Dirac Equation with Unequal Scalar and Vector Potentials under Spin and Pseudospin Symmetry

An approximate solution of the Dirac equation in the D-dimensional space is obtained under spin and pseudospin symmetry limits for the scalar and vector inversely quadratic Yukawa potential within the framework of parametric Nikiforov-Uvarov method using a suitable approximation scheme to the spin-orbit centrifugal term. The two components spinor of the wave function and their energy equations are fully obtained. Some numerical results are obtained for the energy level with various dimensions (D), quantum number (n), vector potential 0 V and scalar potential 0 S . The results obtained under spin symmetry using either 0 V or 0 S are equal to the results obtained using 0 0 V S  . But under the pseudospin symmetry, the results obtained using 0 V or 0 S are not equal to the results obtained using 0 0 V S  .


INTRODUCTION
Dirac equation is a relativistic wave equation derived in particle Physics by Paul Dirac in 1928 which described the spin -1/2 massive particles such as electrons and quarks 1,2 .
This equation is consistent with the principles of quantum mechanics and the theory of special relativity.It was the first theory to completely describe the special relativity in quantum mechanics 3 .In the framework of the Dirac equation, the relativistic symmetries (spin and pseudospin) of the Dirac Hamiltonian has been discovered which have been recently recognized empirically in spherical atomic nuclei and hadronic spectroscopic [4][5][6] .
The pseudospin symmetry was used to explain some certain features such as deformed nuclei 7 , superdeformation and effective shell coupling scheme 8,9 .The spin symmetry is relevant to meson 10 .The pseudospin symmetry refers to a quasi-degeneracy of single nucleon doublets with non-relativistic quantum number ( , , 1 / 2) nj  and ( 1, 2, 3 / 2), nj     where n denote the single nucleon radial number, is the orbital angular quantum number and j is the total angular quantum number 4,5 .The total angular quantum number , js  where 1  is pseudo-angular momentum and s is pseudospin angular momentum 11 .The pseudospin symmetry occurs when the sum of the potential for the repulsive Lorentz vector potential ()  Vr and the attractive Lorentz scalar potential () Sr is a constant, that is ( ) ( ) ( ) r V r S r     constant while the spin symmetry occurs as the difference of the potential between the repulsive Lorentz vector potential () Vr and the attractive Lorentz scalar potential () Sr is a constant, that is ( ) ( ) ( ) r V r S r     constant [12][13][14] .In line with the importance and usefulness of the Dirac equation, a great number of studies have been recently devoted to obtain the analytic solutions of the relativistic Dirac equation with the well-known potential models in the framework of the spin and pseudospin symmetry limits in the presence or absence of tensor potential.For instance, Maghsoodi et al. 15 , studied Dirac particles in the presence of the Yukawa potential plus a tensor interaction in SUSY QM framework.Oyewumi et al. 16 , investigated k states solutions for the fermionic massive spin-1/2 particles interacting with double ring-shaped Kratzer and oscillator potentials.Onate et al. 17 , obtained approximate solutions of the Dirac equation for Second Pӧschl-Teler like scalar and vector potentials with a Coulomb tensor interaction.Eshghi et al. [18][19][20] , studied relativistic Killingbeck energy states under external magnetic fields, bound states of (1+1)-dimensional Dirac equation with kink-like vector potential and delta interaction and the relativistic bound states of a non-central potential.
Hosseinpour and Hassanabadi where  is the depth/strength of the potential,

2
  and  is the screening parameter.

Dirac Equation
The Dirac equation for fermionic massive spin-1/2 particles moving in the field of an attractive scalar potential and repulsive vector potential is given as where E is the relativistic energy of the system, pi    is the three-dimensional momentum operator and M is the mass of the particle,  and  are the 44  usual Dirac matrices.The spinor wave functions can be classified according to their angular momentum , j the spin-orbit quantum number k , and the radial quantum number n as follows  () ()

Parametric Nikiforov-Uvarov Method.
To solve a second-order differential Schrӧdinger-like equation using parametric Nikiforov-Uvarov method, we first consider the differential equation of the form 39-43 According to the parametric Nikiforv-Uvarov method, the condition for eigen energies and eigen functions respectively are 40, 44-47       where the parametric constants in equations ( 9) and (10) above are defined in Appendix 1.

3.1
The Spin Symmetry Limit.
Under the spin symmetry limit, () 0 dr dr   and () , 49 .The sum potential in this case is the summation of the scalar potential and the vector potential.Thus, the interacting potential becomes It is noted that equation ( 6) is a second-order differential equation containing a spin- which has strong singularity at 0. r  Thus, this needs a careful treatment while performing the approximation.Taking into account the spin-orbit centrifugal term, the radial Dirac equation can no longer be solved in a closed form for the exact solutions thus, it becomes obvious to resort to the approximate solutions.Different approximation schemes have been applied to the spin-orbit centrifugal term over the years.
The choice of approximation applied depends on the interacting potential.Considering the potential given in equation ( 11) which also has a strong singularity at 0, r  we resort to use the following approximation scheme for a short potential range [50][51][52][53] that is a good approximation to the centrifugal term   where ( ) ( ) FF ry  and the following are used for mathematical simplicity 1 , () into equations ( 9) and ( 10), we have the energy equation for the spin symmetry as follows 2) 4( ) and the upper component of the wave function for the spatial dimensions as ) , and  
Under the pseudospin symmetry limit, () 0 dr dr   and , () ps rC  48,49 .The sum potential for a case when the scalar potential is not equal to zero, the interacting potential becomes Figure 3: The difference potential   Substituting equations ( 20) and ( 12) into equation ( 7) and using the change of variable as in the spin symmetry, we have where we used the following for mathematical simplicity (2 ) , .
Using the same procedures as in the spin symmetry, we easily obtain the negative component energy of the Dirac equation and its wave function in the form )) (, The upper component of the wave function is given as (

