# Electron Dynamics in Noncommutative Geometry with Magnetic Field and
Zitterbewegung Phenomenon

Mehran Zahiri<EMAIL_ADDRESS>and Mehrdad
<EMAIL_ADDRESS>
Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839,
Iran

(June 20, 2021)

Starting from a gauge invariant Dirac Hamiltonian with noncommutativity of
space sector in the presence of an external uniform magnetic field, the
resulting Dirac equation has been solved for electrons and its corresponding
_zitterbewegung_ (ZBW) phenomenon has been studied. The corresponding energy
spectrum is shown to be different from previous studies wherein the non-gauge
invariant Dirac Hamiltonian has been used. The effects of noncommutativity
alter the amplitude as well as the cyclotron and the ZBW frequencies of the
average velocity of charge carriers. This result is contrary to previous
studies wherein there was no magnetic field and hence, neither the amplitude
nor the frequency of the motion was affected. Moreover, all of the ZBW
frequencies of the Landau energy-levels appear in the results. Also, in weak
magnetic fields, we have calculated the average velocity of charge carriers
for two initially localized spin-up and spin-down cases. The plotted
trajectories reveal difference between these two cases. In addition, the ZBW
phenomenon has been shown that manifests itself as a circular motion whose
direction is spin dependent while accompanied by the cyclotron motion.
PACS number: 02.40.Gh; 03.65.-w; 11.10.Nx; 03.65.Sq; 13.40.Em
Keywords: Noncommutative Geometry; Zitterbewegung; Magnetic Field

##  1 Introduction

One of the outcomes of the Dirac equation is a phenomenon, which has been
dubbed ZBW by Schrödinger [1, 2]. It is a trembling/quivering motion which is
supposed to be endured by a free Dirac particle while indicates that its
velocity and momentum do not coincide and rapidly oscillates with the speed
$c$ around its center of mass whereas moves like a relativistic particle with
velocity ${\bf p}/m$ [3]–[7]. The amplitude of such a motion is predicted to
be of the order of the Compton wavelength, i.e. ${\mathchar
22\relax\mkern-9.0mu\lambda}_{c}=\hbar/(m_{e}c)\simeq
3.86\times{10}^{-13}\,{\rm m}$ for electrons. Through this picture, it has
also been suggested that the spin, and in turn the magnetic moment, of an
electron (as a point charge) is generated as an intrinsic local motion
[8]–[12] such that the direction of its spin is caused via the motion of
particle around a circle of radius ${\mathchar
22\relax\mkern-9.0mu\lambda}_{c}/2$ with frequency $\omega^{\rm[zbw]}(\simeq
1.6\times{10}^{21}\,{\rm s}^{-1}$ for electrons). This result leads to the
intrinsic magnetic moment of particle with the correct gyromagnetic $g$
factor. Even in this regard, by taking the neutrino as a localized wave packet
consisting of positive and negative energies, its chiral ZBW has been studied
[13]. Wherein, it has been claimed that it may explain the ‘missing’ solar
neutrino experiments. Besides, the chiral oscillations have been interpreted
[14] in terms of the ZBW effect. Moreover, the ZBW of massive and massless
scalar (spin-$0$) bosons and massive (spin-$1$) bosons have been analyzed in
Ref. [15]. Furthermore, the matter of an electron in the presence of an
external magnetic field has widely been studied, see, e.g., Refs. [16]–[18],
where its ZBW effect has also been considered by us [19].

In addition, the ZBW phenomenon has been investigated in many different
contexts, one of which is the Dirac materials [20]. Indeed, many of the
relativistic concepts, like the ZBW, have been transferred from the
relativistic quantum mechanics to Dirac materials [20]. Interestingly enough,
the phenomenon of ZBW for electrons has also been simulated using the trapped
ions [21], and it has been shown (e.g., Refs. [22]–[24] and other references
mentioned in Refs. [19, 25]) to occur in non-relativistic cases and in two-
dimensional Dirac materials (like graphene and silicon [24, 26]–[28]) as well.
In the latter cases, as the speed of light is replaced by the Fermi
velocity,333The original Dirac equation is a Lorentz covariant, but, in this
case, the corresponding result is not a Lorentz covariant. it leads to lower-
frequencies and higher-amplitudes for the ZBW motion that give more hopes of
being detected in this framework. In the case of such a detection, it would
indirectly support the occurrence of the ZBW in vacuum as well.

Nevertheless, it has been claimed [29] that the ZBW can be avoided by a
unitary Foldy-Wouthuysen transformation (FW), in which negative energy
components in electron wave functions have been eliminated. In contrast, one
should be reminded that the ZBW can occur when both positive and negative
energy solutions are considered together. Moreover, the results of such a FW
transformation are in contrast with the Darwin term [30] in atomic physics.
Also, it has been speculated that an observable ZBW may only occur in curved
spacetimes [31].

Another concept in high-energy physics, which can lead to a new vision for
spacetime structure, is the noncommutative (NC) geometry, see, e.g., Refs.
[32]–[41]. This topic has been discussed thoroughly in the literature both on
theoretical and phenomenological aspects, see, e.g., Refs. [42, 43] and
references therein. The concept of noncommutativity has also strong motivation
in the framework of the string and M-theories, see, e.g., Refs. [44]–[46].
Indeed, it has been claimed [47] that as the standard linear quantum mechanics
should be a limiting case of an underlying non-linear quantum theory, a
possible approach for such a formalism can be sought through the NC geometry.
In this regard, due to changes raised in this picture, the quantum mechanics
and quantum field theory (QFT) phenomena have been affected and some
researches have been devoted to this sector [48]–[53]. Actually, while
considering the NC geometry, the QFT modifies the relativistic wave equation,
i.e. the Dirac equation, by which, one can calculate the consequences of NC
parameters on different physical phenomena. It is worthwhile to mention that
applying the NC effects into the action of a Dirac electron in an external
electromagnetic field, via the so called the Seiberg-Witten (SW) map
transformations [44], leads to a gauge invariant Dirac equation [52, 53].

The effects of the NC geometry on electron dynamics and the ZBW have been
investigated in vacuum [55], and in two-dimensional Dirac materials, and on
the quantum Hall effect as well, see, e.g., Refs. [56]–[59]. Besides, in Ref.
[60], the effect of $\kappa$-deformation of spacetime on ZBW has also been
considered. Furthermore, in Ref. [61], it has been demonstrated that, for an
almost-commutative spacetime (it has been mentioned that this situation is
different from the NC geometry, see, e.g., Ref. [62]), the upper bound on ZBW
frequency cannot be altered by the presence of an electromagnetic field. Also,
the effects of the NC geometry on electron dynamics in an external magnetic
field in $2+1$ dimensions have been studied in Refs. [38, 54], wherein the
Dirac equation used is not gauge invariant [52, 53]. Nevertheless, to our
knowledge, the effect of NC geometry on ZBW has not been considered in a gauge
invariant Dirac equation. Moreover, a two dimensional Dirac equation does not
accommodate the spin and particle-hole degrees of freedom simultaneously in
one picture.

