∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Departamento de Física, Universidade Federal do Maranhão,
Campus Universitário do Bacanga, 65080-805 São Luís, Maranhão, Brazil

# Quantum dynamics of a spin-1/2 charged particle in the presence of magnetic
field with scalar and vector couplings

Luis B. Castro e1  Edilberto O. Silva e2

(Received: date / Accepted: date)

###### Abstract

The quantum dynamics of a spin-1/2 charged particle in the presence of
magnetic field is analyzed for the general case where scalar and vector
couplings are considered. The energy spectra are explicitly computed for
different physical situations, as well as their dependencies on the magnetic
field strength, spin projection parameter and vector and scalar coupling
constants.

††journal: Eur. Phys. J. C††journal: Eur. Phys. J. C

##  1 Introduction

The study of relativistic quantum systems under the influence of magnetic
field and scalar potentials has attracted attention of researchers in various
branches of physics. It is well known that these potentials can be inserted
into the Dirac equation

$\left[\beta\boldsymbol{\gamma}\cdot\mathbf{p}+\beta
M\right]\psi\left(\mathbf{r}\right)=E\psi\left(\mathbf{r}\right),$ (1)

through three usual substitutions, known as minimal, vector and scalar
couplings, whose representation is denoted, respectively, by

$\displaystyle\mathbf{p}$ $\displaystyle\rightarrow$
$\displaystyle\mathbf{p}-e\mathbf{A,}$ (2) $\displaystyle E$
$\displaystyle\rightarrow$ $\displaystyle E-V\left(r\right),$ (3)
$\displaystyle M$ $\displaystyle\rightarrow$ $\displaystyle
M+S\left(r\right).$ (4)

a variety of relativistic and nonrelativistic effects can be studied. Those
couplings above differ in the manner how they are inserted into the Dirac
equation greiner.rqm.wf . The minimal coupling (2) is useful for studying the
dynamics of a spin-1/2 charged particle in a magnetic field. For example,
using this model, we can study Landau levels Book.1981.Landau , Aharonov-Bohm
effect PR.1959.115.485 , Hall effect AJM.1879.2.287 , and other effects
associated with magnetic field.

It is known that the prescription (3) acts differently on electron and
positron states, respectively, and the eigenvalue spectrum of the particle is
not symmetric. In this case, bound states exist for only one of the two kinds
of particles. In other words, we can say that, for vector coupling, the
potential couples to the charge. In the context of the Dirac equation, this
coupling has been used, for example, to study the influence of a harmonic
oscillator on the Aharonov-Casher problem EPJC.2013.73.2402 , the Aharonov
Bohm effect for a spin-1/2 particle in the case that a $1/r$ potential is
present AoP.1996.251.45 , effects of nongauge potentials on the spin-1/2
Aharonoh-Bohm problem PRD.1993.48.5935 , quasiclassical theory of the Dirac
equation with applications in the physics of heavy-light mesons
PRD.2011.83.076003  and confining potentials with pure vector coupling
PRL.2008.101.190401 . In the Schrödinger theory, It also has important
applications, such as, in the dynamics of an electron in a two-dimensional
quantum ring PRB.1996.53.6947 ; JMP.2012.53.023514 , quantum particles
constrained to move on a conical surface JMP.2012.53.122106  and effect of
singular potentials on the harmonic oscillator AoP.2010.325.2529 .

In the case of the scalar coupling (4), it is added to the mass term of the
Dirac equation and, therefore, it can be interpreted as an effective,
position-dependent mass and, furthermore, it also acts equally on particles
and antiparticles. This coupling has been used, for example, to obtain an
exact solution of the Dirac equation for a charged particle with position-
dependent mass in the Coulomb field PLA.2004.322.72 , to study the
relativistic quantum dynamics of a charged particle in cosmic string spacetime
EPJC.2012.72.2051 ; CQG.1996.13.L41 , scattering of a fermion in the
background of a smooth step potential with a general mixing of vector and
scalar Lorentz structures with the scalar coupling stronger than or equal to
the vector coupling AoP.2014.346.164 , inclusion of the generalized Hulthén
potential in the case of the smooth step mass distribution PLA.2006.352.478
and extension of PT-symmetric quantum mechanics AoP.2008.323.566 . The
coupling (4) also has important applications in nonrelativistic quantum
mechanics. The Schrödinger equation with a position-dependent mass has
attracted a lot of attentions due to a wide range of applications in various
areas of material science and condensed matter physics. For example, to study
the dynamics of an one-dimensional harmonic oscillator CTP.2014.62.790 ,
derivation of the Shannon entropy for a particle with a nonuniform solitonic
mass density AoP.2014.348.153 , context of displacement operator for quantum
systems PRA.2012.85.034102 ; PRA.2014.89.049904 , use of instantaneous
Galilean invariance to derive the expression for the Hamiltonian of an
electron PRA.1995.52.1845 , determination of some potential functions for
exactly solvable nonrelativistic problems PRA.2002.66.042116  and Hermitian,
rotationally invariant one-band Schrödinger Hamiltonian PRB.1999.59.7596 .

The case in which the couplings are composed by a vector (3) and a scalar (4)
potentials, with $S=V$ ($S=-V$), are usually pointed out as necessary
condition for occurrence of exact spin (pseudospin) symmetry. It is known that
the spin and pseudospin symmetries are SU(2) symmetries of a Dirac Hamiltonian
with vector and scalar potentials. The pseudospin symmetry was introduced in
nuclear physics many years ago PLB.1969.30.517 ; NPA.1969.137.129  to account
for the degeneracies of orbital in single-particle spectra. Also, it is known
that the spin symmetry occurs in the spectrum of a meson with one heavy quark
PRL.2001.86.204  and anti-nucleon bound in a nucleus PR.1999.315.231 , and the
pseudospin symmetry occurs in the spectrum of nuclei PRL.1997.78.436 .

