Research Article  Open Access
Hulthén and CoulombLike Potentials as a Tensor Interaction within the Relativistic Symmetries of the ManningRosen Potential
Abstract
The boundstate solutions of the Dirac equation for the ManningRosen potential are presented approximately for arbitrary spinorbit quantum number with the Hulthén and Coulomblike potentials as a tensor interaction. The generalized parametric NikiforovUvarov (NU) method is used to obtain energy eigenvalues and corresponding twocomponent spinors of the two Dirac particles and these are obtained in the closed form by using the framework of the spin symmetry and pspin symmetry concept. We have also shown that tensor interaction removes degeneracies between spin and pspin doublets. Some numerical results are also given.
1. 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 shellmodel 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 possesses not only a spin symmetry but also a symmetry, but a Dirac Hamiltonian in the case of possesses a pseudospin symmetry and a pseudo symmetry. Meng et al. [8] have showed that the pseudospin symmetry is exact under the condition . In addition, Alhaidari et al. [9] have investigated in detail physical interpretation on the threedimensional Dirac equation in the case of spin symmetry limit and pseudospin symmetry limit . 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 [10–13].
The pspin symmetry refers to a quasidegeneracy of single nucleon doublets with nonrelativistic quantum number and , where , , and are single nucleon radial, orbital, and total angular quantum numbers, respectively [1, 2]. The total angular momentum with is a pseudoangular momentum and is pspin angular momentum [14–18].
Motivated by recent work about the spin symmetry and pseudospin symmetry solutions of the Dirac equation for arbitrary spinorbit coupling quantum number with the Eckart potential and PöschlTeller 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 ManningRosen 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 . The wave scattering solutions and bound state energy eigenvalues for the ManningRosen 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 ManningRosen potential by using an approximate to the centrifugal term [24].
For more review of the ManningRosen potential, one can refer to the [25–29].
Tensor potentials were introduced into the Dirac equation with the substitution and a spinorbit coupling is added to the Dirac Hamiltonian [30, 31].
Recently, different kinds of potentials (e.g., Coulomblike, Linear plus Coulomblike, and Yukawa), as tensor couplings, have been used widely in the studies of nuclear properties. In this regard, see [32–39]. Here we study a tensor potential in the Hulthén (2) and Coulomblike (3) form as follows: where is the potential depth, is the screening range parameter, and represents the spatial range. If the potential is used for atoms, then (in the relativistic units ), where is identified as the atomic number. The Hulthén potential behaves like the Coulomb potential near the origin (i.e., or ) but decreases exponentially in the asymptotic region when , so its capacity for bound states is smaller than the Coulomb potential. The Hulthén potential is widely used for the description of the nucleonheavy nucleus interactions (see [40] and the references therein). This potential has been applied to a number of areas such as nuclear and particle physics [41–43], atomic physics [44, 45], molecular physics [46, 47], and chemical physics [48].
Also, the Coulomblike potential [49] for the tensor due to a charge interacting with a charge , distributed uniformly over a sphere of radius , is added: where 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 DiracCoulomb problem [56, 57]. Otherwise, some authors studied relativistic and nonrelativistic equations with different potentials [58–90]. In this paper, we solve approximately the Dirac equation with the ManningRosen potential for the spinorbit quantum number and we obtained the energy eigenvalues and wavefunctions of Dirac particle in the fields of the scalar and vector ManningRosen 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 ManningRosen potential and tensor potentials with arbitrary spinorbit coupling number including tensor interaction under spin and pspin 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.
2. Dirac Equation Including Tensor Coupling
The Dirac equation for fermionic massive spin particles moving in attractive scalar , repulsive vector , and tensor potentials is (in units ) where is the relativistic energy of the system, is the threedimensional momentum operator, and is the mass of the fermionic particle. Further, and are the Dirac matrices given by where is unitary matrix and are threevector spin matrices The total angular momentum operator and spinorbit , where is orbital angular momentum of the spherical nucleons, commute with the Dirac Hamiltonian. The eigenvalues of spinorbit 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 ; spinorbit 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 pspin 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 Schrödingerlike differential equations for the upper and lower radial spinor components, respectively: where and .
The quantum number is related to the quantum numbers for spin symmetry and pspin symmetry as and the quasidegenerate doublet structure can be expressed in terms of a pspin angular momentum and pseudoorbital angular momentum , which can be defined as where . For example, and can be considered as pspin doublets.
2.1. Spin Symmetry Limit
In this section, we will solve Dirac equation under spin symmetry limit with ManningRosen potential and Hulthén potential as a tensor interaction. The exact spin symmetry occurs in Dirac equation when or constant [91–94]. We are taking and tensor potentials as where , , and .
In Figure 1, structure of the ManningRosen potential for various values of screening parameter and is shown.
Under this symmetry, from (13), one obtains where and . Also, and for and , respectively.
2.2. pSpin Symmetry Limit
Within the pseudospin symmetry case, or constant, and pspin symmetry is exact in the Dirac equation [8, 95–97]. In this part, we consider as and tensor potential as Under this symmetry, from (14), we obtain where and . Also, and for and , respectively.
The Dirac equation for the ManningRosen potential given by (18) and (21) cannot be solved analytically. To obtain analytical approximate solutions for the ManningRosen type potential with spinorbit coupling term, we have to use an approximation in order to deal with the spinorbit coupling term [57] as Or equivalently which is only valid for small values of the parameter .
2.3. Basic Equations of Parametric NikiforovUvarov (NU) Method
The NU method is a powerful mathematical tool that recently has been used widely to solve secondorder differential equations [98, 99]. In this section, we briefly describe this method. This technique is based on solving the hypergeometric type secondorder differential equations by means of the special orthogonal functions [100]. For a given potential, the Schrödinger or Schrödingerlike 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 are polynomials, at most of second degree, and is a firstdegree 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 Schrödingerlike 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.

Where , and , , the corresponding wave functions are where , , , and are Jacobi polynomials.
3. Bound States of the ManningRosen Potential with Hulthén Tensor Interaction
3.1. Spin Symmetry Solution
Substituting (22) and (23) into (18) gives Defining a new variable of the form and using (30), we obtain where Comparing (31) and (25), we can easily obtain the coefficients () and analytical expressions (). By using Table 1, we are obtained the coefficients (). The values of all coefficients are shown in Table 2.

In Table 2, , and by using (28), we can obtain the energy eigenvalues of the ManningRosen 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 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, fm^{−1}, , fm^{−1}, and .

We can observe that every pair of orbitals , , and has the same energy in the absence of the tensor potential . Thus, they can be viewed as the spin doublets; that is, the state with and forms a spin doublet with the state with and . On the other hand, in the presence of the tensor potential , 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: , . From Figure 2, we observe that, in the case of (no tensor interaction), members of spin doublets have the same energy. However, in the presence of the tensor potential , these degeneracies are removed.
3.2. pSpin Symmetry Solution
In this subsection we will obtain the energy eigenvalues and the corresponding wave functions for the pspin symmetric limit of the ManningRosen 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 pspin symmetric case, Also, we get the lower component of the Dirac spinor as where is normalization constant and .
The upper component of the Dirac spinor from (11) can be calculated as where . In Table 4, we use the same parameters as in the previous subsection (with ) and we obtain numerical results in the pspin symmetry in the presence and absence of tensor Hulthén form potential. The existence of the degeneracy was observed in the following doublets: , , , , and so on, in the absence of the tensor potential , and they are considered as pspin doublets. In Figure 3, we have investigated the effect of the tensor potential on the pspin doublet splitting by considering the following pairs of orbital: , .