In this work, we use the solutions of the gauge invariant Dirac equation, in
the presence of an external uniform magnetic field, to study the effect of the
NC geometry on the electron ZBW while taking into account the spin degree of
freedom. Accordingly, we propose to employ the $3+1$ dimensional
representation of the Dirac equation. Hence in the next section, we review the
geometry of NC spacetime. The third section has been devoted to consider the
Dirac equation in the presence of an external uniform magnetic field in the NC
background. In the fourth section, we mainly investigate the effects of NC
geometry on the cyclotron and the ZBW motion of an electron in this case. We
study these effects via deriving the average velocity of charge carriers for a
general wave packet in the presence of generic magnetic fields, and then, for
weak magnetic fields with a specific initial condition as well. The summary
comes at the end. Meanwhile through the work, the lower-case Latin indices run
from one to three and the Greek indices run from zero to three.

##  2 Noncommutative Geometry

The NC geometry [63] particularly has interesting features on quantum
mechanics, and one of approaches toward quantum gravity theory is deformation
of the phase-space structure, see, e.g., Ref. [64]. In this geometry, the
fundamental commutators are generalized in the NC phase-space as

$[x_{i},x_{j}]=i\theta_{ij},\qquad\quad[p_{i},p_{j}]=i\beta_{ij}\qquad\quad{\rm
and}\qquad\quad[x_{i},p_{j}]=i\hbar\left(\delta_{ij}+\sigma_{ij}\right),$ (1)

where the NC parameters $\theta_{ij}$ and $\beta_{ij}$ are real constant
antisymmetric sectors with dimensions of length squared and momentum squared,
respectively, and $\sigma_{ij}$s (that can be written as a combination of
$\theta_{ij}$ and $\beta_{ij}$) are dimensionless symmetric constants. Using
the Levi-Civita antisymmetric tensor, one can write these parameters as

${\theta}_{k}\equiv{1\over 2}\,\varepsilon_{ijk}\,{\theta}_{ij}\qquad\quad{\rm
and}\qquad\quad{\beta}_{k}\equiv{1\over 2}\,\varepsilon_{ijk}\,{\beta}_{ij},$
(2)

where the parameters $\theta_{k}$ and ${\beta}_{k}$ are usually supposed to be
very small and being considered up to the first-order [37, 39]. Also, it is
known that the momentum noncommutativity plays the role of a vector potential
for a corresponding external NC uniform magnetic field, and indeed manifests
itself as a shift in it [39, 53, 55]. That is, the effect of the NC parameters
$\beta_{k}$ in the NC geometry is similar to the presence of a uniform
magnetic field in the usual space. Whereas, the space noncommutativity plays
an essential role when the gauge invariance of the Dirac equation matters
[53], which is the case in this work. Meanwhile, for the NC spacetime case of
$\theta_{0\mu}$, it has been shown that the corresponding theory is not
unitary [33].

In classical physics, noncommutativity is described by replacing the ordinary
product with the Moyal star-product between two arbitrary functions of the
phase-space variables as [44, 65]

$(f*g)(\zeta)=\exp\Bigl{[}\frac{i}{2}\alpha^{ab}\partial_{a}^{(1)}\partial_{b}^{(2)}\Bigr{]}f(\zeta^{(1)})g(\zeta^{(2)}){\Bigr{|}}_{\zeta^{(1)}=\zeta=\zeta^{(2)}},$
(3)

where $\zeta^{a}\equiv(x^{\prime i},p^{\prime j})$, $a,b=1,2,\cdots,2n$ in
$n$-dimensional space, and the symplectic structure is a real matrix as

$(\alpha_{ab})=\left(\begin{array}[]{cc}\theta_{ij}&\delta_{ij}+\sigma_{ij}\\\
-\delta_{ij}-\sigma_{ij}&\beta_{ij}\\\ \end{array}\right).$ (4)

However, to introduce such deformations via the Moyal product, one instead can
consider the non-canonical linear transformation

$x_{i}=x^{\prime}_{i}-{\theta_{ij}\over 2\hbar}p^{\prime}_{j}\qquad\quad{\rm
and}\qquad\quad p_{i}=p^{\prime}_{i}+{\beta_{ij}\over 2\hbar}x^{\prime}_{j},$
(5)

which sometimes is called the Bopp-shift method [32, 33, 40], on the usual
phase-space variables, which also gives
$\sigma_{ij}=-\theta_{k(i}\beta_{j)l}\delta^{kl}/4$. Such a transformation
allows an extension of the commutative space to the NC one. The main advantage
of this approach is that the Hamiltonian of system does not need any
modification. That is, when one changes the canonical variables through (5),
the Hamiltonian in the NC case is still assumed to have the same functional
form as in the commutative one, i.e.

$H^{\rm[NC]}\equiv H(x_{i},p_{j})=H\left(x^{\prime}_{i}-{\theta_{il}\over
2\hbar}p^{\prime}_{l},\,p^{\prime}_{j}+{\beta_{jk}\over
2\hbar}x^{\prime}_{k}\right).$ (6)

However, although this function is defined on the commutative space, but the
effects of NC parameters obviously arise through their equations of motion,
see, e.g., Refs. [41, 43, 66, 67].

##  3  External Magnetic Field in NC Background

The Dirac Hamiltonian in $3+1$ dimensions, in the natural units $c=1=\hbar$,
is

$H_{D}=\left(-i\,\gamma_{0}\gamma^{j}\partial_{j}+m\gamma_{0}\right),$ (7)

where $m$ is the rest mass of electron. Meanwhile, the matrices $\gamma_{\mu}$
are defined as444Also, in terms of the Dirac (gamma) matrices, one usually
defines $\alpha_{i}\equiv\gamma_{0}\gamma_{i}$.

$\displaystyle\gamma_{i}=\left(\begin{array}[]{cc}0&\sigma_{i}\\\
-\sigma_{i}&0\\\ \end{array}\right)\qquad\quad\ \textrm{and}\qquad\quad\
\gamma_{0}=\left(\begin{array}[]{cc}I&0\\\ 0&-I\\\ \end{array}\right),$ (12)

where $\sigma_{i}$ and $I$ are the $2\times 2$ Pauli matrices and the unit
matrix, respectively.

