Solutions to the Dirac Equation for Manning-Rosen Plus Shifted Deng-Fan Potential and Coulomb-Like Tensor Interaction Using Nikiforov-Uvarov Method

We solve the Dirac equation for the Manning-Rosen plus shifted Deng-Fan potential including a Coulomb-like tensor potential with arbitrary spin–orbit coupling quantum number κ. In the framework of the spin and pseudospin (pspin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions in closed form by using the Nikiforov–Uvarov method. Also Special cases of the potential as been considered and their energy eigen values as well as their corresponding eigen functions are obtained for both relativistic and non-relativistic scope.


Introduction
In 1957, in an effort to find a suitable diatomic potential to describe the vibrational spectrum, Deng and Fan proposed a molecular exponential-type potential called Generalized Morse potential (Ikot et al., 2014).This potential is also known as Deng-Fan molecular potential, DF, and it is a modification of the Morse potential.This potential has been a subject of numerous studies by researchers in various applications (Oyewumi et al., 2013), and can be used to describe the mobility of nucleons in the mean field produced from the interactions of the nuclei (Maghsoodi et al., 2012).In 2012, Hamzavi et al. proposed a modified form of the DF called Shifted Deng-Fan potential, sDF (Hamzavi and Ikhdair, 2012).In the modification the DF potential shifted by dissociation energy α.It was demonstrated in their work that sDF and Morse potential are closely similar for values of r in the regions r ≈ re and r > re however they differ at r ≈ 0. In recent years numerous studies on the approximate analytical solution of Schrödinger equation with Manning-Rosen potential has been conducted (Awoga et al., 2013).The manning-Rosen potential was put forward to delineate the appropriate properties of diatomic molecules (Sameer Ikhdair, 2011).This potential is classified under exactly solvable and has been intensely studied.When spectrum of continuous range of values of the potential parameter could be determined analytically, such potentials are referred to as exactly solvable potentials (Min-Cang et al., 2010).Exactly solutions of wave equations with certain physical potential are of great interest in quantum mechanics.This is because the analysis makes the abstract understanding of the physical system and provides facts in affirming the impeccability of the quantum theory.This solution is paramount in examining and perfecting models and numerical methods employed for solving complex physical systems (Schiff, 1955).Recently, We reported the solution of the Dirac equation for the Manning-Rosen plus shifted Deng-Fan potential including a Yukawa-like tensor potential with arbitrary spinorbit coupling quantum number κ where, in the framework of the spin and pseudospin (pspin) symmetry, we obtained the energy eigenvalue equation and the corresponding eigenfunctions in closed form by using the Nikiforov-Uvarov method.We also considered special cases of the potential and their energy eigen values as well as their corresponding eigen functions were obtained for both relativistic and non-relativistic scope (Louis et al., 2018).In this work, our aim is to solve the Dirac equation for the Manning-Rosen plus shifted Deng-Fan (MRsDF) potential in the presence of spin and pspin symmetries and by including a Coulomb-like tensor potential.The MRsDF potential takes the following form: Thus eq. ( 1a) can be further expressed as where α is the screening parameter, C, D are potential depths,   is the Dissociation energy, where   is the equilibrium bond length.
This paper is organized as follows.In section 2, we briefly introduce the Dirac equation with scalar and vector potentials with arbitrary spin-orbit coupling quantum number κ including tensor interaction under spin and pspin symmetry limits.The Nikiforov-Uvarov (NU) method is presented in section3.The energy eigenvalue equations and corresponding eigenfunctions are obtained in section 4. In section 5, we discussed some special cases of the potential.Finally, our conclusion is given in section 6.