In this work, we study the quantum dynamics of a spin-1/2 charged particle in
the presence of magnetic field with scalar and vector couplings. This system
has been considered in Ref. AoP.2014.341.153 . The difference between our
approach and the previous one is that, here, we solve the problem in a
rigorous way taking into account other questions which have not been analyzed
by the authors. For example, absence of the term which depends explicitly on
the spin in the equation of motion. Since we are considering the dynamics of a
particle with spin, such a term can not be neglected in equation of motion
PRL.1990.64.503 . Moreover, as the authors make a connection with the
Aharonov-Bohm problem, the presence of this term has important implications on
the physical quantities of interest, such as energy eigenvalues, scattering
matrix and phase shift (see Ref. AoP.2013.339.510  for more details). By
taking into account the term that depends explicitly on the spin into Pauli
equation, we address the system in connection with the spin-1/2 Aharonov-Bohm
problem IJMPA.1991.6.3119  and analyze the following questions: (a) the
existence of isolated solutions to the first order equation Dirac and (b)
general dynamics in all space, including the $r=0$ region. We use the self-
adjoint extension method to determine the most relevant physical quantities,
such as energy spectrum and wave functions by applying boundary conditions
allowed by the system.

The paper is organized as follows. In section 2, we consider the Dirac
equation in $(2+1)$ dimensions with minimal, scalar and vector couplings, and
derive the set of first order differential equations. These equations are
useful to investigate possible isolated solutions to the problem. In section
3, we solve the first order Dirac equation in connection with the Aharanov-
Bohm problem and scalar and vector couplings. We found that, for certain
values assumed by the physical parameters of the system, isolated solutions
exist and discuss the limits of validity of them. In section 4, we derive the
Pauli equation and study the dynamics of the system taking into account exact
symmetry spin and pseudospin limits. In section 5, we briefly discuss some
concepts of the self-adjoint extension method and specify the boundary
conditions at the origin which will be used. In section 6, expressions for the
energy eigenvalues and wave functions are determined for both symmetry limits
and compare them with the results of Ref. AoP.2014.341.153 . We verify that
the presence of the spin element in the equation of motion introduces a
correction in the expressions for the bound state energy eigenvalues. In
section 7, we present our concluding remarks.

##  2 Equation of motion

We begin with the Dirac equation (1) in $(2+1)$ dimensions in polar
coordinates ($\hbar=c=1$)

$\left\\{\beta\boldsymbol{\gamma}\cdot\mathbf{\pi}+\beta\left[M+S\left(r\right)\right]\right\\}\psi\left(\mathbf{r}\right)=\left[E-V\left(r\right)\right]\psi\left(\mathbf{r}\right),$
(5)

where
$\mathbf{\pi}=\left(\pi_{r},\pi_{\varphi}\right)=(-i\partial_{r},-i\partial_{\varphi}/r-eA_{\varphi})$,
$\mathbf{r}=(r,\varphi)$ and $\psi$ is a two-component spinor. The
$\boldsymbol{\gamma}$ matrices in Eq. (5) are given in terms of the Pauli
matrices as EPJC.2014.74.3187

$\displaystyle\beta\gamma^{r}$
$\displaystyle=\sigma_{1}\cos\varphi+s\sigma_{2}\sin\varphi=\left(\begin{array}[]{cc}0&e^{-is\varphi}\\\
e^{+is\varphi}&0\end{array}\right),$ (8) $\displaystyle\beta\gamma^{\varphi}$
$\displaystyle=-\sigma_{1}\cos\varphi+s\sigma_{2}\sin\varphi=\left(\begin{array}[]{cc}0&-ise^{-is\varphi}\\\
ise^{+is\varphi}&0\end{array}\right),$ (11) $\displaystyle\beta$
$\displaystyle=\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$ (14)

where $s$ is twice the spin value, with $s=+1$ for spin “up” and $s=-1$ for
spin “down”. Equation (5) can be written more explicitly as

$\displaystyle e^{-is\varphi}\left[\pi_{r}-is\pi_{\varphi}\right]\psi_{2}$
$\displaystyle=$ $\displaystyle\left[E-M-\Sigma\left(r\right)\right]\psi_{1},$
(15) $\displaystyle
e^{+is\varphi}\left[\pi_{r}+is\pi_{\varphi}\right]\psi_{1}$ $\displaystyle=$
$\displaystyle\left[E+M-\Delta\left(r\right)\right]\psi_{2},$ (16)

where $\Sigma\left(r\right)=V\left(r\right)+S\left(r\right)$ and
$\Delta\left(r\right)=V\left(r\right)-S\left(r\right)$.

If one adopts the following decomposition

$\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)=\left(\begin{array}[]{c}\sum\limits_{m}f_{m}(r)\;e^{im\varphi}\\\
i\sum\limits_{m}g_{m}(r)\;e^{i(m+s)\varphi}\end{array}\right),$ (17)

with $m+1/2=\pm 1/2,\pm 3/2,\ldots$, with $m\in\mathbb{Z}$, and inserting this
into Eqs. (15) and (16), we obtain

$\displaystyle\left[\frac{d}{dr}+\frac{s\left(m+s\right)}{r}-esA_{\varphi}\right]g\left(r\right)$
$\displaystyle=$
$\displaystyle\left[E-M-\Sigma\left(r\right)\right]f\left(r\right),$ (18)
$\displaystyle\left[-\frac{d}{dr}+\frac{sm}{r}-esA_{\varphi}\right]f\left(r\right)$
$\displaystyle=$
$\displaystyle\left[E+M-\Delta\left(r\right)\right]g\left(r\right).$ (19)

Note that the above equations are coupled. However, if $\Sigma\left(r\right)$
or $\Delta\left(r\right)$ is made zero in any of the equations, we can
uncouple them easily. We will see below that this result in important physical
consequences to the physical system in question.