Now, consider the Dirac equation in the NC geometry of space sector (i.e.,
when $\beta_{ij}=0$) with $\theta_{0\mu}=0$, in the presence of an external
uniform magnetic field. Using the gauge invariant Dirac equation [52]

$i\,\partial_{t}\Psi=\left(H_{\mathrm{D}}+H_{\mathrm{SW}}^{[\theta]}\right)\Psi,$
(13)

where

$H_{\mathrm{SW}}^{[\theta]}\equiv\frac{e\,Q}{2}\Big{\\{}\left(\mathbf{E\times\mbox{\boldmath$\pi$}}\right)\cdot{\mbox{\boldmath$\theta$}}+\left[{\mbox{\boldmath$\theta$}}\times\left({{\mbox{\boldmath$\alpha$}}}\times\mathbf{B}\right)\right]\cdot\mathbf{\mbox{\boldmath$\pi$}}\Big{\\}}.$
(14)

This equation is gauge invariant under $U_{\lambda}\left(1\right)$ gauge
transformations. Also,
$\mathbf{\mbox{\boldmath$\pi$}}=\mathbf{\mbox{\boldmath$p$}}-eQ\mathchoice{\mbox{\boldmath$A$}}{\mbox{\boldmath$A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}$, where $e$ is the positive unit
of charge and $Q$ determines its sign, and the corresponding Dirac Hamiltonian
is

$H_{\mathrm{D}}=\mathchoice{\mbox{\boldmath$\alpha$}}{\mbox{\boldmath$\alpha$}}{\mbox{\boldmath$\scriptstyle\alpha$}}{\mbox{\boldmath$\scriptscriptstyle\alpha$}}\cdot\mathbf{\mbox{\boldmath$\pi$}}+m\gamma_{0}.$
(15)

We proceed with the case of a pure magnetic field, wherein Eq. (13) reduces to

$i\,\partial_{t}\Psi=\Big{\\{}H_{\mathrm{D}}+\frac{e\,Q}{2}\left[{\mbox{\boldmath$\theta$}}\times\left({{\mbox{\boldmath$\alpha$}}}\times\mathbf{B}\right)\right]\cdot\mathbf{\mbox{\boldmath$\pi$}}\Big{\\}}\,\Psi.$
(16)

As in Refs. [16]–[18], we consider the motion of an electron in a uniform
classical background magnetic field along the $z$-direction of the coordinate
axis. That is, we choose the background gauge field as

$A^{0}_{\rm B}=A^{y}_{\rm B}=A^{z}_{\rm B}=0\qquad\quad{\rm and}\qquad\quad
A^{x}_{\rm B}=-y{B}.$ (17)

Furthermore, in the case of using the gauge invariant Dirac equation, the
second term in Eq. (16) reads

$H_{\mathrm{SW}}^{[\theta]}=\frac{e\,QB}{2}\left[\alpha_{1}(\pi_{1}\theta_{3}-p_{3}\theta_{1})+\alpha_{2}(p_{2}\theta_{3}-p_{3}\theta_{2})\right].$
(18)

Therefore, although the Landau problem is known to be two dimensional case
[54], it seems that it cannot be in the NC background, when starting from the
gauge invariant Hamiltonian (16). Hence to simplify the problem, we choose
$p_{3}=0$ and indeed assume the momentum being in the plane perpendicular to
$z$-direction. We also take the external magnetic field being perpendicular to
the plane of momentum.

Taking these considerations into account, Eq. (16) can be solved using the
spinor ansatz

$\Psi=e^{-iEt}\left(\begin{array}[]{c}\zeta\\\ \xi\end{array}\right)$ (19)

with $\zeta$ and $\xi$ as 2-component objects, to get the coupled differential
equations

$\displaystyle(E-m)\zeta\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\eta\,\mathchoice{\mbox{\boldmath$\sigma$}}{\mbox{\boldmath$\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}\cdot(-i\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}-eQ\mathchoice{\mbox{\boldmath$A$}}{\mbox{\boldmath$A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}})\xi\,,$
$\displaystyle(E+m)\xi\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\eta\,\mathchoice{\mbox{\boldmath$\sigma$}}{\mbox{\boldmath$\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}\cdot(-i\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}-eQ\mathchoice{\mbox{\boldmath$A$}}{\mbox{\boldmath$A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}})\zeta,$ (20)

where $\eta\equiv 1+eQB\theta_{3}/2$ in the natural units.555When units are
recovered $\eta\equiv c\left(1+eQB\theta_{3}/(2\hbar)\right)$. With the choice
of vector potential (17), this equation is decoupled to

$(E^{2}-m^{2})\zeta=\eta^{2}\left[-\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}^{2}+(eQ{B})^{2}y^{2}-eQ{B}\left(2iy{\partial\over\partial
x}+\sigma_{3}\right)\right]\zeta.$ (21)

Accordingly, one can take the solution

$\zeta=e^{ip_{x}x+ip_{z}z}f(y)$ (22)

with $f(y)$ as a 2-component matrix, and $p_{x}$ and $p_{z}$ as the
eigenvalues of momentum.

Also, without loss of generality, we take $f(y)$ being the eigenstates of
$\sigma_{3}$ with the eigenvalues $s=\pm 1$, namely

$f^{[+]}(y)=\left(\begin{array}[]{c}F^{[+]}(y)\\\
0\end{array}\right)\qquad\quad{\rm and}\qquad\quad
f^{[-]}(y)=\left(\begin{array}[]{c}0\\\ F^{[-]}(y)\end{array}\right).$ (23)

Thus, Eq. (21) leads to

$a^{2}F^{[s]^{\prime\prime}}-\eta^{2}(eQ{B}y+p_{x})^{2}F^{[s]}+(E^{2}-m^{2}+\eta^{2}eQ{B}s)F^{[s]}=0,$
(24)

where prime denotes derivative with respect to the argument. With the aid of a
dimensionless variable

$\chi\equiv\sqrt{e|Q|{B}}\left(y+{p_{x}\over eQ{B}}\right),$ (25)

Eq. (24) becomes a special form of the Hermite equation, namely

$\left({d^{2}\over d\chi^{2}}-\chi^{2}+b^{[s]}\right)F^{[s]}=0,$ (26)

where

$b^{[s]}\equiv{(E^{2}-m^{2})/\eta^{2}+eQ{B}s\over e|Q|{B}}\,.$ (27)

If $b^{[s]}$ being odd integers, then the solution to this equation will
exist. Hence, for $b^{[s]}=2k+1$, with $k=0,1,2,\cdots$, it leads to the
energy eigenvalues

$E^{2}=m^{2}+\eta^{2}(2k+1)e|Q|{B}-\eta^{2}eQ{B}s$ (28)

and the solutions $F_{k}^{[s]}=C_{k}e^{-\chi^{2}/2}H_{k}(\chi)$, where $H_{k}$
is the Hermite polynomials and
$C_{k}\equiv\left[{\sqrt{e|Q|{B}}/(\sqrt{\pi}k!\,2^{k})}\,\right]^{1/2}$ is a
normalization factor. In addition, the functions $F_{k}^{[s]}$ satisfy the
completeness relation

