Hulthén and Coulomb-Like Potentials as a Tensor Interaction within the Relativistic Symmetries of the Manning-Rosen Potential

The bound-state solutions of the Dirac equation for the Manning-Rosen potential are presented approximately for arbitrary spinorbit quantum number κ with the Hulthén and Coulomb-like potentials as a tensor interaction. The generalized parametric Nikiforov-Uvarov (NU) method is used to obtain energy eigenvalues and corresponding two-component spinors of the two Dirac particles and these are obtained in the closed form by using the framework of the spin symmetry and p-spin symmetry concept. We have also shown that tensor interaction removes degeneracies between spin and p-spin doublets. Some numerical results are also given.


Introduction
The spin and pseudospin symmetry concepts introduced in nuclear theory [1,2] have been used to explain the features of deformed nuclei [3] and superdeformation [4] and to establish an effective shell-model coupling scheme [5].Within the framework of the relativistic mean field theory, Ginocchio [6,7] has found that a Dirac Hamiltonian with scalar and vector harmonic oscillator potentials in the case of (() − () = 0) possesses not only a spin symmetry but also a (3) symmetry, but a Dirac Hamiltonian in the case of (() + () = 0) possesses a pseudospin symmetry and a pseudo-(3) symmetry.Meng et al. [8] have showed that the pseudospin symmetry is exact under the condition ((() + ())/ = 0).In addition, Alhaidari et al. [9] have investigated in detail physical interpretation on the threedimensional Dirac equation in the case of spin symmetry limit (()−() = 0) and pseudospin symmetry limit (()+ () = 0).In recent years, by considering the importance of spin and pseudospin symmetries, some authors have contributed many works in this field.For more review of this, one can read the recent works by Wei and Dong [10][11][12][13].
Motivated by recent work about the spin symmetry and pseudospin symmetry solutions of the Dirac equation for arbitrary spin-orbit coupling quantum number  with the Eckart potential and Pöschl-Teller potential carried out by Jia et al. [19,20], in this work we attempt to study the spin symmetry and pseudospin symmetry solutions of the Dirac equation for arbitrary quantum number  with the Manning-Rosen potential: where  and  are two dimensionless parameters and  is the screening parameter which is used for determining the range of potential and has dimension of length [21].It is shown that this potential remains invariant by mapping  ↔ 1 − .The -wave scattering solutions and bound state energy eigenvalues for the Manning-Rosen potential have been calculated by using the function analysis method [22] and path integral approach [23], respectively.Recently, Ikhdair and Sever have also studied the arbitrary -wave solutions of the -dimensional Schrödinger equation with the Manning-Rosen potential by using an approximate to the centrifugal term [24].
Tensor potentials were introduced into the Dirac equation with the substitution →  → →  −im ⋅ r() and a spinorbit coupling is added to the Dirac Hamiltonian [30,31].
Recently, different kinds of potentials (e.g., Coulomb-like, Linear plus Coulomb-like, and Yukawa), as tensor couplings, have been used widely in the studies of nuclear properties.In this regard, see [32][33][34][35][36][37][38][39].Here we study a tensor potential in the Hulthén (2) and Coulomb-like (3) form as follows: where  0 is the potential depth,  is the screening range parameter, and  represents the spatial range.If the potential is used for atoms, then  0 =  (in the relativistic units ℏ =  =  = 1), where  is identified as the atomic number.The Hulthén potential behaves like the Coulomb potential near the origin (i.e.,  → 0 or  ≪ )   () = − 2 / but decreases exponentially in the asymptotic region when  ≫ 0, so its capacity for bound states is smaller than the Coulomb potential.The Hulthén potential is widely used for the description of the nucleon-heavy nucleus interactions (see [40] and the references therein).This potential has been applied to a number of areas such as nuclear and particle physics [41][42][43], atomic physics [44,45], molecular physics [46,47], and chemical physics [48].Also, the Coulomb-like potential [49] for the tensor due to a charge    interacting with a charge   , distributed uniformly over a sphere of radius   , is added: where   = 7.78 fm is the Coulomb radius and   and   denote the charges of the projectile  and the target nuclei , respectively.These potentials have been used in several branches of physics and their discrete and continuum states have been studied by a variety of techniques such as the algebraic perturbation calculations which are based upon the dynamical group structure SO(2,1) [50], the formalism of supersymmetric quantum mechanics within the framework of the variational method [51], the supersymmetry and shape invariance property [52], the asymptotic iteration method [53,54], the NU method [55], and the approach proposed by Biedenharn et al. for the Dirac-Coulomb problem [56,57].Otherwise, some authors studied relativistic and nonrelativistic equations with different potentials .In this paper, we solve approximately the Dirac equation with the Manning-Rosen potential for the spin-orbit quantum number  and we obtained the energy eigenvalues and wavefunctions of Dirac particle in the fields of the scalar and vector Manning-Rosen potential including a Hulthén and Coulomb tensor interaction under consideration of spin symmetry and pseudospin symmetry case.
The organization of this paper is as follows.In Section 2, we briefly introduce the Dirac equation with the Manning-Rosen potential and tensor potentials with arbitrary spinorbit coupling number  including tensor interaction under spin and p-spin symmetric limits.Also the generalized parametric NU method is presented in this section.The energy eigenvalue equations and corresponding wave functions for Hulthén and Coulomb tensor interaction in terms of the Jacobi polynomials are obtained in Sections 3 and 4. We also give some remark graphs and numerical results.We end with conclusion in Section 5.

