Dirac-hyperbolic scarf problem including a coulomb-like tensor potential

The Dirac equation have been solved for the q-deformed hyperbolic Scarf potential coupled to 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 normalized wave functions.


Introduction
One of the important tasks of quantum mechanics is to find exact analytic solutions of the wave equations are only possible for certain potentials of physical interest under consideration since they contain all the necessary information regarding the quantum system.It is well known that the exact solutions of these wave equations are only possible in a few simple cases such as the harmonic oscillator, the coulomb, pseudo harmonic potentials and others (IKHDAIR;SEVER, 2006SEVER, , 2008;;LANDAU;LIFSHITZ, 1977;NEITO, 1979;SCHIFF, 1968).
The exact solution of the Dirac equation with any potential is an important subject in relativistic quantum physics.These solutions are valuable tools in checking and improving models.Many authors have studied the Dirac under the spin and/or pseudo-spin symmetry for various potentials.For exemple, see (ARDA;SEVER, 2009;ESHGHI;MEHRABAN, 2011a and b;MOVAHEDI;HAMZAVI, 2011;PANAHI;BAKHSHI, 2011).
On the other hand, the spin symmetry appears when the magnitude of the scalar and vector potentials are nearly equal, i.e.,

≅
, in the nuclei (i.e., when the difference potential . However, the pseudo-spin symmetry occurs when , 1997).The bound states of nucleons seem to be sensitive to some mixtures of these potentials.The 2) symmetries of the Dirac Hamiltonian (GINOCCHIO, 2005).The spin symmetry is relevant for mesons (PAGE et al., 2001) and the pseudo-spin symmetry has been used to explain the features of deformed nuclei (BOHR et al., 1982), super-deformation (DUDEK et al., 1987), and to establish an effective nuclear shellmodel scheme (ARIMA et al., 1969;HECHT;ADLER, 1969).Summary, such symmetry, the near equality of an attractive scalar potential with a repulsive vector potential is well know in the literature (GINOCCHIO, 1997;2005) of the Dirac equation and has been proved very useful in describing the motion of nucleons in the relativistic mean fields resulting form nucleon-meson interactions, nucleon-nucleon Skyrme-type interactions and QCD sum rules.
According to the report given in the researcher (ESHGHI; MEHRABAN, 2011a), the q-deformed hyperbolic Scarf potential is defined by (1) In this present work, we give the approximate solutions, and corresponding wave functions of the Dirac equation for the q-deformed Hyperbolic Scarf potential (1) under the case of spin that including a coulomb-like tensor potential (AKCAY, 2009; ESHGHI; MEHRABAN, 2011c) (where = c R 7.78 fm is the Coulomb radius, a Z and b Z denote the charges of the projectile a and the target nuclei b, respectively.).We obtain the energy equation and the normalized corresponding spinor wave functions.In order to find the spectrum we use the parametric generalization of the Nikiforov-Uvarov (NU) is a powerful tool to solve of the second order linear differential equations with special orthogonal functions.

NU method
We give a brief description of the conventional NU method (NIKIFOROV; UVAROV, 1988).This method is based on solving the second order differential equation of hypergeometric-type by means of special orthogonal functions where: ) (s ) where: ) (r π is a polynomial of order at most one, and where: n a is a normalization constant and the weight function The function ) (s π and the parameter λ in the above equation are defined as follows The determination of q is the essential point in the calculation of ) (s π .It is simply defined by setting the discriminate of the square root which must be zero.The eigenvalues equation have calculated from the above equation For a more simple application of the method, we develop a parametric generalization of the NU method valid for any potential under consideration by an appropriate coordinate transformation ) (r s s = . The following equation is a general form of the Schrodinger equation written for any potentials (HAMZAVI et al., 2011;ESHGHI, 2011) as ) (12) We may solve this as follows.Comparing ( 12) with (3), yields Substituting these into (9), we find with the following parameters ( ) In Equation ( 14), the function under the square root should be the square of a polynomial according to the NU method.so that ( ) where: For each k the following π 's are obtained.The function for the k-value ( ) We also have from ( ) ( ) 2 ( ) s s s Thus, we impose the following condition to fix the k-value When ( 10) is used with ( 20) and ( 21) the following equation is derived This equation gived the energy spectrum of a given problem.By using ( 8) and together with (7), we have where: and ) , ( β α n P are Jacobi polynomials.By using (5), we get and the total wave function become In some problems the situation appears where 3 0 α = .For such problems, the solution given in (28) becomes as In some cases, one may need a second solution of (12).In this case, if the same procedure is followed, by using ( ) the solution becomes and the energy spectrum is where E is the relativistic energy of the system, ∇ − =   i P is the three-dimensional momentum operator, α and β are the 4 × 4 matrices which have the following forms (GRINER, 2000), respectively where I denotes the 2 2 × identity matrix and σ are three-vector Pauli spin matrices For a particle in a spherical (central) field, the total angular momentum J  The Dirac spinor can be written using the Pauli-Dirac representation where: ( , ) Substituting (37) into (34) and using the following relations (BJORKEN;DRELL, 1964) as Splitting off the angular part and leaving the radial wave function satisfy the following equations [ ] [ ] and where: ( 1) ( 1) 45) and ( 46) can not be solved exactly for 0, 1 k = − and 0,1 k = , because of the spin-orbit centrifugal term.We applied deform hyperbolic functions introduced for the first time by Arai in (ARAI, 1991) sinh 2 where q is real parameter and 0 > q .Substituting (1) and ( 2) into ( 46) and considering spin symmetry (the condition of spin symmetry et al., 1998), we have This equation is describes a particle of spin-1/2 such as the electron in the Dirac theory with qdeformed hyperbolic Scarf potential including a tensor coupling.
We apply the approximation for the centrifugal term of the form as where the dimensionless constant (ANTIA et al., 2010).However, when  by comparing ( 51) with ( 12) we have obtained the parameter set as Using ( 14), ( 16) and ( 52), we calculate the parameters required for the method where: ) (s π   22) and ( 52), the Energy eigenvalue equation for the potential under the consideration following as The corresponding normalized eigen-functions are obtained in terms of the functions, most of the first degree.In this method, if we take the following factorization According to the report given in the researcher(AKCA, 2009;AYDOGDU;SEVER, 2010;  HAMZAVI et al., 2010a and b; HAMZAVI et al.,  2011;ESHGHI; MEHRABAN, 2011a and c;  ZARRINKAMAR et al., 2010), the Dirac equation with the attractive scalar potential ( ) , where L is orbital angular momentum operator.For a given total angular momentum j, the eigenvalues of the radial wave functions of the upper-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 pseudo-orbital angular momentum quantum numbers for spin symmetry l and pseudo-spin symmetry l ~refer to the upper-and lowercomponent respectively.For a given spin-orbit quantum number 1, 2,..., k = ± ± the orbital angular momentum and pseudo-orbital angular momentum are -like equations for the upper and lower components, respectively Scheme become the conventional approximation Scheme suggested by Greene and Aldrich in Ref. (GREENE; ALDRICH, 1976).By using the approximation in (50) and transformation of the form r s α cosh = , we rewrite (48) as follows 2 2 2 d s d ds q s ds q bC q bV q