$\int_{-\infty}^{\infty}F_{k}^{[s]}(\chi)F_{k^{\prime}}^{[s]}(\chi)d\chi=\sqrt{e|Q|{B}}\;\delta_{kk^{\prime}}.$
(29)

By concentrating on the case of electrons with $Q=-1$, the energy eigenvalues
(in the natural units) are

$E_{n}^{[e^{-}]}=\left(m^{2}+2ne\eta^{[e^{-}]2}B\right)^{\frac{1}{2}},$ (30)

which are the relativistic forms of the Landau energy-levels with
$\eta^{[e^{-}]}=1-eB\theta_{3}/2$. Let us compare the obtained spectrum with
the corresponding one (if any) in the literature. For simplicity, we perform
it for the lowest Landau level of (30) (i.e. for $n=1$) and up to the first
order on the space NC parameter, i.e.
$E_{1}^{[e^{-}]}\simeq\left(m^{2}+2eB-2e^{2}B^{2}\theta_{3}\right)^{\frac{1}{2}}$.
In this regard, we noticed two papers [38, 54] almost in this case, wherein
their results in this level are the same and (after changing their notations
and conventions to the ones used here) they obtained
$E_{1}^{[e^{-}]}\simeq\left(m^{2}+2eB+e^{2}B^{2}\theta_{3}\right)^{\frac{1}{2}}$.
It should be noted that the starting point of these two references is the non-
gauge invariant Hamiltonian, and hence, the difference in the energy spectrum
in this case is due to the gauge invariant action that we have employed.

Furthermore, as $\theta_{3}$ is very small,666For example, the upper limit on
$\theta_{3}$ has been indicated [33, 34] to be $\theta_{3}\lesssim
10^{-40}\,{\rm m}^{2}$, and/or a better bound $\theta_{3}\lesssim
10^{-30}\,{\rm m}^{2}$ [35]. the energy eigenvalues actually are

$E_{n}^{[e^{-}]}=\left[(m^{2}+2neB)-2ne^{2}B^{2}\theta_{3}\right]^{1/2}\simeq(m^{2}+2neB)^{1/2}-\frac{ne^{2}B^{2}\theta_{3}}{(m^{2}+2neB)^{1/2}}\,.$
(31)

However in general, the energy levels are two-fold degenerates, i.e. for $s=1$
with $n=k+1$, and for $s=-1$ with $n=k$. Representing the solutions related to
the $n$-th Landau-level by a subscript $n$, one can thus write solutions (23)
for the positive energy for spin-up and spin-down as

$f_{n}^{[+]}(\chi)=\left(\begin{array}[]{c}F_{n-1}(\chi)\\\
0\end{array}\right)\qquad\quad{\rm and}\qquad\quad
f_{n}^{[-]}(\chi)=\left(\begin{array}[]{c}0\\\ F_{n}(\chi)\end{array}\right),$
(32)

where $F_{-1}(\chi)=0=H_{-1}(\chi)$, and hence $n\geq 1$. Solutions (32)
determine the upper-component of the spinor through Eq. (22). The lower-
component, denoted by $\xi$, can be solved by substituting solution $\zeta$
into the second equation of Eqs. (3). Finally, the positive energy solutions
of the corresponding Dirac equations (3) are

$U_{E>0}^{[+]}(p_{x},y,n)=\left(\begin{array}[]{c}F_{n-1}(\chi)\\\\[8.61108pt]
0\\\\[8.61108pt] 0\\\\[8.61108pt]
-\,\eta^{[e^{-}]}{\textstyle\sqrt{2ne{B}}\over\textstyle
E_{n}^{[e^{-}]}+m}F_{n}(\chi)\end{array}\right)\qquad{\rm and}\qquad
U_{E>0}^{[-]}(p_{x},y,n)=\left(\begin{array}[]{c}0\\\\[8.61108pt]
F_{n}(\chi)\\\\[8.61108pt]
-\,\eta^{[e^{-}]}{\textstyle\sqrt{2ne{B}}\over\textstyle
E_{n}^{[e^{-}]}+m}F_{n-1}(\chi)\\\\[8.61108pt] 0\end{array}\right),$ (33)

where $U^{[\pm]}_{E>0}(p_{x},y,n)$ are related to solution (22) via solutions
(32) and the corresponding one for $\xi$.

For the case of positrons, which are positively charged with negative energy
solutions of the Dirac equation, one has to put $Q=1$. In this case, the
energy of the relativistic Landau-levels read

$E_{n}^{[e^{+}]}=-\left(m^{2}+2ne\eta^{[e^{+}]2}B\right)^{\frac{1}{2}}$ (34)

with $\eta^{[e^{+}]}=1+eB\theta_{3}/2$. Hence, it actually is

$E_{n}^{[e^{+}]}=-\left[(m^{2}+2neB)+2ne^{2}B^{2}\theta_{3}\right]^{1/2}\simeq-\left[(m^{2}+2neB)^{1/2}+\frac{ne^{2}B^{2}\theta_{3}}{(m^{2}+2neB)^{1/2}}\right]$
(35)

with two-fold degeneracy for $s=-1$ with $n=k+1$ and for $s=1$ with $n=k$. A
similar procedure (as used for solving the positive energy spinor) can be
adopted to solve for the negative energy spinor, and the solutions are

$U_{E<0}^{[+]}(p_{x},y,n)=\left(\begin{array}[]{c}0\\\\[8.61108pt]
\eta^{[e^{+}]}{\textstyle\sqrt{2ne{B}}\over\textstyle
E_{n}^{[e^{+}]}+m}F_{n}(\hat{\chi})\\\\[8.61108pt]
F_{n-1}(\hat{\chi})\\\\[8.61108pt] 0\end{array}\right)\qquad{\rm and}\qquad
U_{E<0}^{[-]}(p_{x},y,n)=\left(\begin{array}[]{c}\eta^{[e^{+}]}{\textstyle\sqrt{2ne{B}}\over\textstyle
E_{n}^{[e^{+}]}+m}F_{n-1}(\hat{\chi})\\\\[8.61108pt] 0\\\\[8.61108pt]
0\\\\[8.61108pt] F_{n}(\hat{\chi})\end{array}\right),$ (36)

where $U^{[\pm]}_{E<0}(p_{x},y,n)$ are also related to solution (22) and the
corresponding one for $\xi$, and $\hat{\chi}$ is obtained from $\chi$ by
changing the sign of the $p_{x}$-term in definition (25).

##  4  NC Effects on Cyclotron Motion and ZBW

In this section, we intend to investigate the effects of NC geometry on the
cyclotron and the ZBW motion of an electron in the presence of an external
uniform magnetic field. In this respect, we deal with the most general wave
packet, and calculate the average velocity of charge carriers. Then, in two
subsections, we consider these effects first with generic magnetic fields and
subsequently with weak magnetic fields limit while using a specific initial
condition.