Dirac Equation Including Tensor Coupling
The Dirac equation for fermionic massive spin-1/2 particles moving in attractive scalar (), repulsive vector (), and tensor () potentials is (in units ℏ =  = 1) where  is the relativistic energy of the system, →  = − → ∇ is the three-dimensional momentum operator, and  is the mass of the fermionic particle.Further, →  and  are the 4 × 4 Dirac matrices given by → = ( 0 where  is 2 × 2 unitary matrix and →  are three-vector spin matrices The total angular momentum operator  is orbital angular momentum of the spherical nucleons, commute with the Dirac Hamiltonian.The eigenvalues of spin-orbit coupling operator are  = ( + 1/2) > 0 and  = −( + 1/2) < 0 for unaligned spin  =  − 1/2 and the aligned spin  =  + 1/2, respectively.( 2 , ,  2 ,   ) 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: ) , (7) where   ( →  ) is the upper (large) component and   ( →  ) is the lower (small) component of the Dirac spinors.   (, ) and  l  (, ) are spin and p-spin spherical harmonics, respectively, and  is the projection of the angular momentum on the -axis.Substituting ( 7) into (4) and using the following relations: together with the following properties: one obtains two coupled differential equations for upper and lower radial wave functions   () and   () as where Eliminating   () and   () from ( 10) and ( 11), we finally obtain the following two Schrödinger-like differential equations for the upper and lower radial spinor components, respectively: where ( − 1) = l( l + 1) and ( + 1) = ( + 1).
The quantum number  is related to the quantum numbers for spin symmetry  and p-spin symmetry l as and the quasidegenerate doublet structure can be expressed in terms of a p-spin angular momentum s = 1/2 and pseudoorbital angular momentum l, which can be defined as where  = ±1, ±2, . ... For example, (1 1/2 , 0 3/2 ) and (1 3/2 , 0 5/2 ) can be considered as p-spin doublets.

Spin Symmetry Limit.
In this section, we will solve Dirac equation under spin symmetry limit with Manning-Rosen potential and Hulthén potential as a tensor interaction.
The exact spin symmetry occurs in Dirac equation when
In Figure 1, structure of the Manning-Rosen potential for various values of  screening parameter and  is shown.

p-Spin Symmetry Limit.
Within the pseudospin symmetry case, [() + ()]/ = Σ()/ = 0 or Σ() =  ps = constant, and p-spin symmetry is exact in the Dirac equation [8,[95][96][97].In this part, we consider Δ() as and tensor () potential as Under this symmetry, from ( 14), we obtain where γ =   −  −  ps and β 2 = ( +   )( −   +  ps ).Also,  = − l and  = l + 1 for  < 0 and  > 0, respectively.The Dirac equation for the Manning-Rosen potential given by ( 18) and ( 21) cannot be solved analytically.To obtain analytical approximate solutions for the Manning-Rosen type potential with spin-orbit coupling term, we have to use an approximation in order to deal with the spin-orbit coupling term [57] as Or equivalently which is only valid for small values of the parameter .

