Tensor coupling and relativistic spin and pseudospin symmetries of the Pöschl–Teller-like potential

In this research, we have been obtained the Dirac equation for second Pöschl–Teller-like potential including a Coulomb-like tensor interaction with arbitrary spin–orbit coupling quantum number j. Under the condition of spin and pseudospin (p-spin) symmetries, we use the basic concept of the supersymmetric shape invariance formulism in quantum mechanics and the function analysis method to obtain energy eigenvalues and corresponding two-component spinors of the Dirac particle. 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 (V(r) -S(r) = 0) possesses not only a spin symmetry but also a U(3)symmetry, but a Dirac Hamiltonian in the case of (V(r) ? S(r) = 0) possesses a pseudospin symmetry and a pseudo-U(3)symmetry. Meng et al. [8] have showed that the pseudospin symmetry is exact under the condition (d(V(r) ? S(r))/dr = 0). In addition, Alhaidari et al. [9] have investigated in detail physical interpretation on the three-dimensional Dirac equation in the case of spin symmetry limit (V(r) -S(r) = 0) and pseudospin symmetry limit (V(r) ? S(r) = 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].
In this paper, we attempt to study the spin and pseudospin symmetry solutions of the Dirac equation for arbitrary quantum number j with the Pöschl-Teller-like potential. This is given by The potential parameters V 1 and V 2 describe the property of the potential well, V 1 [ V 2 , while a is related to the range of the potential [19][20][21][22][23][24]. The behavior of this potential with respect to four different values of a is shown in Fig. 1.
Also Tensor potentials have been introduced into the Dirac equation with the substitution p ! p À im x b Á r U r ð Þ and a spin-orbit coupling is added to the Dirac Hamiltonian [25,26]. For more review of tensor interaction, one can refer to the [27][28][29][30][31][32][33][34] that authors used different potential and different kinds of tensor potential. Here we study a tensor potential in the Coulomb-like form as follows: where R C = 7.78 fm is the coulomb radius, and Z a and Z b denote the charges of the projectile a and the target nuclei b, respectively. The Potential in Eq. (1) is also one of the important examples for the special case of the multiparameter exponential-type potential model [35,36]. By solving the Klein-Gordon equation and Dirac equation with equal scalar and vector Pöschl-Teller-like potentials, the exact relativistic energy equations have been obtained for the swave bound states (l = 0) [37,38]. Using the conventional approximation scheme proposed by Greene and Aldrich [39] to deal with the centrifugal term, Dong et al. [40,41] have investigated the arbitrary l-wave bound-state solutions of the Schrödinger equation and Klein-Gordon equation with the Pöschl-Teller-like potential in terms of the standard function analysis method. However, as far as we know, one has not reported the investigation of the spin and pseudospin symmetries solutions of the Dirac equation with the Pöschl-Teller-like potential including a Coulomblike potential as a tensor interaction for the arbitrary spinorbit quantum number j. In this paper, we solve approximately the Dirac equation with the Pöschl-Teller-like potential for the spin-orbit quantum number j. Under the condition of spin and pseudospin symmetries, we study the bound-state energy equation and corresponding spinor wave functions in terms of the basic concept of the supersymmetric shape invariance formalism [42,43] and the function analysis method.

