref. \cite {Dja7} for a general form of the potential, and in
the present article, we apply this law of motion for the central
potential case and in particular to the Hydrogen's atom. We
proceeded by three main steps, the first is the establishment of
the 3D-QSHJE and the quantum reduced action for the case of a
central potential (Sec. 2). Then, in the second step, we
established the 3D Quantum law of motion under the central
potential (Sec. 3). Finally, in the third step, we plotted the 3D
quantum trajectories of the Hydrogen's electron (Sec. 5).
Analyzing our results, especially the trajectories obtained in
Sec. 5, for different bound states of the electron, we find
interesting outcomes.

\vskip0.3\baselineskip
The first outcome is the fact that
for the bound state case, the electron is trapped between two radial positions
given by the two roots of the equation
%
%
\begin{equation}
E-V(r)-{l(l+1)\, \hbar^2 \over 2\, m_0\, r^2}=0\, ,
\end{equation}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\def\put(#1,#2)#3{\leavevmode\rlap{\hskip#1\unitlength\raise#2\unitlength\hbox{#3}}}
%  \begin{center}
%    {\includegraphics[height=2.32in,width=2.7in]{Fig11.png}}
%  \end{center}
%  \centerline { \vbox { \hsize=10cm\noindent \small Fig. 11:  The ejection of
%the electron from the atom when it live the trapping zone $(0\leq
%r \leq 2\, a_o)$ in the case of the fundamental bound state
%($E=-13.6$\, eV, $n=1$, $l=0$, $m_l=0$). We chose for the radial
%hidden variables the values $a_r=1.5$ and $b_r=-0.5$. } }
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and, if the electron reach an external position of the trapping zone
delimited by the two roots, the electron will be ejected from the atom. This
fact is well established in Fig. (11). Indeed, this figure shows that after passing
the position $r=2\, a_0=1.05834 \times 10^{-10}$ m, which is one of the two roots of Eq. (118)
for the bound state ($n=1$, $l=0$, $m_l=0$) defining the trapping zone, the electron reach in a
limited time a position of about $10^{10}$ m. This result means that, after living the
trapping zone the electron will be strongly ejected far from the atom. This is in
disagreement with the standard quantum mechanics which suggest that for any position
in the space the atomic electron has a non vanishing probability to be there. Our approach shows
that the electron is trapped in a given trapping zone, otherwise it will be ejected far
from the atom. Note that this result is even valid for all other excited bound states.
A consequence of such a result is a partial localization of the electron in the space
in the case of the bound states.

The second outcome is the existence of nodes from which all the
possible quantum trajectories plotted for the same state pass. In
fact, we established the existence of nodes for two kind of
trajectories, the radial time trajectories (Figs. (2), (4) and
(6)) and the spatial trajectories (Figs. (3), (5), (7) and (10)).
The radial time trajectories are similar to the 1D time
trajectories. So, in Ref. \cite {Dja2}, we established that the
nodes exist independently of the nature of the potential. This is
also valid for the radial time trajectories and the nodes
corresponds to the zeros of the functions ${\cal X}^{(nl)}$ and
to the points where the radial velocity ${\dot r}$ vanishes. The
spatial trajectories, in spite of their complexity, pass throw
points constituting nodes. The number and the positions of these
nodes are established from Eq. (118). When the point $r=0$ is
situated into the trapping zone defined by the two roots $r_1$
and $r_2$ of Eq. (118), the number of nodes is three and their
positions are $r=0$, $r=r_1$ and $r=r_2$. When the point $r=0$ is
out of the trapping zone, the number of nodes is two and their
positions are $r_1$ and $r_2$.

Another important outcome concerns the comparison between the quantum trajectories and the
corresponding purely classical trajectory. We noted in Sec. 5, that such comparison
can be done only for the states for which $m_l\neq 0$, otherwise the classical
conjugate momentum $dM/d\varphi$ with respect to $\varphi$ takes an imaginary value.
This is the reason for what we take such a comparison only for the state for which
$n=2$, $l=1$ and $m_l=1$. For this state, we find that either classical and quantum
trajectories describe a closed curves in the 3D space which are contained between two
radial positions $r=(4-2\, \sqrt{2})\, a_o$ and $r=(4+2\, \sqrt{2})\, a_o$.
The classical trajectory is an ellipse situated into an inclined plan
(Figs. (1) and (8)), while the quantum trajectories have a complex forms and move
in all the space. However, for some values of the hidden variables the quantum
trajectories goes approximately to the classical trajectory with a an apparent
residual deformation (Figs. (8), (9)). This is an obvious result since the quantum
equations of motion reduce to the classical's.