Basic Equations of Parametric Nikiforov-Uvarov (NU)
Method.The NU method is a powerful mathematical tool that recently has been used widely to solve second-order differential equations [98,99].In this section, we briefly describe this method.This technique is based on solving the hypergeometric type second-order differential equations by means of the special orthogonal functions [100].For a given potential, the Schrödinger or Schrödinger-like equations in spherical coordinates are reduced to a generalized equation of hypergeometric type with an appropriate coordinate transformation  →  and then they are solved systematically to find the exact or particular solutions.The main equation which is closely associated with the method is given in the following form [101,102]: where () and σ() are polynomials, at most of second degree, and τ() is a first-degree polynomial.To make the application of the NU method simpler and direct without need to check the validity of solution, we present a shortcut for the method.Hence, firstly we write the general form of the Schrödinger-like equation ( 24) in a more general form as satisfying the wave functions Secondly, we compare (27) with its counterpart equation ( 26) to obtain the following parameter values: Now, following the NU method [101], we obtain the energy equation [102,103] The values for the parametric constants are shown in Table 1.
In Table 2,   = (( + 1/2) 2 + ( 0 /)(( 0 /) + 2 + 1) +  1 / 2 ) 1/2 , and by using (28), we can obtain the energy eigenvalues of the Manning-Rosen potential with spin symmetric limit as To find the corresponding wave functions, referring to Table 2 and (29), we find the functions By using   () = ()  (), we get the upper component of the Dirac spinor as where   is normalization constant.The lower component of the Dirac spinor from (10) can be calculated as Table 1: The values for the parametric constants of the NU method.
Parametric NU constants The specific values for the parametric constants necessary for calculating the energy eigenvalues and eigenfunctions of the spin symmetry limit.

Constant
Analytic value Constant Analytic value where  ̸ = −  +   .Some numerical results of ( 33) are given in Table 3.In Table 3, we obtain numerical results in the spin symmetry in the presence and absence of tensor Hulthén form potential.We took a set of parameter values,  = 5.0 fm −1 ,   = 0,  = 0.025 fm −1 , and  = 0.01.
We can observe that every pair of orbitals ( 1/2 ,  3/2 ), ( 3/2 ,  5/2 ), and ( 5/2 ,  7/2 ) has the same energy in the absence of the tensor potential ( 0 = 0).Thus, they can be viewed as the spin doublets; that is, the state 1 1/2 with  = 1 and  = 1 forms a spin doublet with the 1 3/2 state with  = 1 and  = −2.On the other hand, in the presence of the tensor potential ( 0 ̸ = 0), one can notice that degeneracy between every pair of spin doublets is removed.In Figure 2, we have investigated the effect of the tensor potential on the spin doublet splitting by considering the following pairs of orbital: (1 3/2 , 1 1/2 ), (1 7/2 , 1 5/2 ).From Figure 2, we observe that, in the case of ( 0 = 0) (no tensor interaction), members of spin doublets have the same energy.However, in the presence of the tensor potential ( 0 ̸ = 0), these degeneracies are removed.

p-Spin Symmetry Solution.
In this subsection we will obtain the energy eigenvalues and the corresponding wave functions for the p-spin symmetric limit of the Manning-Rosen potential, that is, solutions of (21).Substituting (22) and ( 23) into (21) gives and further making the change of variables  =  − , we obtain where On the other hand, to avoid repetition in the solution of (38), in the p-spin symmetric case, Also, we get the lower component of the Dirac spinor as where Ñ is normalization constant and η = The upper component of the Dirac spinor from ( 11) can be calculated as where  ̸ =  +  ps .In Table 4, we use the same parameters as in the previous subsection (with  ps = 0) and we obtain numerical results in the p-spin symmetry in the presence and absence of tensor Hulthén form potential.The existence of the degeneracy was observed in the following doublets: (1 1/2 , 0 3/2 ), (1 3/2 , 0 5/2 ), (1 5/2 , 0 7/2 ), (1 7/2 , 0ℎ 9/2 ), and so on, in the absence of the tensor potential ( 0 = 0), and they are considered as p-spin doublets.In Figure 3, we have investigated the effect of the tensor potential on the p-spin doublet splitting by considering the following pairs of orbital: The energy spectrum of spin and p-spin symmetry given by Tables 3 and 4 changes with the relevant quantum numbers as well as the screening parameter .The variation of the energy spectrum ( , ) according to the screening parameter  for some states is shown in Figures 4 and 5.

Bound States of the Manning-Rosen Potential with Coulomb Tensor Interaction
In this stage we will solve Dirac ( 13) and ( 14) that are obtained in Section 2, for Manning-Rosen potential including a Coulomb-like tensor interaction under consideration of spin and pseudospin symmetry case.Under spin symmetry, from (13), for the upper radial spinor component, we obtain where   =  +  0 ,  =  +   −   , and  2 = ( −   )( +   −   ).Also,  =  and  = − − 1 for  < 0 and  > 0, respectively.Also under p-spin symmetry, from (14), for the lower radial spinor component, we have where   =  +  0 , γ =   −  −  ps , and β 2 = ( +   )( −   +  ps ).Also,  = − l and  = l + 1 for  < 0 and  > 0, respectively.22) and ( 23) into (43) and defining a new variable of the form  =  − , we get Here Comparing (45) with relations that are obtained in Section 3.1, we can easily obtain that We can obtain the energy eigenvalues of the Manning-Rosen potential with spin symmetric limit as To find the corresponding wave functions, we can use (34), (35), and (36).Some numerical results of (48) are given in Table 5.In Table 5, we obtain numerical results in the spin symmetry in the presence and absence of tensor Coulomb form potential.We took a set of parameter values:  = 5.0 fm −1 ,   = 0,  = 0.025 fm −1 , and  = 0.01.In Figure 6, we have investigated the effect of the tensor potential on the spin doublet splitting by considering the following pairs of orbital: (1

