Eigen Spectra for q-Deformed Hyperbolic Scarf Potential Including a Coulomb-like Tensor Interaction

We study the Dirac equation for the q-deformed hyperbolic Scarf potential including a coulomb-like tensor potential under the spin symmetry. The parametric generalization of the Nikiforov-Uvarov method is used to obtain the energy eigenvalues equation and the unnormalized wave functins.

The description of phenomena at high energies requires the investigation of relativistic wave equations.Dirac equation describes the dynamics of spin-1/2 particle [1,2] has been used in solving many problems of nuclear and high-energy physics.The spin symmetry appears when the magnitude of the scalar and vector potentials are nearly equal, i.e., ) (

≅
, in the nuclei while the pseudospin symmetry occurs when ) ( ) ( r V r V s v − ≅ [3].The spin symmetry is relevant for mesons [4] and the pseudospin symmetry has been used to explain the features of deformed nuclei [5], superdeformation [6], and to establish an effective nuclear shell-model scheme [7,8].The solutions of the Dirac equation having the pseudospin and spin symmetries have been strongly studied in recent years.The Dirac equation under the spin and/or pseudospin symmetry has been investigated by with different type of potentials such as the Mie-type potential [9], Coulomb-like potential [10], Wood-Saxon potential [11], Eckart potential [12], the generalized Poschl-Teller potential [13], the Makarov potential [14], the Davidson potential [15], the special Hulthen potential [16], the Hyperbolic potential [17], etc.
The q-deformed hyperbolic Scarf potential is defined by q r ln > α [26,27], We write this potential that may be used in decribing the nuclei interactins.Recently, Nikiforov-Uvarov method has been introduced for solving the Schrodinger, Klien-Gordon, Dirac equations.These equations with the well known potentials are solved by using the Nikiforov-Uvarov method [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44].This method has shown its power in calculating the exact energy levels for some solvable quantum systems.Our aim in this paper, is to attempt to study the Dirac equation with q-deformed Hyperbolic Scarf potential (1) for s-wave including a coulomb-like tensor potential (where 78 fm is the Coulomb radius, a Z and b Z denote the charges of the projectile a and the target nuclei b, respectively.)under the spin symmetry.We obtain the bound state energy equation and the corresponding spinor wave functions by using the parametric generalization of the Nikiforov-Uvarov method [20,45].

Dirac equation
According to the report given in the researcher [18][19][20][21][22], the Dirac equation with the scalar potential ( ) s V r , vector potential ( ) v V r and a tensor potential where E is the relativistic energy of the system, is the momentum operator, α and β are the 4 × 4 matrices.For spherical nuclei, the angular momentum J  and the lower-spinor components respectively, m is the projection of the total angular momentum on the z-axis, n is the radial quantum number.The orbital and the pseudoorbital angular momentum quantum numbers for spin symmetry l and pseudo-spin symmetry l ~refer to the upper-and lower-component respectively.
Splitting off the angular part and leaving the radial wave function satisfy the following equations [ ] [ ] 12) and ( ) nk F r in ( 13), one obtains Schrodinger-like equations for the upper and lower components, respectively where ( 1) ( 1) + and ( 1) ( 1) . We applied deform hyperbolic functions introduced for the first time by Arai [46] sinh 2 where q is real parameter and 0 > q .

Spin symmetry for s-wave
Substituting (1) into ( 14 ) [47,48], we obtain the second-order differential equation for the upper component of the Dirac spinor as follows, and r s (18) where Following to the report given in the researcher [20,45], we obtain the parameter set as We calculate the parameters required for the method 2 1 ) ( where Therefore, we obtain the energy eigenvalue equation for the potential under the consideration [20,45], We write the corresponding unnormalized eigenfunctions are obtained in terms of the functions, ] Now, let us give the corresponding upper Dirac spinor.Using [25], one obtains where n c is a normalization constant and it is determined by the condition

Spin symmetry with tensor coupling
Substituting 1 = q in (1), Eqs. ( 1) and ( 2) into ( 14) and considering spin symmetry (the condition of spin symmetry 0 ) [47,48], we obtain the second-order differential equation for the upper component of the Dirac spinor as follows, By using the approximation scheme suggested by Greene and Aldrich [49] r r Following to the report given in the researcher [20,45], we obtain the Energy eigenvalue equation for the potential under the consideration Now, let us give the corresponding lower Dirac spinor.Using [25], one obtains

Conclusion
We have investigated the Dirac equation with q-deformed hyperbolic Scarf potential for swave and including Coulomb-like tensor coupling in the case of spin symmetry.By using the approximation scheme and the parameteric generalization of the Nikiforov-Uvarov method, we have obtained the energy eigenvalues equation and the unnormalized wave functions.
Dirac Hamiltonian, where L is orbital angular momentum.The Dirac spinor can be classified according to their angular momentum j, k, the radial wave functions of the upper-