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

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.


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 through three usual substitutions, known as minimal, vector and scalar couplings, whose representation is denoted, respectively, by 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 [1]. 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 [2], Aharonov-Bohm effect [3], Hall effect [4], and other effects associated with magnetic field. a e-mail: lrb.castro@ufma.br b e-mail: edilbertoos@pq.cnpq.br 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 [5], the Aharonov Bohm effect for a spin-1/2 particle in the case that a 1/r potential is present [6], effects of nongauge potentials on the spin-1/2 Aharonoh-Bohm problem [7], quasiclassical theory of the Dirac equation with applications in the physics of heavy-light mesons [8] and confining potentials with pure vector coupling [9]. In the Schrödinger theory, It also has important applications, such as, in the dynamics of an electron in a two-dimensional quantum ring [10,11], quantum particles constrained to move on a conical surface [12] and effect of singular potentials on the harmonic oscillator [13].
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 [14], to study the relativistic quantum dynamics of a charged particle in cosmic string spacetime [15,16], 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 [17], inclusion of the generalized Hulthén potential in the case of the smooth step mass distribution [18] and extension of PT-symmetric quantum mechanics [19]. 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 [20], derivation of the Shannon entropy for a particle with a nonuniform solitonic mass density [21], context of displacement operator for quantum systems [22,23], use of instantaneous Galilean invariance to derive the expression for the Hamiltonian of an electron [24], determination of some potential functions for exactly solvable nonrelativistic problems [25] and Hermitian, rotationally invariant oneband Schrödinger Hamiltonian [26].
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 [27,28] 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 [29] and anti-nucleon bound in a nucleus [30], and the pseudospin symmetry occurs in the spectrum of nuclei [31].
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. [32]. 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 [33]. 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. [34] 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 [35] 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 sec-tion 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. [32]. 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.

Equation of motion
We begin with the Dirac equation (1) where π = π r , π ϕ = (−i∂ r , −i∂ ϕ /r − eA ϕ ), r = (r, ϕ) and ψ is a two-component spinor. The γ matrices in Eq. (5) are given in terms of the Pauli matrices as [36] β γ r = σ 1 cos ϕ + sσ 2 sin ϕ = 0 e −isϕ e +isϕ 0 , 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 where If one adopts the following decomposition with m + 1/2 = ±1/2, ±3/2, . . ., with m ∈ Z, and inserting this into Eqs. (9) and (10), we obtain Note that the above equations are coupled. However, if Σ (r) or ∆ (r) 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.

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 = ±M, directly in the first order equations in Eqs. (12) and (13). Such a solution is known to be excluded from the Sturm-Liouville problem, and have been investigated under diverse perspectives in the latest years [37][38][39][40][41][42][43][44]. We are seeking for bound-state solutions subjected to the normalization condition In order to determine the isolated bound-state solutions, we consider Σ (r) = 0 in Eq. (12), so that, for E = M, we can write d dr whose general solutions are where a + and b + are constants, and I(r) is given by which, for a given V (r) and A ϕ (r), it can be expressed in terms of the upper incomplete Gamma function [45] Γ Let us now analyze solutions for E = −M, and consider ∆ (r) = 0 in Eq. (13). For this case, we write whose general solution is where 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 with where B 0 is the magnetic field magnitude and φ is the flux parameter. The potentials in Eq. (26) both provide one magnetic field perpendicular to the plane (r, ϕ), namely with where B 1 is an external magnetic field and 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 nonnull. If the radius of the solenoid is r 0 ≈ 0, then the relevant magnetic field is B 2 ∼ δ (r) as in Eq. (30). This situation has not been accomplished in Ref. [32], 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 = ±M. Using Eqs. (29) and (30), we have If φ > 0 and B 0 > 0, for E = M, we have bound-state solutions only in the following cases: and for E = −M, where δ = eB 0 /4 and λ = eφ . Note that the above results are independent of the values of s, m and λ to ensure a bound state. This is because the function e −δ r 2 predominates over the polynomials r m−λ and r m−λ −1 . If we consider B 0 = V (r) = 0, which leads to the usual Aharonov-Bohm effect, the solutions for E = M reads wherẽ For E = −M, we get wherẽ Unlike from cases of Eqs. (32), (33), (34) and (35) (14) diverges.

Equation of motion and analysis of symmetries
In this section, we investigate the dynamics for E = ±M. For this, we choose to work with Eq. (5) in its quadratic form. After application of the operator we get In this stage, it is worthwhile to mention that the Eq. (43) is the correct quadratic form of the Dirac equation with minimal, vector and scalar couplings, because the Pauli term is considered.