RESULTS AND DISCUSSION
The energy eigenvalues equations obtained for spin symmetry and pseudospin symmetry in equations ( 16) and ( 24) respectively are quadratic algebraic equations according to nk E .It therefore becomes necessary to obtain the solutions of these algebraic equations with respect to energy by choosing some numerical values for n and .k In Table 1, we obtained the energy eigenvalues for spin symmetry limit with 1 5, V  and 0 1 S  are equal to the energy obtained with 0 1 V  and 0 5 S  .This is due to the fact that since in the energy equation for the spin symmetry we have 00 This trend is also observed when 0 1, V  0 0 S  and when 0 0, V  0 1 S  .It is also observed that the energy obtained decreases as n increases.Similarly, as D increases, the energy eigenvalue increases.This feature is also seen as the value of 00 VS  increases.In Table 2, we obtained the energy eigenvalues for the pseudospin symmetry limit with 1 5, The same process in the spin symmetry was repeated.In this symmetry, as n increases, the energy obtained equally increases.This same feature is also observed as D increases from 1 to 3. It is readily observed that the energy obtained with 00 1 VS  and that obtained with 00 0 VS  are equal.This is due to the fact that 00 0 VS  since in the energy equation for the pseudospin symmetry we have 00 ( ) VS  . However, the energies obtained with 2 D  are equal to the energies obtained with 6 D  .Similarly, the energies obtained with 3 D  are equal to the energies obtained with 5 D  .In Figures 1-3, we graphically described the sum potential, approximation scheme and difference potential respectively.It can be seen that the three figures are similar.As it can be seen from the Figures, no matter the value of , r the value of () VS  or SV  cannot exceed 0. This is also justified by equations ( 11), ( 12) and (20).In Figures 4 and 5, we plotted energy in spin symmetry and pseudospin symmetry respectively against the potential depth for some states.
Figure 4 shows that the energy for the spin symmetry are purely positive while the revise is seen in Figure 5 for the pseudospin symmetry.

CONCLUSION
In this study, we obtained the bound state solutions of the D-dimensional Dirac equation with spin and pseudospin symmetry in the presence of both the scalar and vector inversely quadratic Yukawa potential.The two spinor components of Dirac equation and their corresponding energy equation are obtained in a closed and compact form in view of the parametric Nikiforov-Uvarov method.Some numerical results are obtained for each of the symmetry.The energies increase as both D and n increases.The results obtained using either vector potential or scalar potential are equivalent to the results obtained by using the sum of the vector and scalar potentials the spin symmetry.But under the pseudospin symmetry, the results obtained using either vector potential or scalar potential are not equivalent to the results obtained using the difference between the vector and scalar potentials.

Y
 are spin and pseudospin spherical harmonics respectively, and m is the projection of the angular momentum on the z  axis.Simplifying equation (3) further gives the two coupled differential equations whose solutions are the upper and lower radial

Fr
from equations (4) and (5) respectively, we obtain the following two Schrӧdinger-like differential equations for the upper and lower radial

Figure 1 :
Figure 1: The sum potential

Figure 2 :
Figure 2: Potential (1) in the presence of approximation scheme given in equation(12) 13) with equation (8), we deduce the values of the parametric constants as shown in Appendix 2. Substituting the values of the parametric constants in Appendix 2

C
Jacobi polynomials.The lower component of the wave function  the lower spinor exist if ME  which gives positive energy eigenvalues for the spin symmetry.Recall that

APPENDIX 2 .
Values of the parametric constants.
22, studied scattering states of Dirac equation in the presence of cosmic string for Coulomb interaction.Ikot et al.22, obtained bound state solutions of the Dirac equation for Eckart potential with Coulomb-like Yukawa-like tensor interactions. Ikt, Maghsoodi, Ibanga, Ituen and Hassanabadi 23 , obtained bound states of the Dirac equation for Modified Mobius square potential within the Yukawa-like tensor interaction.Ikot, Zarrinkamar, Zare and Hassanabadi 24 , investigated relativistic Dirac attractive radial problem with Yukawa-like tensor interaction via SUSY QM.Onate and Onyeaju 25 , studied Dirac particles in the field of Frost-Musulin diatomic potential and the thermodynamic properties via parametric Nikiforov-Uvarov method.Onate and Ojonubah 26 , investigated Dirac equation in the presence of the interaction of hyperbolic and generalized Pӧschl-Teller like potential model in the framework of spin and pseudospin symmetries.Hassanabadi et al.
30, studied Actual and General Manning-Rosen potential under spin and pseudospin symmetries of the Dirac equation.Dong and Ma28, also studied exact solutions of the Dirac equation with a Coulomb potential in 2+1 dimensions. Sot al.29, in their own investigation, obtained k states solutions of the Dirac equation for the Eckart potential with spin and pseudospin symmetry.Ikhdair and Sever30, deduced two approximation schemes to the bound states of the Dirac-Hulthén problem. ,

Table 1 :
Energy for spin

Table 2 :
Energy for pseudospin ( ) Energy of the pseudospin symmetry against the sum of the potential depth for 9/2 5t is readily observed from the table that the energy obtained with 05