##  3 Isolated solutions for the Dirac equation of motion

In this section, we investigate the existence of isolated solutions in the
quantum motion of a fermionic massive charged particle in $(2+1)$ dimensions.
This is accomplished by considering the particle at rest, i.e., $E=\pm M$,
directly in the first order equations in Eqs. (18) and (19). Such a solution
is known to be excluded from the Sturm-Liouville problem, and have been
investigated under diverse perspectives in the latest years EPL.2014.108.30003
; AoP.2013.338.278 ; PS.2008.77.045007 ; IJMPE.2007.16.2998 ;
IJMPE.2007.16.3002 ; JPA.2007.40.263 ; PS.2007.75.170 ; PLA.2006.351.379 . We
are seeking for bound-state solutions subjected to the normalization condition

$\int_{0}^{\infty}\left(|f_{m}(r)|^{2}+|g_{m}(r)|^{2}\right)rdr=1.$ (20)

In order to determine the isolated bound-state solutions, we consider
$\Sigma\left(r\right)=0$ in Eq. (18), so that, for $E=M$, we can write

$\displaystyle\left[\frac{d}{dr}+\frac{s(m+s)}{r}-seA_{\varphi}\right]g_{m}(r)=0,$
(21)
$\displaystyle\left[-\frac{d}{dr}+\frac{sm}{r}-seA_{\varphi}\right]f_{m}(r)=2\left(M-V\left(r\right)\right)g_{m}(r),$
(22)

whose general solutions are

$\displaystyle g_{m}(r)$ $\displaystyle=$ $\displaystyle
a_{+}r^{-s\left(m+s\right)}e^{se\int A_{\varphi}dr},$ (23) $\displaystyle
f_{m}(r)$ $\displaystyle=$
$\displaystyle\left[b_{+}-a_{+}I(r)\right]r^{sm}e^{-se\int A_{\varphi}dr},$
(24)

where $a_{+}$ and $b_{+}$ are constants, and $I(r)$ is given by

$I(r)=\int dr\left[2M-2V\left(r\right)\right]e^{2se\int
A_{\varphi}\left(r\right)dr},$ (25)

which, for a given $V\left(r\right)$ and $A_{\varphi}\left(r\right)$, it can
be expressed in terms of the upper incomplete Gamma function
Book.1972.Abramowitz

$\Gamma(a,x)=\int_{x}^{\infty}t^{a-1}e^{-t}dt,\qquad\Re(a)>0.$ (26)

Let us now analyze solutions for $E=-M$, and consider $\Delta\left(r\right)=0$
in Eq. (19). For this case, we write

$\displaystyle\left[\frac{d}{dr}+\frac{s\left(m+s\right)}{r}-esA_{\varphi}\left(r\right)\right]g\left(r\right)=-2\left[M+V\left(r\right)\right]f\left(r\right),$
(27)
$\displaystyle\left[-\frac{d}{dr}+\frac{sm}{r}-esA_{\varphi}\left(r\right)\right]f\left(r\right)=0,$
(28)

whose general solution is

$\displaystyle f_{m}(r)$ $\displaystyle=$ $\displaystyle a_{-}r^{sm}e^{-se\int
A_{\varphi}dr},$ (29) $\displaystyle g_{m}(r)$ $\displaystyle=$
$\displaystyle\left[b_{-}-a_{-}H(r)\right]r^{-s\left(m+s\right)}e^{se\int
A_{\varphi}dr},$ (30)

where

$H(r)=\int dr\left[2M-2V\left(r\right)\right]e^{-2se\int A_{\varphi}dr}.$ (31)

Now, lets us consider the particular case where the particle moves in a
constant magnetic field and in the presence of Aharonov-Bohm effect. The
vector potential in the Coulomb gauge is

$\mathbf{A=A}_{1}+\mathbf{A}_{2},$ (32)

with

$\mathbf{A}_{1}=\frac{B_{0}r}{2}\boldsymbol{\hat{\varphi}},~{}~{}~{}\mathbf{A}_{2}=\frac{\phi}{r}\boldsymbol{\hat{\varphi}},$
(33)

where $B_{0}$ is the magnetic field magnitude and $\phi$ is the flux
parameter. The potentials in Eq. (32) both provide one magnetic field
perpendicular to the plane $\left(r,\varphi\right)$, namely

$\mathbf{B}=\mathbf{B}_{1}+\mathbf{B}_{2},$ (34)

with

$\displaystyle\mathbf{B}_{1}$ $\displaystyle=$
$\displaystyle\mathbf{\nabla}\times\mathbf{A}_{1}=B_{0}\mathbf{\hat{z}},$ (35)
$\displaystyle\mathbf{B}_{2}$ $\displaystyle=$
$\displaystyle\mathbf{\nabla}\times\mathbf{A}_{2}=\phi\frac{\delta(r)}{r}\mathbf{\hat{z}},$
(36)

where $\mathbf{B}_{1}$is an external magnetic field and $\mathbf{B}_{2}$ is
the magnetic field due to a solenoid. If the solenoid is extremely long, the
field inside is uniform, and the field outside is zero. However, in a general
dynamics, the particle is allowed to access the $r=0$ region. In this region,
the magnetic field is non-null. If the radius of the solenoid is $r_{0}\approx
0$, then the relevant magnetic field is $\mathbf{B}_{2}\sim\delta(r)$ as in
Eq. (36). This situation has not been accomplished in Ref. AoP.2014.341.153 ,
which is crucial to give meaning to the term that explicitly depends of the
spin, namely, the Pauli term that appearing in the second order differential
equation. This issue will be considered later when we treat solutions for the
case $E\neq\pm M$.