In this regard, the solution of Eq. (16) can be written as a general wave
packet expanded in terms of momentum eigenfunctions via solutions (33) and
(36) as [8]

$\\!\\!\\!\\!\\!\Psi(x,y,z,t)=\int\sum_{\rm
n,s}c_{n}(p_{x},z)\Bigl{[}e^{-ip_{x}x}a_{s}U_{E>0}^{[s]}(p_{x},y,n)\exp(-i\omega_{n}^{[e^{-}]}t)+e^{ip_{x}x}a^{\prime}_{s}U_{E<0}^{[s]}(p_{x},y,n)\exp(i\omega_{n}^{[e^{+}]}t)\Bigr{]}d{p}_{x},$
(37)

where $\omega_{n}^{{[e^{\pm}]}}=|E_{n}^{[e^{\pm}]}|$, and $a_{s}$ and
$a^{\prime}_{s}$ are arbitrary coefficients. Also, following Ref. [27], we
choose
$c_{n}(p_{x},z)=\sqrt{1/(\pi\sqrt{\sigma}D_{p_{x}}\,D_{z})}\exp\left[\frac{-(p_{x}^{2}-p_{0_{x}}^{2})}{2D_{p_{x}}^{2}}-\frac{z^{2}}{2D_{z}^{2}}\right]\exp\left[\frac{-(n-n_{0})^{2}}{2\sigma}\right]$
with $D_{p_{x}}$ and $D_{z}$ respectively indicating the widths of wave packet
in the corresponding momentum and spatial coordinates, $\sigma$ is the
standard deviation, and the fixed values $p_{0_{x}}$ and $n_{0}$ are the
expectations of distribution of momentum and Landau-level, respectively.
Actually, solution $\Psi$ is a linear combination of the spin-up and spin-down
amplitudes of the free particle Dirac waves with positive/negative energies,
whose momentum is distributed around $p_{0_{x}}$.

Now, we derive the average of the velocity operator $\langle\dot{{\bf
r}}\rangle=\int\Psi^{\ast}(x,y,z,t)(c\mbox{\boldmath$\alpha$})\Psi(x,y,z,t)\,dxdydz$.
However, not to digress from the main issue, we put the results in the
Appendix. As it is obvious in those averages (i.e., Eqs. (55)), there are two
types of frequencies involved in these relations. To explore their nature and
find out how these are affected by the noncommutativity of space sector, we
look through the case when only the positive energy charge carriers are
involved. In this respect, after some manipulations, the average velocity
operators are

$\displaystyle\langle\dot{x}\rangle=\sqrt{2eB}\,\eta^{[e^{-}]}\\!\\!\int\\!\sum_{\rm
n=1}\Big{\\{}\\!\\!\\!\\!\\!\\!$ $\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!V_{n,n+1}K_{n}^{[e^{-}]}F_{n}^{2}(\chi)\sqrt{n}\cos{[(\omega_{n+1}^{[e^{-}]}-\omega_{n}^{[e^{-}]})t]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!V_{n,n-1}K_{n-1}^{[e^{-}]}F^{2}_{n-1}(\chi)\sqrt{(n-1)}\cos{[(\omega_{n}^{[e^{-}]}-\omega_{n-1}^{[e^{-}]})t]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!i\,V_{n,n+1}K_{n}^{[e^{-}]}F_{n}^{2}(\chi)\sqrt{n}\sin{[(\omega_{n+1}^{[e^{-}]}-\omega_{n}^{[e^{-}]})t]}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!iV_{n,n-1}K_{n-1}^{[e^{-}]}\,F_{n-1}^{2}(\chi)\sqrt{(n-1)}\sin{[(\omega_{n}^{[e^{-}]}-\omega_{n-1}^{[e^{-}]})t]}\Big{\\}}\,dy,$
(39) $\displaystyle\langle\dot{y}\rangle=\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{-}]}\\!\\!\int\\!\sum_{\rm
n=1}\Big{\\{}i\,V_{n,n+1}K_{n}^{[e^{-}]}F_{n}^{2}(\chi)\sqrt{n}\cos{[(\omega_{n+1}^{[e^{-}]}-\omega_{n}^{[e^{-}]})t]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!iV_{n,n-1}K_{n-1}^{[e^{-}]}\,F_{n-1}^{2}(\chi)\sqrt{(n-1)}\cos{[(\omega_{n}^{[e^{-}]}-\omega_{n-1}^{[e^{-}]})t]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!V_{n,n+1}K_{n}^{[e^{-}]}F_{n}^{2}(\chi)\sqrt{n}\sin{[(\omega_{n+1}^{[e^{-}]}-\omega_{n}^{[e^{-}]})t]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!V_{n,n-1}K_{n-1}^{[e^{-}]}F_{n-1}^{2}(\chi)\sqrt{(n-1)}\sin{[(\omega_{n}^{[e^{-}]}-\omega_{n-1}^{[e^{-}]})t]}\Big{\\}}\,dy,$

and $\langle\dot{z}\rangle=0$ as expected. However, after making a shift
$n\rightarrow n+1$ in the second and forth terms of each relation, the
imaginary terms cancel out. Then, using the completeness relation (29), these
relations read

$\displaystyle\langle\dot{x}\rangle\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-2\sqrt{2eB}\sum_{\rm
n=1}V_{n,n+1}\,\eta^{[e^{-}]}\\!K_{n}^{[e^{-}]}\sqrt{n}\cos{[(\omega_{n+1}^{[e^{-}]}-\omega_{n}^{[e^{-}]})t]},$
$\displaystyle\langle\dot{y}\rangle\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-2\sqrt{2eB}\sum_{\rm
n=1}V_{n,n+1}\,\eta^{[e^{-}]}\\!K_{n}^{[e^{-}]}\sqrt{n}\sin{[(\omega_{n+1}^{[e^{-}]}-\omega_{n}^{[e^{-}]})t]}.$
(41)

As it is expected, in this case, the frequencies, which govern the
oscillation, are the cyclotron (intraband) frequencies, i.e. $\omega^{[\rm
c]}_{n}\equiv\omega_{n+1}^{[e^{\pm}]}-\omega_{n}^{[e^{\pm}]}$, whereas the ZBW
(interband) frequencies, $\omega^{[\rm
zbw]}_{n}\equiv\omega_{n+1}^{[e^{\pm}]}+\omega_{n}^{[e^{\mp}]}$, are absent.
The result is similar to the case of graphene [26], wherein intraband
frequencies are due to the band quantization by the magnetic field and are
absent in the field-free situation. Whereas, the interband frequencies, which
appear due to the interference of the positive and negative energy states, are
the characteristic of ZBW.

