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 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][2][3][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 lasermatter interaction (for a review, see [5] and references therein) and condensed matter physics [6].
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 [33] and recently used by many authors to solve the Schrodinger equation for different potential forms [34][35][36][37][38]. 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 [34]. In this letter, 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.

Review to Dirac Equation including Tensor Coupling
The Dirac equation which describes a nucleon in repulsive vector ( ) V r and attractive scalar   S r and a tensor   U r potentials is written as where ( ) M r is the effective mass of the fermionic particle, E is the relativistic energy of the system, p i       is the three-dimensional momentum operator.   and  are the 4 4  Dirac matrices give as where I is 2 2  unitary matrix and   are three-vector spin matrices The total angular momentum operator J  and spin-orbit and with the following properties , , , one obtains two coupled differential equations for upper and lower radial wave Eqs. (7), we finally obtain the following two Schrödinger-like differential equations for the upper and lower radial spinor components, respectively: 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 l and p-spin symmetry l  as and the quasi-degenerate doublet structure can be expressed in terms of a p-spin angular momentum 1 2 s   and pseudo-orbital angular momentum l  , which is defined as (9a). In this stage, we take the vector potential in the form of an attractive Coulomblike field [18] as where q  is being a vector dimensionless real parameter coupling constant and c  is being a constant with . J fm dimension. Also, it is convenient to take the mass function [18] as where 0 m and 1 m stand for the rest mass of the fermionic particle and the perturbed mass, respectively. Further, b is the dimensionless real constant to be set to zero for the constant mass case and 0  is the Compton-like wavelength in fm units. Further, the tensor interaction takes the simple form: where c R is the coulomb radius, a Z and b Z stand for the charges of the projectile particle a and the target nucleus , b respectively.
Substituting Eqs. (12) with  is a constant and then inserting into Eq. (17a), we have Now, to obtain a finite wave function when r   , if we take     in equation The LTM [44,45] leads to an equation Equation (21) is a first-order differential equation and therefore we may directly make use of the integral to get the expression where N is a constant. Noting that  (24) where N  is a constant. In terms of a simple extension of the inverse Laplace transformation [44,45], we can immediately obtain where we have used [45] Inserting the parameters in Eqs. (16) and (17b) which is identical to Ref. [18] In case when 0, The wave function (28) where we have used In tables 1 to 3 with 1 c    , we give some numerical results for the energy eigenvalues from energy formula (31).

p-spin symmetry case
To avoid repetition in the solution of Eq. (9b), 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 [18]: which is identical to Ref. [18] The lower-spinor component of wave function as   The upper-spinor wave function can be obtained via

Numerical Results
In tables 1 to 3, we see that energies of bound states such as:           The increase in the energies is slight when the strength of T is large when its small when the strength T is small. In Table 3, for constant values of s C and , T the energy increases when 1 m 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 s C increases, the energy increasing.
Finally, we plot the relativistic energy eigenvalues under spin and pspin symmetry limitations in figures 1 to 4. In fig. 2, we plot the energy eigenvalues of spin symmetry limit versus the perturbated mass 1 m . It is seen that when 1 m increases, the energy increases too. In fig. 2, we have shown the variation of the energy as a function of T . We can see the degeneracy removes between spin doublets and also they become far from each other, when the parameter T increases. In figs. 3 to 4, we plot the energy states of the pseudospin symmetry limit for different levels as functions of parameters 1 m and T , respectively. The variation of energy can also be seen from these figures.

Conclusion
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 0. T 