Spin and Pseudospin Symmetries with Trigonometric Pöschl-Teller Potential including Tensor Coupling

We study approximate analytical solutions of the Dirac equation with the trigonometric Pöschl-Teller (tPT) potential and a Coulomb-like tensor potential for arbitrary spin-orbit quantum number κ under the presence of exact spin and pseudospin (pspin) symmetries. The bound state energy eigenvalues and the corresponding two-component wave functions of the Dirac particle are obtained using the parametric generalization of the Nikiforov-Uvarov (NU) method. We show that tensor interaction removes degeneracies between spin and pseudospin doublets. The case of nonrelativistic limit is studied too.


Introduction
The Dirac equation, which describes the motion of a spin-1/2 particle, has been used in solving many problems of nuclear and high-energy physics.The spin and the p-spin symmetries of the Dirac Hamiltonian had been discovered many years ago; however, these symmetries have recently been recognized empirically in nuclear and hadronic spectroscopes [1].Within the framework of Dirac equation, pspin symmetry used to feature the deformed nuclei and the super deformation to establish an effective shell-model [2][3][4], whereas spin symmetry is relevant for mesons [5].The spin symmetry occurs when the scalar potential () is nearly equal to the vector potential () or equivalently () ≈ (), and p-spin symmetry occurs when () ≈ −() [6,7].The p-spin symmetry refers to a quasidegeneracy of single nucleon doublets with nonrelativistic quantum number (, ,  =  + 1/2) and ( − 1,  + 2,  =  + 3/2), where , , and  are single nucleon radial, orbital, and total angular quantum numbers, respectively [8,9].The total angular momentum is given by  = l + s, where l =  + 1 pseudoangular momentum and s is p-spin angular momentum [10,11].Liang et al. [12] investigated the symmetries of the Dirac Hamiltonian and their breaking in realistic nuclei in the framework of perturbation theory.Guo [13] used the similarity renormalization group to transform the spherical Dirac operator into a diagonal form, and then the upper (lower) diagonal element became an operator describing Dirac (anti-)particle, which holds the form of the Schrödinger-like operator with the singularity disappearing in every component.Chen and Guo [14] investigated the evolution toward the nonrelativistic limit from the solutions of the Dirac equation by a continuous transformation of the Compton wavelength .Lu et al. [15] recently showed that the p-spin symmetry in single particle resonant states in nuclei is conserved when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign.
Tensor potentials were introduced into the Dirac equation with the substitution ⃗  → ⃗  −   ⋅ r (), and a spin-orbit coupling is added to the Dirac Hamiltonian [16,17].Lisboa et al. [18] have studied a generalized relativistic harmonic oscillator for spin-1/2 particles by considering a Dirac Hamiltonian that contains quadratic vector and scalar potentials together with a linear tensor potential, under the conditions of pseudospin and spin symmetry.Alberto et al. [19] studied the contribution of the isoscalar tensor coupling to the realization of pseudospin symmetry in nuclei.Akcay showed that the Dirac equation with scalar and vector quadratic potentials and a Coulomb-like tensor potential can be solved exactly [20]; also, he exactly solved Dirac equation with tensor potential containing a linear and Coulomb-like terms too [21].Aydogdu and Sever obtained exact solution of Dirac equation for the pseudoharmonic potential in the presence of linear tensor potential under the pseudospin symmetry and showed that tensor interactions remove all degeneracies between members of pseudospin doublets [22].Zhou et al. solved Dirac equation approximately for Hulthén potential including Coulomb-like tensor potential with arbitrary spin-orbit coupling number  under spin and pseudospin symmetry limit [10].Aydogdu and Sever solved approximately for the Woods-Saxon potential and a tensor potential with the arbitrary spin-orbit coupling quantum number  under pseudospin and spin symmetry [23].Very recently, Hamzavi et al. gave exact solutions of the Dirac equation for Mie-type potential and positiondependent mass Coulomb potential with a Coulomb-like tensor potential [24,25] and pseudoharmonic potential with linear plus Coulomb-like tensor potential [26].
