Research Article | Open Access
Exact Solutions of a Spatially Dependent Mass Dirac Equation for Coulomb Field plus Tensor Interaction via Laplace Transformation Method
The spatially dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction potential under the spin and pseudospin (p-spin) symmetric limits by using the Laplace transformation method (LTM). Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number . Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit and in the absence of tensor interaction .
Dirac equation has become one of the most appealing relativistic wave equations for spin-1/2 particles. However, solving such a wave equation is still a very challenging problem even if it has been derived more than 80 years ago and has been utilized profusely. It is always useful to investigate the relativistic effects [1–4]. For example, in the relativistic treatment of nuclear phenomena the Dirac equation is used to describe the behavior of the nuclei in nucleus and also in solving many problems of high-energy physics and chemistry. For this reason, it has been used extensively to study the relativistic heavy ion collisions, heavy ion spectroscopy, and more recently in laser-matter interaction (for a review, see  and references therein) and condensed matter physics .
On the other hand, systems with position-dependent mass (PDM) have been found to be very useful in studying the physical properties of various microstructures [7–14]. Recently, there has been increased interest in searching for analytical solutions of the Dirac equation with PDM and with constant mass under the spin and p-spin symmetries [15–34].
Here, we shall attempt to solve the Dirac equation by using the Laplace transform method (LTM). The LTM is an integral transform and is comprehensively useful in physics and engineering  and recently used by many authors to solve the Schrödinger equation for different potential forms [36–40]. This method could be a nearly new formalism in the literature and serve as a powerful algebraic treatment for solving the second-order differential equations. As a result, the LTM describes a simple way for solving of radial and one-dimensional differential equations. The other advantage of this method is that a second-order equation can be converted into more simpler form whose solutions may be obtained easily . In this paper, we obtain solution of the Dirac equation both PDM and tensor interaction for attractive scalar and repulsive vector Coulomb potential under the spin and p-spin symmetry limits. We give some numerical results.
2. Review to Dirac Equation including Tensor Coupling
The Dirac equation which describes a nucleon in repulsive vector and attractive scalar and a tensor potentials is written as where is the effective mass of the fermionic particle, is the relativistic energy of the system, is the three-dimensional momentum operator. and are the Dirac matrices give as where is unitary matrix and are three-vector spin matrices The total angular momentum operator and spin-orbit , where is orbital angular momentum, of the spherical nucleons commute with the Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are and for unaligned spin and the aligned spin , respectively. () can be taken as the complete set of the conservative quantities. Thus, the spinor wave functions can be classified according to their angular momentum , spin-orbit quantum number and the radial quantum number can be written as follows: where is the upper (large) component and is the lower (small) component of the Dirac spinors. and are spin and p-spin spherical harmonics, respectively, and is the projection of the angular momentum on the -axis. Substituting (2.4) into (2.1) and using the following relations  as and with the following properties: one obtains two coupled differential equations for upper and lower radial wave functions and as where Eliminating and from (2.5e) and (2.5f), we finally obtain the following two Schrödinger-like differential equations for the upper and lower radial spinor components, respectively: where and . These radial wave functions are required to satisfy the necessary boundary conditions. The spin-orbit quantum number is related to the quantum numbers for spin symmetry and p-spin symmetry as and the quasi-degenerate doublet structure can be expressed in terms of a p-spin angular momentum and pseudo-orbital angular momentum , which is defined as where . For example, and can be considered as p-spin doublets. In the next section, we will consider the spin and p-spin symmetry cases.
3. Relativistic Bound State Solutions
3.1. Spin Symmetry Case
Equation (2.5i) cannot be solved analytically because of the last term . Therefore, in solving the mathematical relation [42, 43], we can then exactly solve (2.5i). In this stage, we take the vector potential in the form of an attractive Coulomb-like field  as where is being a vector dimensionless real parameter coupling constant and is being a constant with dimension. Also, it is convenient to take the mass function  as where and stand for the rest mass of the fermionic particle and the perturbed mass, respectively. Further, is the dimensionless real constant to be set to zero for the constant mass case and is the Compton-like wavelength in fm units. Further, the tensor interaction takes the simple form: where is the coulomb radius, and stand for the charges of the projectile particle and the target nucleus , respectively.