Besides, in another work [55], we have shown that, in the absence of magnetic
field, neither the amplitude nor the frequency of the ZBW motion is affected
by the noncommutativity. However, as it is obvious, when an external uniform
magnetic field is introduced, the amplitude and both the cyclotron and the ZBW
frequencies are altered. In this case, let us find out estimations of the
order of the corrections caused due to the space noncommutativity on these
frequencies in the following subsections, where in the latter one, we also pay
more attention to the ZBW phenomenon itself.

###  4.1 Generic Magnetic Fields

In this short subsection, we just consider an external uniform generic
magnetic field and mainly focus only on the effects of the NC term. For this
purpose, using relations (31) and (35), and due to the smallness of the NC
parameter, we have

$\delta^{\rm[NC]}\omega_{n}^{[e^{\pm}]}\approx\pm\frac{2neB}{m}\delta\eta,$
(42)

where $\delta\eta\equiv eB\theta_{3}/2$ in the natural units. Consequently,
the NC effects on the cyclotron frequency is
$\delta^{\rm[NC]}\omega^{[c]}\approx\pm\frac{2eB}{m}\delta\eta$ (i.e.,
independent of the Landau-level), and on the ZBW frequencies are
$\delta^{\rm[NC]}\omega_{n}^{[\rm zbw]}\approx\pm
2(2n+1)\frac{eB}{m}\delta\eta$. For a generic magnetic field of about
$10\,{\rm Tesla}$, the corrections on all frequencies, when units are
recovered and with the upper bound $\theta_{3}\approx 10^{-30}\,{\rm m}^{2}$,
amount to the order of $\approx 10^{-1}\,{\rm s}^{-1}$, which is negligible
since

$\displaystyle\omega_{n}^{[\rm zbw]}+\delta^{\rm[NC]}\omega_{n}^{[\rm
zbw]}\\!\\!\\!\\!$ $\displaystyle\approx$
$\displaystyle\\!\\!\\!\\!\omega_{n}^{[\rm zbw]}\left(1\pm 10^{-22}\right),$
(43) $\displaystyle\omega_{n}^{[\rm c]}+\delta^{\rm[NC]}\omega_{n}^{[\rm
c]}\\!\\!\\!\\!$ $\displaystyle\approx$
$\displaystyle\\!\\!\\!\\!\omega_{n}^{[\rm c]}\left(1\pm 10^{-14}\right).$
(44)

However, for a very strong external magnetic field of order $10^{10}\,{\rm
Tesla}$, e.g., at the surface of neutron stars, the corrections on all
frequencies become of the order of $10^{17}\,{\rm s}^{-1}$, which though is
still small but noticeable, namely

$\displaystyle\omega_{n}^{[\rm zbw]}+\delta^{\rm[NC]}\omega_{n}^{[\rm
zbw]}\\!\\!\\!\\!$ $\displaystyle\approx$
$\displaystyle\\!\\!\\!\\!\omega_{n}^{[\rm zbw]}\left(1\pm 10^{-4}\right),$
(45) $\displaystyle\omega_{n}^{[\rm c]}+\delta^{\rm[NC]}\omega_{n}^{[\rm
c]}\\!\\!\\!\\!$ $\displaystyle\approx$
$\displaystyle\\!\\!\\!\\!\omega_{n}^{[\rm c]}\left(1\pm 10^{4}\right).$ (46)

This result may perhaps lead to a possible detection of the NC effects on the
electron dynamics in this situation. In addition, the amplitudes of the
average velocity oscillation are also affected by the noncommutativity of
space sector, whose amount, for a very strong external magnetic field of order
$10^{10}\,{\rm Tesla}$, is of the order $\delta\eta\approx 10^{3}\,{\rm
ms}^{-1}$.

###  4.2  Weak Magnetic Fields Limit

As we have indicated in the previous subsection, an interesting point is that
the NC effect on the cyclotron and the ZBW frequencies, and on the amplitude
of the oscillations, is magnetic field dependent. However, it is very small
even for a strong magnetic field. Thus, in this subsection, to simplify the
calculations and lending more attention to the ZBW phenomenon itself, we
choose to work in the weak magnetic fields limit. Besides, in this
approximation, as the NC effects are quite negligible, we also neglect the
corresponding terms in relations (31) and (35). Accordingly, in this limit, it
remains

$\omega_{n}^{[e^{\pm}]}\approx m+\frac{neB}{m}\equiv\omega_{n}.$ (47)

Therefore, the cyclotron frequency is independent of the Landau-level, i.e.
$\omega^{[c]}\approx eB/m$, as expected, and the ZBW frequencies are

$\omega_{n}^{[\rm zbw]}\approx 2m+(2n+1)\frac{eB}{m}.$ (48)

Furthermore, in the weak-field limit, one has $K_{n}^{[e^{\pm}]}\approx
1/(2m)\equiv K$ and $\eta^{[e^{\pm}]}\approx 1$. Also, we have neglected the
terms including $eBK_{n}^{2}$, in this limit.

To proceed while being more concrete, we choose a specific initial condition
at $t=0$ for solutions (33) that being a localized spin-up electron in the
$z$-direction in the form of a Gaussian wave packet in phase space, while its
center is at rest in the origin, namely

$\Psi(x,y,z,0)=\int\sum_{n=1}c_{n}(p_{x},z)F_{n-1}(\chi)\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ 0\\\ \end{array}\right)e^{{-i{p}_{x}x}}dp_{x}.$ (49)

Thus, in this case, the wave packet (37), which consists of both positive and
negative energies, reads

$\Psi(x,y,z,t)\simeq\int\sum_{n=1}c_{n}(p_{x},z)\left(\\!\\!\\!\\!\begin{array}[]{c}F_{n-1}(\chi)e^{-i\omega_{n}t}\\\\[8.61108pt]
0\\\\[8.61108pt] 0\\\\[8.61108pt]
\,K{\textstyle\sqrt{2ne{B}}}F_{n}(\chi)(e^{i\omega_{n}t}-e^{-i\omega_{n}t})\end{array}\\!\\!\\!\\!\right)e^{{-i{p}_{x}x}}dp_{x}.$
(50)

This rough solution satisfies the Dirac equation when the aforementioned
approximations are taken into account. Now, let us calculate the average
velocity of charge carriers, and get

$\displaystyle\langle\dot{x}\rangle\\!\\!\\!$ $\displaystyle\simeq$
$\displaystyle\\!\\!\\!2\sqrt{2eB}K\sum_{n=1}V_{n,n+1}\sqrt{n}\Big{[}\cos{(\omega_{n+1}+\omega_{n})t}-\cos{(\omega_{n+1}-\omega_{n})t}\Big{]}$
(51) $\displaystyle\simeq$ $\displaystyle\\!\\!\\!2\sqrt{2eB}K\sum_{\rm
n=1}V_{n,n+1}\sqrt{n}\Big{[}\cos(\omega_{n}^{[\rm zbw]}t)-\cos(\omega^{[\rm
c]}t)\Big{]}$

and

$\langle\dot{y}\rangle\simeq 2\sqrt{2eB}K\sum_{\rm
n=1}V_{n,n+1}\sqrt{n}\Big{[}\sin(\omega^{[\rm zbw]}_{n}t)-\sin(\omega^{[\rm
c]}t)\Big{]}.$ (52)