Using Eqs. (35) and (36), we have

$\int A_{\varphi}dr=\frac{B_{0}r^{2}}{4}+\phi\ln r.$ (37)

If $\phi>0$ and $B_{0}>0$, for $E=M$, we have bound-state solutions only in
the following cases:

$\displaystyle\left(\begin{array}[]{c}f_{m}(r)\\\ g_{m}(r)\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{c}1\\\
0\end{array}\right)b_{+}r^{m-\lambda}e^{-\delta
r^{2}},~{}\Bigg{\\{}\begin{tabular}[]{l} $s=+1,$\\\ $a_{+}=0,$\end{tabular},$
(44) $\displaystyle\left(\begin{array}[]{c}f_{m}(r)\\\
g_{m}(r)\end{array}\right)$ $\displaystyle=\left(\begin{array}[]{c}0\\\
1\end{array}\right)a_{+}r^{m-\lambda-1}e^{-\delta
r^{2}},~{}\Bigg{\\{}\begin{tabular}[]{l} $s=-1,$\\\
$b_{+}=M=V=0,$\end{tabular}$ (51)

and for $E=-M$,

$\displaystyle\left(\begin{array}[]{c}f_{m}(r)\\\ g_{m}(r)\end{array}\right)$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}0\\\
1\end{array}\right)b_{-}r^{m-\lambda-1}e^{-\delta
r^{2}},~{}\Bigg{\\{}\begin{tabular}[]{l}$s=-1,$\\\ $a_{-}=0,$\end{tabular}$
(58) $\displaystyle\left(\begin{array}[]{c}f_{m}(r)\\\
g_{m}(r)\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
0\end{array}\right)a_{-}r^{m-\lambda}e^{-\delta
r^{2}},~{}\Bigg{\\{}\begin{tabular}[]{l} $s=+1,$\\\
$b_{+}=M=V=0,$\end{tabular}$ (65)

where $\delta=eB_{0}/4$ and $\lambda=e\phi$. Note that the above results are
independent of the values of $s$, $m$ and $\lambda$ to ensure a bound state.
This is because the function $e^{-\delta r^{2}}$ predominates over the
polynomials $r^{m-\lambda}$ and $r^{m-\lambda-1}$. If we consider
$B_{0}=V\left(r\right)=0$, which leads to the usual Aharonov-Bohm effect, the
solutions for $E=M$ reads

$\displaystyle g_{m}(r)$ $\displaystyle=$ $\displaystyle
a_{+}r^{s\left[\lambda-\left(m+s\right)\right]},\text{ }$ (66) $\displaystyle
f_{m}(r)$ $\displaystyle=$
$\displaystyle\left[b_{+}-a_{+}\tilde{I}(r)\right]r^{s\left(m-\lambda\right)},$
(67)

where

$\tilde{I}(r)=\frac{2M}{2s\lambda+1}r^{2s\lambda+1}\,.$ (68)

For $E=-M$, we get

$\displaystyle f_{m}(r)$ $\displaystyle=a_{-}r^{s\left(m-\lambda\right)},$
(69) $\displaystyle g_{m}(r)$
$\displaystyle=\left[b_{-}-a_{-}\tilde{H}(r)\right]r^{s\left[\lambda-\left(m+s\right)\right]},$
(70)

where

$\tilde{H}(r)=\frac{2M}{-2s\lambda+1}r^{-2s\lambda+1},$ (71)

Unlike from cases of Eqs. (44), (51), (58) and (65), if we impose that $B_{0}$
and $V\left(r\right)$ are zero, there no exist bound-state solutions of
square-integrable. In other words, for any values of $s$, $m$ and $\lambda$ in
Eqs. (66)-(71), the integral (20) diverges.

##  4 Equation of motion and analysis of symmetries

In this section, we investigate the dynamics for $E\neq\pm M$. For this, we
choose to work with Eq. (5) in its quadratic form. After application of the
operator

$\beta\left[\left(M+S\left(r\right)\right)+\beta\left(E-V\left(r\right)\right)+\boldsymbol{\gamma}\cdot\boldsymbol{\pi}\right],$
(72)

we get

$\displaystyle\left\\{\mathbf{p}^{2}-2e\left[\left(\mathbf{A}_{1}+\mathbf{A}_{2}\right)\cdot\mathbf{p}\right]+e^{2}\left(\mathbf{A}_{1}+\mathbf{A}_{2}\right)^{2}\right\\}\psi\left(\mathbf{r}\right)$
$\displaystyle+\left\\{\left[M+S\left(r\right)\right]^{2}-\left[E-V\left(r\right)\right]^{2}-es\boldsymbol{\sigma}\cdot\left(\mathbf{B}_{1}+\mathbf{B}_{2}\right)\right\\}\psi\left(\mathbf{r}\right)$
$\displaystyle-\left(\frac{\partial S\left(r\right)}{\partial
r}\sigma_{2}+i\frac{\partial V\left(r\right)}{\partial
r}\sigma_{1}\right)\psi\left(\mathbf{r}\right)=0.$ (73)

In this stage, it is worthwhile to mention that the Eq. (73) is the correct
quadratic form of the Dirac equation with minimal, vector and scalar
couplings, because the Pauli term is considered.

