Relativistic Energy Analysis Of Five Dimensional q-Deformed Radial Rosen-Morse Potential Combined With q-Deformed Trigonometric Scarf Non-Central Potential Using Asymptotic Iteration Method (AIM)

In this work, we study the exact solution of Dirac equation in the hyper-spherical coordinate under influence of separable q-Deformed quantum potentials. The q-deformed hyperbolic Rosen-Morse potential is perturbed by q-deformed non-central trigonometric Scarf potentials, where whole of them can be solved by using Asymptotic Iteration Method (AIM). This work is limited to spin symmetry case. The relativistic energy equation and orbital quantum number equation lD-1 have been obtained using Asymptotic Iteration Method. The upper radial wave function equations and angular wave function equations are also obtained by using this method. The relativistic energy levels are numerically calculated using Mat Lab, the increase of radial quantum number n causes the increase of bound state relativistic energy level both in dimension D = 5 and D = 3. The bound state relativistic energy level decreases with increasing of both deformation parameter q and orbital quantum number nl.


Introduction
Dirac equation as relativistic wave equation was formulated by P.A.M Dirac in 1928, the exact solution of Dirac equation for some quantum potentials plays fundamental role in relativistic quantum mechanics. [1] In order to investigate nuclear shell model, study of spin symmetry and pseudo-spin symmetry solutions of Dirac equations have been an important field of study in nuclear physics. The concept of spin symmetry and pseudo-spin symmetry limit with nuclear shell model has been used widely in explaining a number of phenomena in nuclear physics and related field. [2] In nuclear physics, spin symmetry and pseudo-spin symmetry concepts have been used to study the aspect of deformed and super deformation nuclei. The concept of spin symmetry has been applied to the level of meson and antinucleon. [3] Pseudo-spin symmetry has been observed in deformed nuclei and can be enhanced in heavy proton-rich nuclei. [4] Solution of Dirac equation for some potentials under limit case of spin symmetry and pseudo-spin symmetry have been investigated intensively whether in three- [5,6] two-or one- [7][8][9][10][11][12][13] dimensional space, and some D-dimensional with spherically Symmetric Spacetimes [14][15]. However, The D-dimensional Dirac equation with (D-1)-dimensional separable non-central potential has not been investigated yet, therefore it may be worthy to investigate Dirac equation in 5 dimensions with separable 4-dimensional noncentral potential in this study.
In this recent years, some researchers have studied solution of Dirac equation with quantum potentials with different application and methods. These investigations include: Eckart potential and trigonometric Manning-Rosen potential using Asymptotic Iteration Method (AIM), [5] qdeformed hyperbolic Pöschl-Teller potential and trigonometric Scarf II non-central potential using Nikiforov-Uvarov method, [6] q-deformed Trigonometric Scarf potential with q-deformed Trigonometric Tensor Coupling potential for Spin and Pseudo-spin Symmetries Using Romanovski Polynomial, [7] generalized nuclear Wood-Saxon potential under relativistic spin symmetry limit, [8] relativistic bound states of particle in Yukawa field with Coulomb tensor interaction, [9] Rosen-Morse potential including the spin-orbit centrifugal term using Nikiforov-Uvarov (NU) method, [3] pseudo-spin symmetric solution of the Morse potential for any  state using AIM, [11] Scalar, Vector, and Tensor Cornell Interaction using Ansatz method, [12] Scalar and Vector generalized Isotonic Oscillators and Cornell Tensor Interaction using Ansatz method, [13] Mie-type potentials for energy dependent pseudoharmonic potential via SUSYQM, [16] trigonometric Scarf potential in D-dimension for spin and pseudo-spin symmetry using Nikiforov-Uvarov (NU) method, [15] Coulombic potential and its thermodynamics properties in Ddimensional space using NU method, [18] hyperbolic tangent potential and its application in material properties in D-dimensional space, [19] and others.
Asymptotic Iteration Method (AIM) have small deviation for determination of eigen energies and eigen functions of Dirac equation. The separable D-dimensional quantum potentials is not studied yet by some researchers. In this paper we use Asymptotic Iteration Method (AIM) to solve the Dirac equation under influence of separable D-dimensional quantum potentials. The relativistic energy levels can be obtained from calculation of relativistic energy equation using Matlab R2013a. In section 2 we present basic theory of Dirac equation in hyper-spherical coordinate with D-dimensional separable quantum potential. In this section also included deformed quantum potential which is proposed by Dutra in 2005. [26] In section 3 we present asymptotic iteration method. Result and disccussion are included in section 4, and in section 5 we present the special case in 3dimensional space. in the last section we present conclusion.

Dirac equation with separable qdeformed quantum potetial in The Hyper-spherical Coordinates
For single particle, Dirac equation with vector potential ( )and scalar potential ( )in the hyper-spherical coordinate can be expressed as follows (in the unit ℏ = = 1): [13,14] where ⃑, , and are D-dimensional momentum operator, total relativistic energy and Relativistic mass of the particle respectively.

The Radial Part
The D-dimensional Dirac equation with qdeformed hyperbolic Rosen-Morse potential plus q-deformed trigonometric Scarf non-central potentials can be resolved into the form of radial part and angular part equations. The radial part of D-dimensional Dirac equation in this case can be expressed as

The Angular Part
The angullar part of 5-dimensional that obtained from Eqs. (14)(15) can be resolved into four parts, and for , we get Eqs. (21)(22)(23)(24) are angular part of Dirac equation for 1 until 4 respectively. The D-dimensional relativistic wave functions and orbital quantum numbers are obtained from those equations.