\vskip0.35\baselineskip To conclude, we stress that our
deterministic 3D approach of quantum mechanics introduced in Ref.
\cite {Dja7} and the establishment of the 3D quantum trajectories
for a central potential (atomic potential) presented in the
present article are a generalization of the deterministic 1D
approach of quantum mechanics introduced in Refs. \cite
{Dja1,Dja2}. This deterministic approach, in the frame of its
validity, will restore an objective reality, and permit a
construction of a deterministic theory of quantum gravity. Such a
construction will be an important step to start a unification of
the fundamental interactions even the gravity.





%\newpage


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip\baselineskip \noindent {\bf APPENDIX}
%\vskip\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip0.5\baselineskip \noindent {\bf  The angular behaviour of
the Hydrogen's electron} \vskip0.5\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this appendix, we plot many angular trajectories corresponding
to several states of the Hydrogen's electron. (See Figs. (12))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%  \begin{center}
%    {\includegraphics[height=2.25in,width=3.5in]{Fig12a.png}}
%  \end{center}

% \begin{center}
%    {\includegraphics[height=2.25in,width=3.5in]{Fig12b.png}}
%  \end{center}

%  \begin{center}
%    {\includegraphics[height=2.25in,width=3.5in]{Fig12c.png}}
%  \end{center}
%  \centerline { \vbox { \hsize=10cm\noindent \small Fig12: The
%above figures show the relation between $\vartheta$ and $\varphi$
%in the electron trajectories for different $l$ and $m_l$. For many
%values of the hidden variables, we plot $\vartheta$($\varphi$).
%(a): $l=0$, $m_l=0$. (b): $l=1$, $m_l=0$ (c): $l=1$, $m_l=1$ . }}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%\newpage

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip3\baselineskip \noindent {\bf REFERENCES}
%\vskip\baselineskip
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}
% ref 1
\bibitem{Dja1}
A. Bouda and T. Djama, Phys. Lett. A 285 (2001) 27-33,
quant-ph/0103071.

% ref 2
\bibitem{Dja2}
A. Bouda and T. Djama, Phys. scripta 66 (2002) 97-104,
quant-ph/0108022.

% ref 3
\bibitem{Dja3}
T. Djama, "Relativistic Quantum Newton's Law and photon's
Trajectories ", quant-ph/0111121.

% ref 4
\bibitem{Dja4}
T. Djama, "Nodes in the Relativistic Quantum  Trajectories  and
photon's trajectories ", quant-ph/0201003.

% ref 5
\bibitem{Dja5}
T. Djama, "The Relativistic Quantum Stationary Hamilton Jacobi
Equation for Particle with  Spin 1 / 2 ", quant-ph/0311057.

% ref 6
\bibitem{Dja6}
T. Djama, "The Relativistic Quantum Law of motion for  a Particle
with  Spin 1 / 2 ", quant-ph/0311059.

% ref 7
\bibitem{Dja7}
T. Djama, " The 3D Quantum Law of motion ", quant-ph/0311082.

%ref 8
\bibitem{FM1}
A.~E.~Faraggi and M.~Matone, {\it Int. J. Mod. Phys.} A 15, 1869
(2000); hep-th/9809127.

%ref 9
\bibitem{Coh1}
C. Cohen-Tannoudji, B. Diu et F. Lalo{\"e}, M{\'e}canique
quantique (Hermann, 1977), tome 1.

%ref 10
\bibitem{Nik}
A. Nikiforov et V. Ouvarov, Fonctions sp{\'e}ciales de la
physique math{\'e}matique (Mir, 1983).









\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Figures}
\begin{itemize}

\item[Fig.\,1:] Classical 3D trajectory of the Hydrogen's
                electron. In order to consider the atomic scalars,
                we have chosen in Eqs (71-73), $E=-13.6/4$\, eV,
                $\alpha=2\, \hbar^2$ and $\beta=(\sqrt{3}/2)\, \hbar^2$.
                It is an ellipse, and we see clearly that the motion is
                done in a plan.


\item[Fig.\,2:] Quantum radial time trajectories for the
                Hydrogen's electron in the fundamental state
                ($E=-13.6$\, eV,$n=1$, $l=0$, $m_l=0$). For all
                the curves, we have chosen $r(t=0)= 2a_o$ ($a_o=0.52917\,
                .\, 10^{-10}$ m  is the Bohr radius).

\item[Fig.\,3:] Quantum 3D spatial trajectories of the Hydrogen's
                electron in the fundamental state ($E=-13.6$\, eV, $n=1$,
                $l=0$, $m_l=0$).

\item[Fig.\,4:] Quantum radial time trajectories for the Hydrogen's
                electron in the first excited state ($E=-13.6/4$\,
                eV, $n=2$,$l=0$, $m_l=0$). For all the curves,
                we have chosen $r(t=0)= 0$.