###  4.1 Exact spin symmetry limit: $S=V$

The condition for establishing the exact symmetry boundary implies that the
solution is of the form

$\psi_{1}=\sum\limits_{m}f_{m}(r)\;e^{im\varphi}.$ (74)

So, by making $S=V$ (or equivalently $\Delta\left(r\right)=0$ PRC.1999.59.154
; PRC.1998.58.R628 ) in Eq. (16) and using the solution (74) in Eq. (73), the
equation for $f_{m}\left(r\right)$ is found to be

$\displaystyle\left[-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}+\frac{m^{2}}{r^{2}}-2e\left(\frac{B_{0}r}{2}+\frac{\phi}{r}\right)\frac{m}{r}+\frac{e^{2}B_{0}^{2}r^{2}}{4}\right]f_{m}\left(r\right)$
$\displaystyle+\left[\frac{e^{2}\phi^{2}}{r^{2}}+e^{2}B_{0}\phi-
es\left(B_{0}+\phi\frac{\delta(r)}{r}\right)\right]f_{m}\left(r\right)$
$\displaystyle+\left[M^{2}-E^{2}+2\left(E+M\right)V\right]f_{m}\left(r\right)=0.$
(75)

Assuming $V\left(r\right)$ as in Ref. AoP.2014.341.153 , i.e., of the form

$V\left(r\right)=a\,r^{2}+\frac{b}{r^{2}},$ (76)

Eq. (75) becomes

$Hf_{m}\left(r\right)=k^{2}f_{m}\left(r\right),$ (77)

with

$H=H_{0}-es\phi\frac{\delta(r)}{r},$ (78)
$H_{0}=-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}+\frac{\nu^{2}}{r^{2}}+\eta^{2}\text{
}r^{2},$ (79)

where

$\displaystyle\nu^{2}$ $\displaystyle=$
$\displaystyle\left(m-e\phi\right)^{2}+2b\left(E+M\right),$ (80)
$\displaystyle\eta^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{2}B_{0}^{2}}{4}+2a\left(E+M\right),$ (81)
$\displaystyle k^{2}$ $\displaystyle=$ $\displaystyle
meB_{0}-e^{2}B_{0}\phi+esB_{0}+\left(E^{2}-M^{2}\right).$ (82)

As pointed out in Ref. AoP.2014.341.153 , the potential $V(r)$ in Eq. (76)
describes an anharmonic oscillator. This model is a particular case of a
proposed in Ref. PRB.1996.53.6947  to study the Landau quantization and the
Aharonov-Bohm effect in a two-dimensional ring as an exactly soluble model.
The model considered in Ref. PRB.1996.53.6947  has an advantage because,
besides the model considered here, it also describes other physical systems,
such as a one-dimensional ring, a straight 2D wire, a single quantum dot for
and an isolated antidot.

###  4.2 Exact pseudospin symmetry limit: $S=-V$

In this case, the condition for establishing the exact pseudospin symmetry
limit implies that the resolution is related to the down component of the
spinor in Eq. (17), namely

$\psi_{2}=i\sum\limits_{m}g_{m}(r)\;e^{i(m+s)\varphi}.$ (83)

By making $S=-V$ (or equivalently $\Sigma\left(r\right)=0$ in Eq. (15) and
again using Eq. (83) in Eq. (73), the equation for $g_{m}\left(r\right)$ can
be found

$\tilde{H}g_{m}\left(r\right)=\tilde{k}^{2}g_{m}\left(r\right),$ (84)

with

$\tilde{H}=\tilde{H}_{0}-es\phi\frac{\delta(r)}{r},$ (85)
$H_{0}=-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}+\frac{\tilde{\nu}^{2}}{r^{2}}+\tilde{\eta}^{2}\text{
}r^{2},$ (86)

where

$\displaystyle\tilde{\nu}^{2}$ $\displaystyle=$
$\displaystyle\left(m+s-e\phi\right)^{2}+2b\left(E-M\right),$ (87)
$\displaystyle\tilde{\eta}^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{2}B_{0}^{2}}{4}+2a\left(E-M\right),$ (88)
$\displaystyle k^{2}$ $\displaystyle=$
$\displaystyle\left(m+s\right)eB_{0}-e^{2}B_{0}\phi+esB_{0}-\left(M^{2}-E^{2}\right).$
(89)

##  5 Self-adjoint extension analysis

In this section, we review some concepts on the self-adjoint extension
approach. An operator $\mathcal{O}$, with domain $\mathcal{D}({\mathcal{O}})$,
is said to be self-adjoint if and only if $\mathcal{O}=\mathcal{O}^{\dagger}$
and $\mathcal{D}(\mathcal{O})=\mathcal{D}(\mathcal{O}^{\dagger})$,
$\mathcal{O}^{\dagger}$ being the adjoint of operator $\mathcal{O}$. For
smooth functions, $\xi\in C_{0}^{\infty}(\mathbb{R}^{2})$ with $\xi(0)=0$, we
should have $H\xi=H_{0}\xi$, and it is possible to interpret the Hamiltonian
$H_{0}$ (79) as a self-adjoint extension of
$H_{0}|_{C_{0}^{\infty}(\mathbb{R}^{2}/\\{0\\})}$ crll.1987.380.87 ;
JMP.1998.39.47 ; LMP.1998.43.43 . The self-adjoint extension approach
consists, essentially, in extending the domain of $\mathcal{D}(\mathcal{O})$
in order to match $\mathcal{D}(\mathcal{O}^{\dagger})$. From the theory of
symmetric operators, it is a well-known fact that the symmetric radial
operator $H_{0}$ is essentially self-adjoint for $\nu\geq 1$, while for
$\nu<1$, it admits an one-parameter family of self-adjoint extensions
Book.1975.Reed.II , $H_{0,\lambda_{m}}$, where $\lambda_{m}$ is the self-
adjoint extension parameter. To characterize this family, we will use the
approach in JMP.1985.26.2520 ; Book.2004.Albeverio , which is based in a
boundary conditions at the origin. All the self-adjoint extensions
$H_{0,\lambda_{m}}$ of $H_{0}$ are parameterized by the boundary condition at
the origin

$\Psi_{0}=\lambda_{m}\Psi_{1},$ (90)

with

$\displaystyle\Psi_{0}={}$ $\displaystyle\lim_{r\rightarrow
0^{+}}r^{\nu}f_{m}(r),$ (91) $\displaystyle\Psi_{1}={}$
$\displaystyle\lim_{r\rightarrow
0^{+}}\frac{1}{r^{\nu}}\left[f_{m}(r)-\Psi_{0}\frac{1}{r^{\nu}}\right],$ (92)

where $\lambda_{m}\in\mathbb{R}$ is the self-adjoint extension parameter. For
$\lambda_{m}=0$, we have the free Hamiltonian (without the $\delta$ function)
with regular wave functions at the origin, and for $\lambda_{m}\neq 0$ the
boundary condition in Eq. (90) permit an $r^{-\nu}$ singularity in the wave
functions at the origin.