Review of Asymptotic Iteration Method (AIM)
Asymptotic Iteration Methods (AIM) is an alternative method which have accuraccy and high efficiency to determine eigen energies and eigen functions for analytically solvable hyperbolic like potential. Asymptotic Iteration method is also giving solution for exactly solvable problem. [18] AIM is used to solve The second orde homogen linear Eq. as follows: [5,[21][22][23][24] where 0 ( ) ≠ 0and prime symbol refers to derivation along x. The others parameter n is interpreted as radial quantum number. Variabel 0 ( )and 0 ( )is a variabel that can be differentiated along x. To get the solution of Eq. (25), we have to differentiate Eq. (25) along x, and then we get Asymptotic Iteration Method (AIM) and can be applied exactly in the different problem if the wave function have been known and fullfill boundary condition zero (0) and infinity (∞).
Eq. (2) can be simply iterated until (k+1) and (k+2), k = 1, 2, 3, ... and then we get which is called as recurrence relation. Eigen value can be found using equation given as and The wave function of Eq. (2) is the solution of Eq.(34) which is given as [21] The q-deformed hyperbolic and trigonometric functions are used as one of the parameters in the modified Rosen-Morse potential and non central Scarf Trigonometric potential are defined by Arai [25] The translation of spatial variable in Eq. (41) can be used to map the energy and wave function of non-deformed potential toward deformed potential of Scarf potential.

Radial Part
The radial part of Dirac equation in hyperspherical space when Cs=0 can be expressed as Eq. (42) can not be solved directly when ℓ −1 ′ ≠ 0, in this condition we use approximation to solve centrifugal term with Pekeris approximation. Because we have used q-deformed quantum potential, the Pekeris approximation in this condition can be expressed as [3] where is equilibrium distance that can be derived by potential parameter. Let we assume Eq. (53) is the second differential equation that can be manipulated into the form as in the Eq.
(31-32), together with Eqs. (55) and (56) we get The eigen value of Eq. (54) can be found by using where is n th eigen value when n = 0, 1, 2, ... and if n is radial quantum number. By using Eqs.
(37), we obtain ( ) as From Eq. (63) we determine radial wave function for various n as shown in Table 7. The unnormalized radial wave function for D =5, with C is normalization constant.

Solution of Angular Part
In this study, the four angular part of Dirac equations are presented in Eqs. (21)(22)(23)(24), so we have to solve each equation of angular Dirac equation using AIM.

Equation of Angular
and substitute   Eq. (66) has two regular singular points for 1 = 0 and 1 = 1, and then the solution of 1 ( 1 ) is set as If we replace 1 ( 1 ) in Eq. (65) with Eq. (66) and simplify it by using appropriate variabel substitution as follows The unnormalized angular wave function for θ2, θ3, and θ4 are in the same pattern with angular wave function for θ1 so we obtain the last three angular wave function by simple change of parameters in Eq. (74) with parameters in Eqs. (78-83).
The relativistic energy levels are calculated numerically using Matlab program R2013a. Table  1 shows the relativistic energy levels as a function of deformation parameter q, the relativistic energy decreases when deformation parameter increases. Here we apply the value of q from 0.2 until 1.2 with step 0.2. The relativistic energy levels as a function of orbital quantum numbers nl are also shown in Table 1. The magnitude of Energies decrease when the orbital quantum numbers increase.The relativistic energy levels as a function of radial quantum number n are shown in Table 2. The relativistic energies increase when radial quantum number increases. The relativistic energies numerically also change as a function of potential parameters and where = 1,2,3,4 are the component of i-th non-central potential. The relativistic energies increase when both potential parameters in each potential component increase. This suggests that the bounded energies become less bounded with increasing of potential paremeters. The unnormalized angular wave function are listed in Table 5. The unnnormalized radial wave functions are plotted by using Eqs. (50) and (63) are shown in the Fig. 1. From the Fig. 1 (a to c) it is seen that amplitude of the wave funtion become increases when the orbital quantum number increases. This suggests that probability to find particles is larger for higher orbital quantum number .    Table 9. In the Table 6, the relativistic energy levels decrease with the increase of both deformation parameter q and orbital quantum number nl, where nl1 is orbital quantum number for θ1 and nl2 is orbital quantum number for θ2. From Table  7 we can conclude that the value of relativistic energies increase with the increase of the radial quantum number n and either with the presence/absence of angular potential parameter. The angular potential parameters a and b influence to the relativistic energy level, where the relativistic energy levels increase by increasing of the angular potential parameter.The unnormalized upper radial wave functions are listed in the Table 9 when the radial quantum numbers are n = 0 and n= 1 and ′ is the normalization constant.The unnormalized angular wave functions as a functions of θ1and θ2 are listed in Table 10. Without the presence of the non-central potential, degeneracy energy spectra occurs in the spin doublets, with quantum numbers (n,ℓ, = ℓ + 1/2) and (n,ℓ, = ℓ − 1/2) , where n, ℓ and are the radial, the orbital and the total angular momentum quantum numbers respectively, for example in (np1/2,np3/2) for ℓ = 1 but for difference values of = 1/2 and 3/2, (nd3/2,nd5/2) for ℓ = 2 with ( = 3/2 and 5/2), (nf5/2,nf7/2) for ℓ = 3 with ( = 5/2 and 7/2), and (ng7/2,ng9/2) for ℓ = 4 with ( = 7/2 and 9/2).The degeneracy energies can be removed with the presence of non-central potential, by changing ℓ → , where = −ℓ − 1 and = ℓ for < 0 and > 0.