If we specify the initial wave packet in the same procedure, but for a
localized spin-down electron as

$\Psi(x,y,z,0)=\int\sum_{n=1}c_{n}(p_{x},z)F_{n}(\chi)\left(\begin{array}[]{c}0\\\
1\\\ 0\\\ 0\\\ \end{array}\right)e^{{-i{p}_{x}x}}dp_{x},$ (53)

then, we will obtain

$\displaystyle\langle\dot{x}\rangle\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!2\sqrt{2eB}K\sum_{\rm
n=1}V_{n,n-1}\sqrt{n}\Big{[}\cos(\omega^{[\rm zbw]}_{n}t)-\cos(\omega^{[\rm
c]}t)\Big{]},$ $\displaystyle\langle\dot{y}\rangle\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-2\sqrt{2eB}K\sum_{\rm
n=1}V_{n,n-1}\sqrt{n}\Big{[}\sin(\omega^{[\rm zbw]}_{n}t)+\sin(\omega^{[\rm
c]}t)\Big{]}.$ (54)

Figure 1: The trajectory of the average velocity of charge carriers, relations
(51) and (52) for the initially localized spin-up electron, has been depicted
only for Landau-level centered around $n=1$ with $\sigma=3$ and
$2\sqrt{2eB}K\equiv 1$. It consists of the cyclotron and the ZBW motions for
the first $10^{-6}$ seconds.

Figure 2: The trajectory of the average velocity of charge carriers, relations
(4.2) for the initially localized spin-down electron, has been depicted only
for Landau-level centered around $n=1$ with $\sigma=3$ and $2\sqrt{2eB}K\equiv
1$. It consists of the cyclotron and the ZBW motions for the first $10^{-6}$
seconds.

As it is obvious, in the both cases, both of the cyclotron and the ZBW
frequencies are involved. The trajectories of the average velocity of charge
carriers have been plotted in Figs. 1 and 2 for the both initially localized
spin-up and spin-down cases. Of course, the plots have been drawn only for
Landau-level centered around $n=1$ as the main part of the issue, with the
data of $\sigma=3$ and $2\sqrt{2eB}K\equiv 1$. The figures reveal difference
between the spin-up and spin-down cases. Besides, the ZBW phenomenon leads to
a circular motion whose direction is opposite for these two cases. Also, due
to the approximations used, the motions are unaffected from the NC parameter.
Interestingly, when electrons are prepared in the form of wave packets, with
$B\neq 0$, all the ZBW frequencies for the Landau-levels appear. Also, both of
the interband and intraband frequencies are at work. Furthermore, although in
the case at hand, the amplitude of the oscillation is proportional to the
external magnetic field, but it is already known [18] that one cannot simply
put $B=0$ and expect to get back to the free Dirac solutions.

##  5 Conclusions

The issue of dynamics of an electron in the presence of an external uniform
magnetic field in the NC background, in vacuum, has been considered. First,
the related $(3+1)$-dimensional Dirac equation, achieved from the gauge
invariant action, has been solved. Then, it has been shown that the solutions
and the energy spectrum are both affected by the NC effects of the space
sector. However, the change in the energy spectrum is affected is different
from previous results in Refs. [38, 54], where the non-gauge invariant
Hamiltonian has been employed. In this regard, the ZBW motion in the NC
background, without any external magnetic field, has been investigated [55]
before and it has been indicated that neither the frequency nor the amplitude
of the oscillation gets affected by the NC effects. Also in Ref. [38], the
change of the spectrum due to the NC effects has been mentioned however, the
solutions of the Dirac equation and dynamics of the electron have not been
considered in detail.

In this research, by using the extracted solutions of the Dirac equation, we
have calculated the average velocity of charge carriers first for a general
wave packet consisted of both positive and negative energies, and then, just
for a positive energy electron. An interesting result of these parts is that
both the amplitude and the frequency of the ZBW motion are affected by the NC
space parameter. Even for a positive energy electron, regardless of the ZBW
phenomenon, the space NC effects are present and manifest themselves as a
shift in the cyclotronic motion of electron, which, although very small, are
in principle detectable. By the way, as mentioned, since the momentum
noncommutativity leads to a NC uniform magnetic field, we have not considered
its effects in this work.

We have given some estimations for these kinds of shifts as well. Indeed, the
calculations suggest that the stronger the external magnetic field is (e.g.,
at the surface of neutron stars) the relatively larger are the shifts of the
corresponding frequencies. In this regard, the amplitude of the oscillation
also changes. These results may pave the way for a new class of experiments
that can probably exhibit the NC effects on the motion of an electron in a
very strong external uniform magnetic field. It should be emphasized that
these effects are absent in a field-free region.

To concentrate on the ZBW phenomenon, we have also calculated the average
velocity of charge carriers in weak magnetic fields limit for two initially
localized spin-up and spin-down cases. In these cases, due to the mentioned
approximations, although the NC effects have been neglected, the ZBW manifests
itself as a circular motion whose direction is spin dependent. This issue is
in line with studies which suggest that the spin of electron might be
generated by such a circular motion [8, 19, 55] in vacuum. We have also drawn
the plots of the trajectory of the average velocity of the charge carriers for
the corresponding Landau-levels centered around a specific level for the both
cases. The figures reveal some difference between the spin-up and spin-down
cases. Besides, the cyclotron frequency came out to have the same direction
for the both cases.