Dirac equation including tensor coupling
The Dirac equation for fermionic massive spin-1/2 particles moving in attractive scalar S(r), repulsive vector V(r) and tensor U(r) potentials is (in units ⁄ = c = 1) where E is the relativistic energy of the system, p ¼ Ài r is the three-dimensional momentum operator and M is the mass of the fermionic particle. Further, ã and b are the 4 9 4 Dirac matrices given by where I is 2 9 2 unitary matrix and r are three-vector spin matrices The total angular momentum operator J and spin-orbit where L is orbital angular momentum of the spherical nucleons, commute with the Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are j ¼ j þ 1 2 À Á [ 0 and j ¼ À j þ 1 2 À Á \0 for unaligned spin j ¼ l À 1 2 and the aligned spin j ¼ l þ 1 2 , respectively. (H 2 , K, J 2 , J z ) can be taken as the complete set of the conservative quantities. Thus, the spinor wave functions can be classified according to their angular momentum j; spinorbit quantum number j and the radial quantum number n can be written as follows: where f nj r ð Þ is the upper (large) component and g nj r ð Þ is the lower (small) component of the Dirac spinors. Y jm l (h, /) and Yl jm h; / ð Þ are spin and p-spin spherical harmonics, respectively, and m is the projection of the angular momentum on the z-axis. Substituting Eq. (6) into Eq. (3) and using the following relations: Fig. 1 The behavior of the Pöschl-Teller-like potential with respect to four different values of a, and for V 1 = 0.5, V 2 = 0.1 together with the following properties: one obtains two coupled differential equations for upper and lower radial wave functions F nj (r) and G nj (r) as: are the difference and the sum potentials, respectively. Eliminating F nj (r) and G nj (r) from Eqs. (9), we finally obtain the following two Schrödinger-like differential equations for the upper and lower radial spinor components, respectively: where j j À 1 ð Þ¼ll þ 1 À Á and j(j ? 1) = l(l ? 1). The quantum number j is related to the quantum numbers for spin symmetry l and p-spin symmetryl as: and the quasidegenerate doublet structure can be expressed in terms of a p-spin angular momentums ¼ 1=2 and pseudoorbital angular momentuml, which can be defined as: where j = ± 1, ± 2, …. For example, (1s 1/2 , 0d 3/2 ) and (1p 3/2 , 0f 5/2 ) can be considered as p-spin doublets.

P-spin symmetry limit
Within the pseudospin symmetry case, (d[V(r) ? S(r)]/ dr = dR(r)/dr = 0) or R(r) = C ps = constant and p-spin symmetry is exact in the Dirac equation [8,[48][49][50]. In this part, we consider D(r) as the Pöschl-Teller-like potential, the equation obtained for the lower component of the Dirac spinor, G nj (r), becomes Equations (13) and (14) can be solved analytically only for the case of k j = -1 and k j = 1 due to the pseudocentrifugal terms, k j (k j ? 1)/r 2 and k j (k j -1)/r 2 , respectively. Using the approximation scheme suggested by Greene and Aldrich [39], we can express approximately the pseudocentrifugal term in the following form [51][52][53][54]: This is a good approximation for small values of the parameter a and it breaks down for large values of a. For the case of ar ( 1, one can show that (see Fig. 2) where c 0 = 1/12 is a dimensionless constant.
Using the basic concept of the supersymmetric shape invariance formulism [42,43], we solve Eq. (17). The ground-state upper component F 0,j (r) can be written as: where W(r) is called a superpotential in supersymmetric quantum mechanics [43]. Substituting Eq. (18) into Eq. (17), we have the following equation for W(r) whereẼ 0;j is the ground-state energy. Considering the compatibility between the superpotential function W(r) and the right-hand side of Eq. (19), we write the superpotential W(r) in the following form: Substituting Eq. (20) into (18)   Substituting Eq. (20) into Eq. (19) and comparing equal powers of two sides in Eq. (19), we obtain the following relationships: Considering the regularity conditions, A [ 0 and B \ 0, we obtain the coefficients A and B by solving Eqs. (23) and (24), Using the expression given in Eq. (20), we construct the following two supersymmetric partner potentials: Setting (a 0 , b 0 ) = (A, B), one can get the following shapeinvariant relationship, The energy eigenvalues of the potential V -(r) can be determined using the shape invariance approach [42]. The energy eigenvalues of the potential V -(r) are given bỹ where the quantum number n = 0, 1, 2, …. From Eqs. (19) and (27), we have the following relation: From Eqs. (17) and (31), we can find the solution for E nj in Eq. (17), where we have employed the relationẼ 0;j ¼ À A þ B ð Þ 2 . Substituting Eqs. (25) and (26) into Eq. (32) and using Þa 2 c 0 , we can find the energy eigenvalue equation of the relativistic Pöschl-Teller-like potential under the condition of spin symmetry, where the quantum number n ¼ 0; 1; 2; . . .; \ 1 a A þ B ð Þ. Using the recursion operator approach [55,56], we can determine the excited state upper components from the superpotential W(r) given in Eq. (20) and the ground-state upper component, F 0,j (r) given in Eq. (21).
To find the corresponding wave functions, we take the function analysis method to calculate the unnormalized excited state upper components. Substituting Eq. (32) into Eq. (17), we obtain This equation is the well-known differential equation satisfied by the hypergeometric function 2 F 1 (a, b; c; s), i.e., where a ¼ Àn; b ¼ n À 2A a À 2B a , and c ¼ 1 2 À 2B a . Using the original variable r, the upper component F nj (r) corresponding to energy level E nj can be expressed as follows: where A and B are given in Eqs. (25) and (26), respectively. Using Eq. (9a) and the expression of F nj (r) given in Eq. (37), we obtain the lower spinor component G nj (r) corresponding to the upper component F nj (r) and energy level E nj , where E nj = -M ? C s . From Eqs. (37) and (38) Table 1. In Fig. 3, we have investigated the effect of the tensor potential on the bound states.