Exact spin symmetry limit: S = V
The condition for establishing the exact symmetry boundary implies that the solution is of the form So, by making S = V (or equivalently ∆ (r) = 0 [46,47]) in Eq. (10) and using the solution (44) in Eq. (43), the equation for f m (r) is found to be Assuming V (r) as in Ref. [32], i.e., of the form Eq. (45) becomes with where As pointed out in Ref. [32], the potential V (r) in Eq. (46) describes an anharmonic oscillator. This model is a particular case of a proposed in Ref. [10] 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. [10] 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.

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. (11), namely By making S = −V (or equivalently Σ (r) = 0 in Eq. (9) and again using Eq. (53) in Eq. (43), the equation for g m (r) can be found with wherẽ  [48][49][50]. The selfadjoint extension approach consists, essentially, in extending the domain of D(O) in order to match D(O † ). From the theory of symmetric operators, it is a well-known fact that the symmetric radial operator H 0 is essentially self-adjoint for ν ≥ 1, while for ν < 1, it admits an one-parameter family of self-adjoint extensions [51], H 0,λ m , where λ m is the selfadjoint extension parameter. To characterize this family, we will use the approach in [52,53], which is based in a boundary conditions at the origin. All the self-adjoint extensions H 0,λ m of H 0 are parameterized by the boundary condition at the origin with where λ m ∈ R is the self-adjoint extension parameter. For λ m = 0, we have the free Hamiltonian (without the δ function) with regular wave functions at the origin, and for λ m = 0 the boundary condition in Eq. (60) permit an r −ν singularity in the wave functions at the origin.

The bound state energy and wave function
In this section, we determine the energy spectrum by solving Eq. (47). For r = 0, the equation for the component f m (r) can be transformed by the variable change ρ = ηr 2 resulting in Due to the boundary condition in Eq. (60), we seek for regular and irregular solutions for Eq. (63). Studying the asymptotic limits of Eq. (63) leads us to the following regular (+) (irregular (−)) solution With this, Eq. (63) is rewritten as Equation (63) is of the confluent hypergeometric equation type In this manner, the general solution for Eq. (63) is with In Eq. (67), F(a, b, z) is the confluent hypergeometric function of the first kind [45] 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. (60). Doing this, one finds the following relation between the coefficients a m and b m : We note that lim r→0 + r 2−2ν diverges if ν ≥ 1. This condition implies that b m must be zero if ν ≥ 1 and only the regular solution contributes to f m (r). For ν < 1, when the operator  H 0 is not self-adjoint, there arises a contribution of the irregular solution to f m (r) [34,[54][55][56][57]. 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. 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 From Eq. (69), for ν < 1, we have b m a m = λ m η ν .
Using this result into Eq. (70), one finds with n a nonnegative integer, n = 0, 1, 2, . . .. By manipulation of Eq. (73), we obtain 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 ν ≥ 1 or when the δ interaction is absent, only the regular solution contributes for the bound state wave function (b m = 0), and the energy is given by Eq. (74) using the plus sign. The unnormalized bound state wave functions for our problem are for S = V , and for S = −V , where N f and where N g are normalization constants, and The self-adjoint extension is related with the presence of the δ interaction. In this manner, the self-adjoint extension parameter must be related with the δ interaction coupling constant φ s. In fact, as shown in Refs. [34,58] (see also Refs. [54,59]), from the regularization of the δ interaction, it is possible to find such a relationship. Using the regularization method, one obtains the following equation for the bound state energy By comparing Eqs. (72) and (82), this relation is found to be where r 0 is a very small radius which comes from the δ regularization [34,58]. The result of Eq. (83) provides an explicit formula for the self-adjoint extension parameter λ m . We have, therefore, derived the most important quantities for the system without any arbitrary parameter coming from the self-adjoint extension method.

Nonrelativisitic limit
Let us now examine the nonrelativistic limit of Eq. (43) by setting E = M + E , with M E , for both cases S = V and S = −V . After applying this limit, we find where Using the ansatz of Eq. wherē On the other hand, for S = −V , the term involving the potential is now identically zero. The resulting equation is given by 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 E = 1 2M eB 0 (2n + 1 ± |m − eφ |) The corresponding wave functions are given by for S = −V .

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 (r) and vector V (r) 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. [32], 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 δ 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. [32].