## Appendix

The averages of the components of the velocity operator are

$\displaystyle\langle\dot{x}\rangle\\!\\!\\!$ $\displaystyle=$ (55)
$\displaystyle\\!\\!\\!\\!\\!\\!\int\sum_{\rm
n,n^{\prime}=1}\\!\\!V_{n,n^{\prime}}\\!\Big{\\{}\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\\!\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\hat{\chi})+F_{n-1}(\chi)F_{n^{\prime}}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!2eB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n}^{[e^{+}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
(56) $\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!2eB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n}^{[e^{-}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\hat{\chi})+F_{n-1}(\chi)F_{n^{\prime}}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{-}]}K_{n}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\chi)+F_{n-1}(\chi)F_{n^{\prime}}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\eta^{[e^{+}]}K_{n}^{[e^{+}]}\Big{[}F_{n-1}(\hat{\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{-}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\chi)+F_{n-1}(\chi)F_{n^{\prime}}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n-1}(\hat{\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!i\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n}(\chi)-F_{n-1}(\hat{\chi})F_{n^{\prime}}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\Big{[}F_{n-1}({\chi})F_{n^{\prime}}(\hat{\chi})-F_{n^{\prime}-1}(\chi)F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
(60) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!2ieB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{-}]}K_{n}^{[e^{+}]}\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n}(\chi)-F_{n-1}(\hat{\chi})F_{n^{\prime}}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
(61) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!2ieB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n}^{[e^{-}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n-1}({\chi})F_{n^{\prime}}(\hat{\chi})-F_{n^{\prime}-1}(\chi)F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{-}]}K_{n}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\chi)+F_{n-1}(\chi)F_{n^{\prime}}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\sqrt{2eB}\,\eta^{[e^{+}]}K_{n}^{[e^{+}]}\Big{[}F_{n-1}(\hat{\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{-}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\chi)+F_{n-1}(\chi)F_{n^{\prime}}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\sqrt{2eB}\,\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n-1}(\hat{\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]\Big{\\}}\,dy,$
(64) $\displaystyle\langle\dot{y}\rangle\\!\\!\\!$ $\displaystyle=$ (65)
$\displaystyle\\!\\!\\!\\!\\!\\!\int\sum_{\rm
n,n^{\prime}=1}\\!\\!V_{n,n^{\prime}}\Big{\\{}-\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\Big{[}F_{n-1}({\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\chi)F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!2eB\eta^{[e^{+}]}\eta^{[e^{-}]}K_{n}^{[e^{+}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
(66) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!2eB\eta^{[e^{+}]}\eta^{[e^{-}]}K_{n}^{[e^{-}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n-1}({\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\chi)F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{-}]}K_{n}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\chi)-F_{n-1}(\chi)F_{n^{\prime}}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{+}]}K_{n}^{[e^{+}]}\Big{[}F_{n-1}(\hat{\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{-}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\chi)-F_{n-1}(\chi)F_{n^{\prime}}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n-1}(\hat{\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!i\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n}(\chi)-F_{n-1}(\hat{\chi})F_{n^{\prime}}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\Big{[}-F_{n-1}({\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\chi)F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
(70) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!2ieB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{-}]}K_{n}^{[e^{+}]}\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n}(\chi)-F_{n-1}(\hat{\chi})F_{n^{\prime}}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
(71) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!2ieB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n}^{[e^{-}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}-F_{n-1}({\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}-1}(\chi)F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{-}]}K_{n}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\chi)-F_{n-1}(\chi)F_{n^{\prime}}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{+}]}K_{n}^{[e^{+}]}\Big{[}F_{n-1}(\hat{\chi})F_{n^{\prime}}(\hat{\chi})-F_{n^{\prime}-1}(\hat{\chi})F_{n}(\hat{\chi}))\Big{]}\cos[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{-}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n}(\chi)-F_{n-1}(\chi)F_{n^{\prime}}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n-1}(\hat{\chi})F_{n^{\prime}}(\hat{\chi})-F_{n^{\prime}-1}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]\Big{\\}}\,dy,$
(74) $\displaystyle\langle\dot{z}\rangle\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\int\sum_{\rm
n,n^{\prime}=1}\\!\\!V_{n,n^{\prime}}\Big{\\{}\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n-1}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}-1}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!\Big{[}F_{n}({\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}}(\chi)F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!2eB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{-}]}K_{n}^{[e^{+}]}\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n-1}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}-1}(\chi)\Big{]}\cos[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!2eB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{+}]}K_{n}^{[e^{-}]}\Big{[}F_{n}({\chi})F_{n^{\prime}}(\hat{\chi})+F_{n^{\prime}}(\chi)F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{-}]}K_{n}^{[e^{-}]}\Big{[}-F_{n^{\prime}-1}(\chi)F_{n-1}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}-1}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{+}]}K_{n}^{[e^{+}]}\Big{[}F_{n}(\chi)F_{n^{\prime}}(\chi)-F_{n^{\prime}}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{-}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}-F_{n^{\prime}-1}(\chi)F_{n-1}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}-1}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sqrt{2eB}\,\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n}(\chi)F_{n^{\prime}}(\chi)-F_{n^{\prime}}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\cos[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!i\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n-1}(\chi)-F_{n-1}(\hat{\chi})F_{n^{\prime}-1}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\Big{[}F_{n}({\chi})F_{n^{\prime}}(\hat{\chi})-F_{n^{\prime}}(\chi)F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
(79) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!2ieB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n}^{[e^{+}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\hat{\chi})F_{n-1}(\chi)-F_{n-1}(\hat{\chi})F_{n^{\prime}-1}(\chi)\Big{]}\sin[(\omega_{n}^{[e^{+}]}+\omega_{n^{\prime}}^{[e^{-}]})t]$
(80) $\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!2ieB\eta^{[e^{-}]}\eta^{[e^{+}]}K_{n}^{[e^{-}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n}({\chi})F_{n^{\prime}}(\hat{\chi})-F_{n^{\prime}}(\chi)F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{-}]}+\omega_{n^{\prime}}^{[e^{+}]})t]$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{-}]}K_{n}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n-1}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}-1}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\eta^{[e^{+}]}K_{n}^{[e^{+}]}\Big{[}F_{n}(\chi)F_{n^{\prime}}(\chi)-F_{n^{\prime}}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]$
(82) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{-}]}K_{n^{\prime}}^{[e^{-}]}\Big{[}F_{n^{\prime}-1}(\chi)F_{n-1}(\chi)+F_{n-1}(\hat{\chi})F_{n^{\prime}-1}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{-}]}-\omega_{n^{\prime}}^{[e^{-}]})t]$
(83) $\displaystyle-$
$\displaystyle\\!\\!\\!\\!\\!i\,\sqrt{2eB}\,\eta^{[e^{+}]}K_{n^{\prime}}^{[e^{+}]}\Big{[}F_{n}(\chi)F_{n^{\prime}}(\chi)-F_{n^{\prime}}(\hat{\chi})F_{n}(\hat{\chi})\Big{]}\sin[(\omega_{n}^{[e^{+}]}-\omega_{n^{\prime}}^{[e^{+}]})t]\Big{\\}}\,dy,$
(84)

where $K_{n}^{[e^{\pm}]}\equiv 1/(E_{n}^{[e^{\pm}]}+m)\,$ and
$V_{n,n^{\prime}}\equiv\int
c_{n^{\prime}}(p_{x},z)c_{n}(p_{x},z)dp_{x}dz=\frac{1}{\sqrt{\sigma}}e^{\frac{-(n-n_{0})^{2}}{2\sigma}}e^{\frac{-(n^{\prime}-n_{0})^{2}}{2\sigma}}$.

## Acknowledgements

We thank the Research Council of Shahid Beheshti University for financial
assistance.

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