The tPT potential has been proposed for the first time by Pöschl and Teller [27] in 1933 to describe the diatomic molecular vibration.Chen [28] and Zhang and Wang [29] have studied the relativistic bound state solutions for the tPT potential and hyperbolical PT (Second PT) potential, respectively.Liu et al. [30] studied the tPT potential within the framework of the Dirac theory.Recently, Candemir [31] investigated the analytical s-wave solutions of Dirac equation for tPT potential under the p-spin symmetry condition.Very recently, Hamzavi and Rajabi [32] studied the exact wave solution ( = 0) of the Schrödinger equation for the vibrational tPT potential.The tPT takes the form: where the parameters  1 and  2 describe the property of the potential well, while the parameter  is related to the range of this potential [30].We find out that this potential has a minimum value at  0 = (1/)tan −1 ( 4 √ 1 / 2 ).The case when  1 =  2 , the minimum value is at  0 = /4 ∈ (0, ∞) for  > 0. The second derivative which determines the force constants at  =  0 is given by for any  value, and thus which means that () at  =  0 has a relative minimum for  > 0. When  1 =  2 =  then minimum value is ( 0 ) = 4 and  2 / 2 | = 0 = 32 2 .In Figures 1(a Here the potential has a minimum value at  0 = 0.27027/.The curve is nodeless in  ∈ (0, /2).For example, with  = 0.30 fm −1 ,  0 = 2.8303 fm and minimum potential ( 0 = 2.8303 fm) = 15.746fm −1 .It is worthy to note that in the limiting case when  → 0, the tPT potential can be reduced to the Kratzer potential [33,34] where   is the equilibrium intermolecular separation and   is the dissociation energy between diatomic molecules.In our case,   =  1 ,  =  2 , and   = 1/.In the case of  = 0 it reduces to the molecular potential which is called the modified Kratzer potential proposed by Simons et al. [35] and Molski and Konarski [36].In the case of  = −  , this potential turns into the Kratzer potential, which includes an attractive Coulomb potential and a repulsive inverse square potential introduced by Kratzer in 1920 [37].
The aim of the present work is to extend our previous work [32] to the relativistic case and  ̸ = ± 1 (rotational case) including a Coulomb-like tensor potential.We introduce a convenient approximation scheme to deal with the strong singular centrifugal term.The ansätz of this approximation possesses the same form of the potential and is singular as the centrifugal term  −2 .
Since the relativistic solution is indispensable, we need to solve the Dirac equation with flexible parameters tPT potential model.However, the Dirac-tPT problem can no longer be solved in a closed form due to the existence of spinorbit coupling term ( ± 1) −2 and it is necessary to resort to approximation methods.Therefore, we use an approximation scheme to deal with this term and solve approximately the Dirac equation with the tPT potential for arbitrary spinorbit quantum number .In the presence of spin and p-spin symmetric limitation, we obtain the approximate relativistic bound state solutions including the energy eigenvalue equations and the corresponding unnormalized upper-and lowerspinor components of the wave functions using the concepts of parametric generalization of the NU method [38] since the relativistic corrections are not neglected.
The structure of the paper is as follows.In Section 2, in the context of spin and p-spin symmetry, we briefly introduce the Dirac equation with scalar and vector tPT and also tensor potential potentials for arbitrary spin-orbit quantum number .The parametric generalization of the NU method is displayed in the appendix.In the presence of the spin and p-spin symmetry, the approximate energy eigenvalue equations and corresponding two-component wave functions of the Dirac-tPT problem are obtained, and effect of tensor potential is shown in this section.The nonrelativistic limit of the problem is discussed in this section too.Finally, our final concluding remarks are given in Section 3.