\item[Fig.\,5:] Quantum 3D spatial trajectories of the Hydrogen's electron
                in the first excited state ($E=-13.6/4$\, eV, $n=2$, $l=0$,
                $m_l=0$).

\item[Fig.\,6:] Quantum radial time trajectories for the Hydrogen's
                electron in the second excited state ($E=-13.6/4$\,
                eV, $n=2$,$l=1$,$m_l=0$). For all the curves, we have
                chosen $r(t=0)= 0$.


\item[Fig.\,7:] Quantum 3D spatial trajectories of the Hydrogen's electron
                in the second excited state ($E=-13.6/4$\, eV, $n=2$, $l=1$,
                $m_l=0$).

\item[Fig.\,8:] This figure shows a comparison between the purely 3D
                classical trajectory to the one of the possible 3D
                quantum trajectories of the Hydrogen's electron for
                the state ($E=-13.6/4$\, eV, $n=2$,$l=1$, $m_l=1$).
                For the corresponding values of the hidden variables
                ($a_r=3.1$, $b_r=-0.4$, $a_\vartheta=0.6$,$b_\vartheta=-0.2$,
                $a_\varphi=1$, $b_\varphi=0$), we see easily from this picture,
                that the quantum trajectory can be considered as a deformation of
                the elliptic purely classical 3D trajectory.More comparison is
                done in Fig. 9, where projections of the two trajectories into
                $YZ$ and $XZ$ plans are done.

\item[Fig.\,9:] Projection onto the $YZ$ and $XZ$ plans of the spatial
                trajectory for the state ($E=-13.6/4$\, eV, $n=2$, $l=1$,
                $m_l=1$)($a_r=3.1$, $b_r=-0.4$,$a_\vartheta=0.6$,
                $b_\vartheta=-0.2$, $a_\varphi=1$,$b_\varphi=0$), and it
                corresponding purely classical trajectory. the quantum
                trajectory seems to be a deformation of the classical one.

\item[Fig.\,10:] Quantum 3D spatial trajectories of the Hydrogen's
                 electron in the third excited state ($E=-13.6/4$\,
                 eV, $n=2$, $l=1$,$m_l=1$).

\item[Fig.\,11:] The ejection of the electron from the atom when it
                 live the trapping zone $(0\leq r \leq 2\, a_o)$ in
                 the case of the fundamental bound state
                 ($E=-13.6$\, eV, $n=1$, $l=0$, $m_l=0$). We chose
                 for the radial hidden variables the values $a_r=1.5$
                 and $b_r=-0.5$.

\item[Fig.\,12:] The above figures show the relation between $\vartheta$
                 and $\varphi$ in the electron trajectories for different
                 $l$ and $m_l$. For many values of the hidden variables,
                 we plot $\vartheta$($\varphi$). (a): $l=0$, $m_l=0$.
                 (b): $l=1$, $m_l=0$ (c): $l=1$, $m_l=1$ .

\end{itemize}



\end {document}

quant-ph0404175.extracted_bib
\bibitem{Dja1}
A. Bouda and T. Djama, Phys. Lett. A 285 (2001) 27-33,
quant-ph/0103071.

% ref 2

\bibitem{Dja2}
A. Bouda and T. Djama, Phys. scripta 66 (2002) 97-104,
quant-ph/0108022.

% ref 3

\bibitem{Dja3}
T. Djama, "Relativistic Quantum Newton's Law and photon's
Trajectories ", quant-ph/0111121.

% ref 4

\bibitem{Dja4}
T. Djama, "Nodes in the Relativistic Quantum  Trajectories  and
photon's trajectories ", quant-ph/0201003.

% ref 5

\bibitem{Dja5}
T. Djama, "The Relativistic Quantum Stationary Hamilton Jacobi
Equation for Particle with  Spin 1 / 2 ", quant-ph/0311057.

% ref 6

\bibitem{Dja6}
T. Djama, "The Relativistic Quantum Law of motion for  a Particle
with  Spin 1 / 2 ", quant-ph/0311059.

% ref 7

\bibitem{Dja7}
T. Djama, " The 3D Quantum Law of motion ", quant-ph/0311082.

%ref 8

\bibitem{FM1}
A.~E.~Faraggi and M.~Matone, {\it Int. J. Mod. Phys.} A 15, 1869
(2000); hep-th/9809127.

%ref 9

\bibitem{Coh1}
C. Cohen-Tannoudji, B. Diu et F. Lalo{\"e}, M{\'e}canique
quantique (Hermann, 1977), tome 1.

%ref 10

\bibitem{Nik}
A. Nikiforov et V. Ouvarov, Fonctions sp{\'e}ciales de la
physique math{\'e}matique (Mir, 1983).