##  6 The bound state energy and wave function

In this section, we determine the energy spectrum by solving Eq. (77). For
$r\neq 0$, the equation for the component $f_{m}(r)$ can be transformed by the
variable change $\rho=\eta r^{2}$ resulting in

$\rho
f_{m}^{\prime\prime}(\rho)+f_{m}^{\prime}(\rho)-\left(\frac{{\nu}^{2}}{4\rho}+\frac{\rho}{4}-\frac{k^{2}}{4\eta}\right)f_{m}(\rho)=0,$
(93)

Due to the boundary condition in Eq. (90), we seek for regular and irregular
solutions for Eq. (93). Studying the asymptotic limits of Eq. (93) leads us to
the following regular ($+$) (irregular ($-$)) solution

$f_{m}(\rho)=\rho^{\pm\frac{\nu}{2}}\mathrm{e}^{-\frac{\rho}{2}}F(\rho).$ (94)

With this, Eq. (93) is rewritten as

$\rho
F^{\prime\prime}(\rho)+\left(1\pm\nu-\rho\right)F^{\prime}(\rho)-\left(\frac{1\pm\nu}{2}-\frac{k^{2}}{4\gamma}\right)F(\rho)=0.$
(95)

Equation (93) is of the confluent hypergeometric equation type

$zF^{\prime\prime}(z)+(b-z)F^{\prime}(z)-aF(z)=0.$ (96)

In this manner, the general solution for Eq. (93) is

$\displaystyle f_{m}(r)$ $\displaystyle=$ $\displaystyle
a_{m}\rho^{\frac{\nu}{2}}\mathrm{e}^{-\frac{\rho}{2}}\;F\left(d_{+},1+\nu,\rho\right)$
(97) $\displaystyle+$ $\displaystyle
b_{m}\rho^{-\frac{\nu}{2}}\mathrm{e}^{-\frac{\rho}{2}}\;F\left(d_{-},1-\nu,\rho\right),$

with

$d_{\pm}=\frac{1\pm\nu}{2}-\frac{k^{2}}{4\eta}.$ (98)

In Eq. (97), $F(a,b,z)$ is the confluent hypergeometric function of the first
kind Book.1972.Abramowitz  and $a_{m}$ and $b_{m}$ are, respectively, the
coefficients of the regular and irregular solutions.

In this point, we apply the boundary condition in Eq. (90). Doing this, one
finds the following relation between the coefficients $a_{m}$ and $b_{m}$:

$\lambda_{m}\eta^{\nu}=\frac{b_{m}}{a_{m}}\left[1+\frac{\lambda_{m}k^{2}}{4(1-\nu)}\lim_{r\rightarrow
0^{+}}r^{2-2\nu}\right].$ (99)

We note that $\lim_{r\rightarrow 0^{+}}r^{2-2\nu}$ diverges if $\nu\geq 1$.
This condition implies that $b_{m}$ must be zero if $\nu\geq 1$ and only the
regular solution contributes to $f_{m}(r)$. For $\nu<1$, when the operator
$H_{0}$ is not self-adjoint, there arises a contribution of the irregular
solution to $f_{m}(r)$ AoP.2013.339.510 ; JPG.2013.40.075007 ;
AoP.2008.323.1280 ; TMP.2009.161.1503 ; EPJC.2014.74.2708 . In this manner,
the contribution of the irregular solution for system wave function steams
from the fact that the operator $H_{0}$ is not self-adjoint.

Figure 1: Plots of the energy ($\Delta(r)=0$) as a function of the magnetic
field $B_{0}$ for $s=1$ and different values of $n$ and $m$: $n=0$ [solid
line], $n=1$ [dashed line] and $n=2$ [dotted line]. Figure 2: Plots of the
energy ($\Delta(r)=0$) as a function of the magnetic field $B_{0}$ for $s=-1$
and different values of $n$ and $m$: $n=0$ [solid line], $n=1$ [dashed line]
and $n=2$ [dotted line].

For $f_{m}(r)$ be a bound state wave function, it must vanish at large values
of $r$, i.e., it must be normalizable. So, from the asymptotic representation
of the confluent hypergeometric function, the normalizability condition is
translated in

$\frac{b_{m}}{a_{m}}=-\frac{\Gamma(1+\nu)}{\Gamma(1-\nu)}\frac{\Gamma(d_{-})}{\Gamma(d_{+})}.$
(100)

From Eq. (99), for $\nu<1$, we have

$\frac{{b_{m}}}{{a_{m}}}=\lambda_{m}\eta^{\nu}.$ (101)

Using this result into Eq. (100), one finds

$\frac{\Gamma(d_{+})}{\Gamma(d_{-})}=-\frac{1}{\lambda_{m}\gamma^{\nu}}\frac{\Gamma(1+\nu)}{\Gamma(1-\nu)}.$
(102)

Equation (102) implicitly determines the bound state energy for the system for
different values of the self-adjoint extension parameter. Two limiting values
for the self-adjoint extension parameter deserves some attention. For
$\lambda_{m}=0$, when the $\delta$ interaction is absent, only the regular
solution contributes for the bound state wave function. In the other side, for
$\lambda_{m}=\infty$, only the irregular solution contributes for the bound
state wave function. For all other values of the self-adjoint extension
parameter, both regular and irregular solutions contributes for the bound
state wave function. The energy for the limiting values are obtained from the
poles of gamma function, namely,