The Dirac Equation With Tensor Coupling Potential
The Dirac equation for fermionic massive spin-1/2 particles moving in the field of an attractive scalar potential (), a repulsive vector potential () and a tensor potential() (in units ħ = c = 1) is where  is the relativistic binding energy of the system,  = − ⃗⃗ is the three-dimensional momentum operator and  is the mass of the fermionic particle. ⃗ and  are the 4×4 usual Dirac matrices given by where  is the 2×2 unitary matrix and  ⃗ are three-vector spin matrices The eigenvalues of the spin-orbit coupling operator are  = ( + , respectively.The set ( 2 , ,  2 ,   ) can be taken as the complete set of conservative quantities with  ⃗ being the total angular momentum operator and  = ( ⃗.  ⃗⃗ + 1) is the spin-orbit where  ⃗⃗ is the orbital angular momentum of the spherical nucleons that commutes with the Dirac Hamiltonian.Thus, the spinor wave functions can be classified according to their angular momentum , the spin-orbit quantum number  and the radial quantum number .Hence, they can be written as follows: where  , ( ⃗) is the upper (large) component andg , ( ⃗) is the lower (small) component of the Dirac spinors.
(, ) and    ̂(, ) are spin and pspin spherical harmonics, respectively, and  is the projection of the angular momentum on the  − .Substituting equation (5) into equation ( 2) and making use of the following relations together with the properties one obtains two coupled differential equations whose solutions are the upper and lower radial wave functions  , () and  , () as where After eliminating  , () and  , () in equations ( 8), we obtain the following two Schrodinger-like differential equations for the upper and lower radial spinor components: respectively, where ( − 1) =  ̂( ̂+ 1) and ( + 1) = ( + 1).

Pseudospin Symmetry Limit
Lucha and Schober (2011) showed that there is a connection between pspin symmetry and near equality of the time component of a vector potential and the scalar potential, () ≈ −().After that, Ikhdair (2012) derived that if where  = − ̃ and  =  ̃+ 1 for κ < 0 and κ > 0, respectively.Also,  ̃= (  −  −   ) and  ̃2 = ( +   )( −   +   ) .( 17b) to obtain the analytic solution, we use an approximation for the centrifugal term as (Louis et al., 2018) (Ita et al., 2018) 1 Finally, for the solutions to equations ( 16) and ( 17) with the above approximation, we will employ the NU method, which is briefly introduced in the following section

The Nikiforov-Uvarov (NU) Method
The NU method is based on the solutions of a generalized second order linear differential equation with special orthogonal functions.The hypergeometric (Ita et al., 2018) method has shown its power in calculating the exact energy levels of all bound states for some solvable quantum systems.
Where σ(s) and (s) are polynomials at most second degree and  ̃(s) is first degree polynomials.The parametric generalization of the N-U method is given by the generalized hypergeometric-type equation The parameters obtainable from equation ( 4) serve as important tools to finding the energy eigenvalue and eigenfunctions.They satisfy the following sets of equation respectively and   is the orthogonal polynomials.

Solutions to the Dirac Equation
We will now solve the Dirac equation with the MRsDF potential and tensor potential by using the NU method.

The Spin Symmetric Case
To obtain the solution to equation ( 16), by using the transformation  =  ; , we rewrite it as follows: Eq. ( 26) is further simplified as where   =  +  + 1, Comparing eq. ( 27) with eq. ( 20), we obtain (28) and from eq. ( 25), we further obtain In addition, the energy eigenvalue equation can be obtained by using eq.( 23) as follows: By substituting the explicit forms of    2 after equation ( 16) into equation ( 30), one can readily obtain the closed form for the energy formula.
On the other hand, to find the corresponding wave functions, referring to equation ( 29) and eq.( 24), we obtain the upper component of the Dirac spinor from eq. 24 as 2 :  (  ;1) (1 − ) where  , is the normalization constant.The lower component of the Dirac spinor can be calculated from equation (8a) where   ≠ − +   .

The Pseudospin Symmetric Case
To avoid repetition in the solution of equation ( 17), we follow the same procedures explained in section 4.1and hence obtain the following energy eigenvalue equation: where   ≠  +   .

Discussion
In this section, we are going to study some special cases of the energy eigenvalues given by Eqs. ( 31) and ( 35) for the spin and pseudospin symmetries, respectively.