Dirac Equation including Tensor Coupling
The Dirac equation for fermionic massive spin-1/2 particles moving in an attractive scalar potential (), a repulsive vector potential (), and a tensor potential where  is the relativistic energy of the system, ⃗  = − ⃗ ∇ is the three-dimensional momentum operator, and  is the mass of the fermionic particle [64].⃗  and  are the 4 × 4 usual Dirac matrices given as where  is 2 × 2 unitary matrix and ⃗  are three-vector spin matrices The total angular momentum operator ⃗  and spin-orbit operator  = ( ⃗ ⋅ ⃗ +1), where ⃗  is orbital angular momentum operator, 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  and can be written as follows: where   ( ⃗ ) is the upper (large) component and   ( ⃗ ) is the lower (small) component of the Dirac spinors.   (, ) and  l  (, ) are spin and pseudospin spherical harmonics, respectively, and  is the projection of the angular momentum on the -axis.Substituting (7) into (4) and using the following relations: with the following properties: one obtains two coupled differential equations for upper and lower radial wave functions   () and   () as where Eliminating   () and   () from (10a) and (10b), we 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 pseudospin symmetry l as and the quasidegenerate doublet structure can be expressed in terms of a pseudospin angular momentum s = 1/2 and pseudoorbital angular momentum l, which is defined as where  = ±1, ±2, . ... For example, (1 1/2 , 0 3/2 ) and (1 3/2 , 0 5/2 ) can be considered as pseudospin doublets.
where   =  +  + 1 is new spin-orbit centrifugal term and  =  and  = − − 1 for  < 0 and  > 0, respectively.The Schrödinger-like equation (15a) that results from the Dirac equation is a second-order differential equation containing a spin-orbit centrifugal term   (  + 1) −2 which has a strong singularity at  = 0, and needs to be treated very carefully while performing the approximation.In absence of tensor interaction, (15a) has an exact rigorous solution only for the states with  = −1 because of the existence of the centrifugal term (+1)/ 2 .However, when this term taken into account, the corresponding radial Dirac equation can no longer be solved in a closed form, and it is necessary to resort to approximate methods.Over the last few decades, several schemes have been used to calculate the energy spectrum.The main idea of these schemes relies on using different approximations of the spin-orbit centrifugal coupling term   (  + 1)/ 2 .So, we need to perform a new approximation for the spin-orbit term as a function of the tPT potential parameters.Therefore, we resort to use an appropriate approximation scheme to deal with the centrifugal potential term as where  0 = 1/12 is a dimensionless shifting parameter and  ≪ 1.The approximation ( 16) is done on the basis that sin() = − 3 /3!+ 5 /5!− 7 /7!+⋅ ⋅ ⋅ and in the limit when  → 0, sin() ≈ .To show the validity and accuracy of our choice to the approximation scheme (16), we plot the centrifugal potential term 1/ 2 and its approximations:  2 /sin 2 () and  2 ( 0 +1/sin 2 ()) in Figure 2. As illustrated, the three curves coincide together and show how accurate this replacement is.Thus, employing such an approximation scheme, we can then write (15a) as Followed by making a new change of variables () = sin 2 (), this allows us to decompose the spin-symmetric Dirac equation ( 17) into the Schrödinger-type equation satisfying the upper-spinor component  , (), where   () ≡  , () has been used.If the previous equation is compared with (A.2), we can obtain the specific values for constants   ( = 1, 2, 3) as In order to obtain the bound state solutions of (17), it is necessary to calculate the remaining parametric constants, that is,   ( = 4, 5, . . ., 13), by means of the relation (A.5).Their specific values are displayed in Table 1 for the relativistic tPT potential model.Further, using these constants along with (A.7), we can readily obtain the energy eigenvalue equation for the Dirac-tPT problem as or equivalently To show the procedure of determining the energy eigenvalues from ( 21), we take a set of physical parameter values,  = 10 fm −1 ,  1 = 5.0 fm −1 ,  2 = 3.0 fm −1 ,   = 0 fm −1 , and  = 0.8, 0.6, 0.4, 0.2, 0.04, and 0.02 [30].
members of spin doublets have same energy.However, in the presence of the tensor potential  ̸ = 0, these degeneracies are removed.We can also see that spin doublet splitting increases with increasing .On the other hand, in order to establish the upper-spinor component of the wave functions  , (), namely, (15a), the relations (A.8), (A.9), (A.10), and (A.11) are used.Firstly, we find the first part of the wave function as Secondly, we calculate the weight function as which gives the second part of the wave function as where  (,)  () are the orthogonal Jacobi polynomials.Finally the upper spinor component for arbitrary  can be found through the relation (A.11) or where and   is the normalization constant.Further, the lowerspinor component of the wave function can be calculated by using where  ̸ = −  +   and in the presence of the exact spin symmetry (  = 0), only positive energy states do exist.
On the other hand, the upper-spinor component of the Dirac wave function can be calculated by where  ̸ =  +   , and in the presence of the exact p-spin symmetry (  = 0), only negative energy states do exist.