p-Spin Symmetry Solution.
Substituting ( 22) and ( 23) into ( 44) and defining a new variable of the form  =  − , we get Here Comparing (49) with relations that are obtained in Section 3.2, we can easily obtain that We can obtain the energy eigenvalues of the Manning-Rosen potential with p-spin symmetric limit as Also, we get the lower and upper component of the Dirac spinor as (41) and (42).In Table 6, we use the same parameters as in the previous subsection (with  ps = 0) and we obtain numerical results in the p-spin symmetry in the presence and absence of tensor Coulomb form potential.In Figure 7, we have investigated the effect of the tensor potential on the pspin doublet splitting by considering some pairs of orbitals.Also the variation of the energy spectrum ( , ) of spin and p-spin symmetry given by Tables 5 and 6, according to the screening parameter  for some states, is shown in Figures 8  and 9.

Conclusion
In this paper, we obtained the approximate analytical solutions of the Dirac equation for the Manning-Rosen potential with the consideration of Hulthén and Coulomb-like potentials as tensor interaction within the framework of spin and pseudospin symmetry limits using the generalized parametric NU technique.We have obtained the energy levels in a closed form and the corresponding wave functions in terms of the Jacobi polynomials.Also we obtained the energy eigenvalues in the absence ( 0 = 0) and the presence ( 0 = 0.1, 0.2) of the Hulthén tensor interaction and the energy eigenvalues in the absence ( 0 = 0) and the presence ( 0 = 1) of the Coulomb-like tensor interaction for various values of the quantum numbers  and .The results are reported in Tables 3-6 under the condition of the spin and pseudospin symmetries, respectively, where we can see the way the tensor interaction affects the degeneracy of the system.Figures 2, 3, 6, and 7 present the effect of the Hulthén and Coulomb-like tensor interaction on the bound states, respectively.Obviously, the degeneracy between the members of doublet states in spin and pseudospin symmetries is removed by these tensor interaction potentials.We represent the effects of the screening parameter  on the bound states for Hulthén and Coulomb-like tensor interaction in Figures 4, 5, 8, and 9, respectively.It is seen that if the parameter increases, for both tensor interactions, the bound states become more bounded in both for symmetry limits.Therefore, the results of our work find many applications in both nuclear and Hadron physics and they obviously indicate the degeneracy-removing role of tensor interaction [104], which plays a significant role in the spectroscopes.

− 1 )Figure 2 :
Figure 2: Contribution of the tensor Hulthén potential parameter to the energy levels in the case of spin symmetry.

E n,𝜅 (fm − 1 )Figure 3 :
Figure 3: Contribution of the tensor Hulthén potential parameter to the energy levels in the case of p-spin symmetry.

(fm − 1 )Figure 4 :
Figure 4: The variation of the energy spectrum  , for tensor Hulthén potential in the case of spin symmetry according to the screening parameter .Here we selected  0 = 0.1,  = 0.01.

Figure 6 :
Figure 6: Contribution of the tensor Coulomb potential parameter to the energy levels in the case of spin symmetry.

(fm − 1 )Figure 8 :(fm − 1 )Figure 9 :
Figure 8: The variation of the energy spectrum  , for tensor Coulomb potential in the case of spin symmetry according to the screening parameter .Here we selected  0 = 0.1,  = 0.01.

Table 3 :
The bound state energy eigenvalues in unit of fm −1 of the spin symmetry Hulthén potential for several values of  and .Here the first and second states column according to (, ).

Table 4 :
The bound state energy eigenvalues in unit of fm −1 of the p-spin symmetry Hulthén potential.Here the first states column according to (, ) and second states column ( + 2,  + 1).

Table 5 :
The bound state energy eigenvalues in unit of fm −1 of the spin symmetry Coulomb potential for several values of  and .Here the first and second states column according to (, ).

Table 6 :
The bound state energy eigenvalues in unit of fm −1 of the p-spin symmetry Coulomb potential.Here the first states column according to (, ) and second states column ( + 2,  + 1).
Figure 5: The variation of the energy spectrum  , for tensor Hulthén potential in the case of p-spin symmetry according to the screening parameter .Here we selected  0 = 0.1,  = 0.01.