Substituting (3.1)–(3.3) into (2.5i) considering the spin symmetry case where ., that is, [44, 45]. Thus, the equation obtained for the upper component of the Dirac spinor becomes Further, defining the new parameters and introducing , then (3.4) turns into the formSetting with is a constant and then inserting into (3.6a), we have Now, to obtain a finite wave function when , if we take in (3.7) then it becomes The LTM [46, 47] leads to an equation Equation (3.10) is a first-order differential equation and therefore we may directly make use of the integral to get the expression where is a constant. Noting that is a multi-valued function and the wave functions are required to be single-valued, we must take which gives single-values wave functions. Using this requirement and further expanding (3.11) into series, we obtain where is a constant. In terms of a simple extension of the inverse Laplace transformation [46, 47], we can immediately obtain and from , we then obtain where is a constant. On the other hand, the confluent hyper-geometric functions is defined as a series expansion  So, we find the upper-spinor component of wave function as where is normalization constant. By using the normalization condition given as , and the relation between the Laguerre polynomials and confluent hyper-geometric functions as , the normalization constant in (3.17) is written as where we have used  Inserting the parameters in (3.5) and (3.6b) into (3.12), one obtain the transcendental energy eigenvalue equation as follows which is identical to  when and the tensor interaction is removed, that is, . For the constant mass case, we have In case when , , , , and , the above equation reduces to the non-relativistic energy limit (in units of ): The wave function (3.17) is finite in the entire range and at the boundaries, that is, and .
3.2. p-Spin Symmetry Case
To avoid repetition in the solution of (2.5j), the negative energy solution for p-spin symmetry can be obtained directly from those of the above positive energy solution for spin symmetry by using the parameter mapping : Following the previous results with the above transformations, we finally arrive at the transcendental energy equation as which is identical to  when and the tensor interaction is removed, that is, . For the constant mass case, we have the lower-spinor component of wave function as where , and is normalization constant The upper-spinor wave function can be obtained via
4. Numerical Results
In Tables 1 to 3, we see that energies of bound states such as: , , , (where each pair is considered as a spin doublet) in the absence of the tensor potential are degenerate but in the presence of the tensor potential, the degeneracies are removed. Also, we investigate the effects of the and parameters on the bound states under the conditions of the spin symmetry limits for . The results are given in Tables 1 to 3. It is readily seen that if and parameters increases, the value of the bound state energy eigenvalues of this potential increases for several states. It is shown that the energy eigenvalues decrease with decreasing and when increase with increasing the tensor strength for and . In Table 3, for constant values of and , the energy increases when increasing. The decrease in the energy values is large without tensor interaction while small in presence of tensor interaction and then being large. Further, when increases, the energy increasing.
Finally, we plot the relativistic energy eigenvalues under spin and p-spin symmetry limitations in Figures 1 to 4. In Figure 2, we plot the energy eigenvalues of spin symmetry limit versus the perturbated mass . It is seen that when increases, the energy increases too. In Figure 2, we have shown the variation of the energy as a function of . We can see the degeneracy removes between spin doublets and also they become far from each other, when the parameter increases. In Figures 3 to 4, we plot the energy states of the pseudospin symmetry limit for different levels as functions of parameters and , respectively. The variation of energy can also be seen from these figures.
In this paper, the relativistic equation for particles with spin 1/2 was solved exactly with both spatially-dependent mass and tensor interaction for attractive scalar and repulsive vector Coulomb potentials under the spin symmetry limit via the Laplace transformation method. Some numerical results are given for specific values of the model parameters. Effects of the tensor interaction on the bound states were presented that tensor interaction removes degeneracy between two states in spin doublets. We also investigated the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetry limits for .
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