The Nonrelativistic Limiting Case.
In this section, we study the energy eigenvalue equation ( 21) and upper-spinor component of wave function (26) of the Dirac-tPT problem under the nonrelativistic limits:   = 0,  → ,   −  ≃   , and  +   ≃ 2.Thereby, we obtain the energy equation of the Schrödinger equation with any arbitrary orbital state for the tPT potential as In the limit when  → 0, the vibration-rotation energy formula (36) reduces into a constant value: Further, there is no loss of generality if  0 = 0; then, (36) becomes where  = 0, 1, 2, . . .and  = 0, 1, 2, . . .are the vibration and rotation quantum numbers, respectively.To obtain numerical energy eigenvalues for the present potential model, we take the following set of parameter values; namely,  = 10 fm −1 ,  1 = 5.0 fm −1 ,  2 = 3.0 fm −1 , and  = 1.2, 0.8, 0.4, 0.2, 0.02, and 0.002 [30].As seen from Table 6, in the limit when potential range parameter  approaches zero, the energy eigenvalues approach a constant value given by (37).Also, we can get the radial wave functions of the Schrödinger equation with tPT potential as Inserting  = sin 2 () in the previous equation, we can obtain where   is a normalization factor to be calculated from the normalization conditions.

Concluding Remarks
In this work, we have studied the bound state solutions of the Dirac equation with trigonometric Pöschl-Teller and Coulomb-like tensor potentials for any spin-orbit quantum number .By making an appropriate approximation to deal with the spin-orbit centrifugal (pseudo-centrifugal) coupling term, we have obtained the approximate energy eigenvalue equation and the unnormalized two components of the radial wave functions expressed in terms of the Jacobi polynomials using the NU method.It is found that tensor interaction removes degeneracies between each pair of pseudospin or spin doublets.

Figure 1 :
Figure 1: (a) A plot of the tPT potential for  = 0.2 fm −1 .(b) A plot of the tPT potential for  = 0.8 fm −1 .

Figure 3 :
Figure 3: Effect of tensor potential on spin doublets.

Figure 4 :
Figure 4: Effect of tensor potential on p-spin doublets.

Table 1 :
The specific values of the parametric constants for the spin symmetric Dirac-tPT problem.

Table 2 :
The bound state energy eigenvalues in units of fm −1 of the spin symmetry tPT potential for several values of  and  with  = 0.8.

Table 3 :
The spin symmetric bound state energy eigenvalues in units of fm −1 of the tPT potential for several values of  and  with  = 10.0,  1 = 5.0,  2 = 3.0,   = 0, and  = 0.
The reason is that term 2 gives different contributions to each level in the spin doublet because  takes different values for each state in the spin doublet.

Table 4 :
The bound state energy eigenvalues in units of fm −1 of the p-spin symmetry tPT potential for several values of  and  with  = 0.8.

Table 5 :
The bound state energy eigenvalues in units of fm −1 of the p-spin symmetry tPT potential for several values of  and  with  = 0.