$\left\\{\begin{array}[]{lll}d_{+}=-n&\mbox{for }\lambda_{m}=0,&\mbox{(regular
solution)},\\\ d_{-}=-n&\mbox{for }\lambda_{m}=\infty,&\mbox{(irregular
solution)},\end{array}\right.$ (103)

with $n$ a nonnegative integer, $n=0,1,2,\ldots$. By manipulation of Eq.
(103), we obtain

$\displaystyle E^{2}-M^{2}$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{e^{2}B_{0}^{2}}{4}+2a\left(E+M\right)}$ (104)
$\displaystyle\times$
$\displaystyle\left[2n+1\pm\sqrt{\left(m-e\phi\right)^{2}+2b\left(E+M\right)}\right]$
$\displaystyle+$ $\displaystyle e^{2}B_{0}\phi-meB_{0}-esB_{0},\;\;S=V,$
$\displaystyle E^{2}-M^{2}$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{e^{2}B_{0}^{2}}{4}+2a\left(E-M\right)}$ (105)
$\displaystyle\times$
$\displaystyle\left[2n+1\pm\sqrt{\left(m+s-e\phi\right)^{2}+2b\left(E-M\right)}\right]$
$\displaystyle+$ $\displaystyle
e^{2}B_{0}\phi-\left(m+s\right)eB_{0}-esB_{0},\;\;S=-V.$

As a illustration the profiles of the energy under the exact spin symmetry
limit ($S=V$) as a function of the magnetic field $B_{0}$ and with spin
projection parameter $s=1$ and $s=-1$ are shown in Fig. 1 and Fig. 2,
respectively. From Fig. 1 and Fig. 2 we can note that the ground state $n=0$
correspond to the lowest energies, as it should be for particle energy levels.

In particular, it should be noted that for the case when $\nu\geq 1$ or when
the $\delta$ interaction is absent, only the regular solution contributes for
the bound state wave function ($b_{m}=0$), and the energy is given by Eq.
(104) using the plus sign. The unnormalized bound state wave functions for our
problem are

$\displaystyle f_{m}(r)$ $\displaystyle=$ $\displaystyle
N_{f}\left(A_{1}\right)^{\pm\frac{1}{4}\sqrt{A_{2}}}r^{\pm\sqrt{A_{2}}}e^{-\frac{1}{2}\sqrt{A_{1}}r^{2}}$
(106) $\displaystyle\times$
$\displaystyle\,F\left(-n,1\pm\sqrt{A_{2}},\sqrt{A_{1}}r^{2}\right),$

for $S=V$, and

$\displaystyle g_{m}(r)$ $\displaystyle=$ $\displaystyle
N_{g}\left(B_{1}\right)^{\pm\frac{1}{4}\sqrt{B_{2}}}r^{\pm\sqrt{B_{2}}}e^{-\frac{1}{2}\sqrt{B_{1}}r^{2}}$
(107) $\displaystyle\times$
$\displaystyle\,F\left(-n,1\pm\sqrt{B_{2}},\sqrt{B_{1}}r^{2}\right),$

for $S=-V$, where $N_{f}$ and where $N_{g}$ are normalization constants, and

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\frac{e^{2}B_{0}^{2}}{4}+2a\left(E+M\right),$ (108)
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\left(m-e\phi\right)^{2}+2b\left(E+M\right),$ (109)
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\frac{e^{2}B_{0}^{2}}{4}+2a\left(E-M\right),$ (110)
$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\left(m+s-e\phi\right)^{2}+2b\left(E-M\right)$ (111)

The self-adjoint extension is related with the presence of the $\delta$
interaction. In this manner, the self-adjoint extension parameter must be
related with the $\delta$ interaction coupling constant $\phi s$. In fact, as
shown in Refs. PRD.2012.85.041701 ; AoP.2013.339.510  (see also Refs.
PLB.2013.719.467 ; JPG.2013.40.075007 ), from the regularization of the
$\delta$ interaction, it is possible to find such a relationship. Using the
regularization method, one obtains the following equation for the bound state
energy

$\frac{\Gamma(d_{+})}{\Gamma(d_{-})}=-\frac{1}{r_{0}^{2\nu}}\left(\frac{\phi
s+\alpha\nu}{\phi
s-\alpha\nu}\right)\frac{1}{\gamma^{\nu}}\frac{\Gamma\left(1+\nu\right)}{\Gamma\left(1-\nu\right)}.$
(112)

By comparing Eqs. (102) and (112), this relation is found to be

$\frac{1}{\lambda_{m}}=\frac{1}{r_{0}^{2\nu}}\left(\frac{\phi
s+\alpha\nu}{\phi s-\alpha\nu}\right)$ (113)

where $r_{0}$ is a very small radius which comes from the $\delta$
regularization PRD.2012.85.041701 ; AoP.2013.339.510 . The result of Eq. (113)
provides an explicit formula for the self-adjoint extension parameter
$\lambda_{m}$. We have, therefore, derived the most important quantities for
the system without any arbitrary parameter coming from the self-adjoint
extension method.