P-spin symmetry solution
In this subsection, we will obtain the energy eigenvalues and the corresponding wave functions for the p-spin symmetric limit by substituting Eq. (16) into Eq. (14) that Table 1 The spin symmetric bound-state energy levels (in unit of fm -1 ) of the Coulomb potential taking several values of n and j l n, j \ 0 (l,  Archive of SID leads us to obtain the following Schrödinger-like equation for the lower spinor component: whereẼ nj is defined asẼ nj ¼ E nj þ M ð Þ E nj À M À C ps À Á Àk j k j À 1 ð Þa 2 c 0 . We write the super-potentialW r ð Þ in the following form: that leads us to obtain the ground-state upper component To avoid repetition in our solution to Eq. (14), We follow the same procedures explained in the previous section to obtain the energy eigenvalue equation, and the corresponding wave functions for the lower Dirac spinor as: Finally, the upper-spinor component of the Dirac equation can also be obtained via Eq. (9b) as: where E nj = M ? C ps . By taking a set of physical parameter values, C ps = 2, a = 1.2, M = 10, V 1 = 5, V 2 = 3, c 0 = 1/12, when n = 1 and j = 2, Eq. (44) yields the following values of E 1,2 : -9.92187466, 11.814382. We choose E 1,2 = -9.92187466 as the solution of Eq. (44), and find that the values ofÃ and B areÃ ¼ 6:32825 andB ¼ À3:73088, respectively. These values satisfy the regularity conditions:Ã [ 0;B\0 and A [ ÀB. If we take E 1,2 = 11.814382 as the solution of Eq. (44), the values ofÃ andB areÃ ¼ 0:379142 and B ¼ À1:25016, which do not satisfy the regularity condition:Ã [ ÀB. Therefore, we can only take the negative energy value E 1,2 = -9.92187466 as the solution of Eq. (44). Using the same parameter values of a, M, V 1 , V 2 and C ps , the numerical solutions of Eq. (44) for the other values of n and j are presented in Table 2. Table 2 The p-spin symmetric bound-state energy levels (in unit of fm -1 ) of the Coulomb potential taking several values of n and j l n, j \ 0 (l, j) E n,j\0 , Also in Fig. 4, we have investigated the effect of the tensor potential on the p-spin doublet splitting by considering some pairs of orbitals.

Conclusion
In this paper, we have approximately studied the boundstate solutions of the Dirac equation for the Pöschl-Tellerlike potential with a Coulomb-like tensor interaction within the framework of spin and pseudospin symmetry limits. By employing an improved approximation scheme to deal with the pseudocentrifugal term 1/r 2 and the SUSYQUM technique, We have obtained the energy levels in a closed form and the corresponding wave functions in terms of the hypergeometric function 2 F 1 (a, b; c; s). Some numerical values of the energy levels are reported in Tables 1 and 2 under the condition of the spin and p-spin symmetries, respectively. Obviously, the degeneracy between the members of doublet states in spin and p-spin symmetries is removed by tensor interaction. The p-spin spectra of the present potential are identical to those ones obtained in Ref. [57] as the potential parameters U 0C = 0, a = 0.15, M = 1.0, C ps = -5.