##  7 Nonrelativisitic limit

Let us now examine the nonrelativistic limit of Eq. (73) by setting
$E=M+\mathcal{E}$, with $M\gg\mathcal{E}$, for both cases $S=V$ and $S=-V$.
After applying this limit, we find

$H\psi=2M\mathcal{E}\psi,$ (114)

where

$H=\left(\mathbf{p}-e\mathbf{A}\right)^{2}-es\boldsymbol{\sigma}\cdot\mathbf{B}+2M\left[S\left(r\right)+V\left(r\right)\right].$
(115)

Using the ansatz of Eq. (17) in Eq. (114), again, we get the equation for
$f_{m}\left(r\right)$ (for $S=V$)

$\displaystyle\left[-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}+\frac{\bar{\nu}^{2}}{r^{2}}+\bar{\eta}^{2}\text{
}r^{2}-es\phi\frac{\delta(r)}{r}\right]f_{m}\left(r\right)$
$\displaystyle-\mathcal{E}f_{m}\left(r\right)=0.$ (116)

where

$\displaystyle\bar{\nu}^{2}$ $\displaystyle=$
$\displaystyle\left(m-e\phi\right)^{2}+4Mb,$ (117)
$\displaystyle\bar{\eta}^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{2}B_{0}^{2}}{4}+4Ma,$ (118) $\displaystyle\bar{k}^{2}$
$\displaystyle=$ $\displaystyle meB_{0}-e^{2}B_{0}\phi+esB_{0}+2M\mathcal{E}.$
(119)

On the other hand, for $S=-V$, the term involving the potential is now
identically zero. The resulting equation is given by

$\displaystyle\left[-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}+\frac{\breve{\nu}^{2}}{r^{2}}+\breve{\eta}^{2}\text{
}r^{2}-es\phi\frac{\delta(r)}{r}\right]g_{m}\left(r\right)$
$\displaystyle-\mathcal{E}g_{m}\left(r\right)=0,$

where

$\displaystyle\breve{\nu}^{2}$ $\displaystyle=$
$\displaystyle\left(m+s-e\phi\right)^{2},$ (120)
$\displaystyle\breve{\eta}^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{2}B_{0}^{2}}{4},$ (121) $\displaystyle\breve{k}^{2}$
$\displaystyle=$ $\displaystyle
eB_{0}\left(m+s\right)-e^{2}B_{0}\phi+esB_{0}+2M\mathcal{E},$ (122)

In order to determine the energy spectrum, we use the same technique above.
Performing the same steps as for the relativistic case, one obtains the energy
levels

$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\frac{1}{M}\sqrt{\frac{e^{2}B_{0}^{2}}{4}+4Ma}\left[2n+1\pm\sqrt{\left(m-e\phi\right)^{2}+4Mb}\right]$
(123) $\displaystyle+$
$\displaystyle\frac{1}{2M}\left[e^{2}B_{0}\phi-\left(m+s\right)eB_{0}-esB_{0}\right],\;\;S=V,$
$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\frac{1}{2M}eB_{0}\left(2n+1\pm\left|m-e\phi\right|\right)$
(124) $\displaystyle+$
$\displaystyle\frac{1}{2M}\left[e^{2}B_{0}\phi-\left(m+s\right)eB_{0}-esB_{0}\right],\;\;S=-V,$

The corresponding wave functions are given by

$\displaystyle
f_{m}(r)=\left(\frac{e^{2}B_{0}^{2}}{4}+4Ma\right)^{\pm\frac{1}{4}\sqrt{\left(m-e\phi\right)^{2}+4Mb}}$
$\displaystyle\times\,r^{\pm\sqrt{\left(m-e\phi\right)^{2}+4Mb}}e^{-\frac{1}{2}\sqrt{\frac{e^{2}B_{0}^{2}}{4}+4Ma}r^{2}}$
$\displaystyle\times\,F\left(-n,1\pm\sqrt{\left(m-e\phi\right)^{2}+4Mb},\sqrt{\frac{e^{2}B_{0}^{2}}{4}+4Ma}r^{2}\right),$
(125)

for $S=V$,

$\displaystyle f_{m}(r)$
$\displaystyle=\left[\frac{e^{2}B_{0}^{2}}{4}\right]^{\pm\frac{1}{4}\left|m-e\phi\right|}r^{\pm\left|m-e\phi\right|}e^{-\frac{1}{4}eB_{0}r^{2}}\,$
$\displaystyle\times\,F\left(-n,1\pm\left|m-e\phi\right|,\frac{1}{2}eB_{0}r^{2}\right),$
(126)

for $S=-V$.

##  8 Conclusions

In this paper, we have studied the relativistic quantum dynamics of a spin-1/2
charged particle with minimal, vector and scalar couplings. The minimal
coupling was chosen as being one which leads to the spin-1/2 Aharonov-Bohm
effect. In a first attempt, we have solved the equation of first order Dirac.
We verified that there are isolated solutions for the system for some special
cases. These solutions depend on the values assumed by the spin projection
parameter $s$, as well as on the choice of the scalar $S\left(r\right)$ and
vector $V\left(r\right)$ potential functions.

In contrast to what was addressed in the literature, we have considered the
correct quadratic form of the Dirac equation with minimal, vector and scalar
couplings. As we have mentioned before, in approach of Ref. AoP.2014.341.153 ,
the authors have not taken into account the term that depends explicitly on
the spin in the Pauli equation of motion. We have revisited the dynamics of
the system in detail and show that the correct approach should involve the
spin element. In this order, we have derived the Pauli equation and studied
the dynamics of the system taking into account exact symmetry spin and
pseudospin limits. Because of the equation of motion includes a $\delta$
function, we have used the self-adjoint extension method to specify the proper
boundary conditions at the origin. The analytical solutions of the model allow
a calculation of expressions for the energy eigenvalues and wave functions for
both symmetry limits. We verify that the presence of the spin element in the
equation of motion introduces a correction in the expressions for the bound
state energy and wave functions, a fact that does not occur in Ref.
AoP.2014.341.153 .

## Acknowledgments

This work was supported by the CNPq, Brazil, Grants No. 482015/2013-6
(Universal), No 455719/2014-4 (Universal), No. 306068/2013-3 (PQ),
304105/2014-7 (PQ) and FAPEMA, Brazil, Grants No. 00845/13 (Universal), No.
01852/14 (PRONEM).

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