Subur Pramono, 1 A. Suparmi, 2 and Cari Cari Introduction

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1 Advances in High Energy Physics Volume 6 Article ID 7934 pages Research Article Relativistic Energy Analysis of Five-Dimensional -Deformed Radial Rosen-Morse Potential Combined with -Deformed Trigonometric Scarf Noncentral Potential Using Asymptotic Iteration Method Subur Pramono A. Suparmi and Cari Cari Physics Department Graduate Program Sebelas Maret University Jl. Ir. Sutami 36A Kentingan Surakarta 576 Indonesia Physics Department Faculty of Mathematics and Fundamental Science Sebelas Maret University Jl. Ir. Sutami 36A Kentingan Surakarta 576 Indonesia Correspondence should be addressed to Subur Pramono; suburpramono.6@gmail.com Received 8 March 6; Revised July 6; Accepted 3 July 6 Academic Editor: Edward Sarkisyan-Grinbaum Copyright 6 Subur Pramono et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. The publication of this article was funded by SCOAP 3. We study the exact solution of Dirac euation in the hyperspherical coordinate under influence of separable -deformed uantum potentials. The -deformed hyperbolic Rosen-Morse potential is perturbed by-deformed noncentral trigonometric Scarf potentials where all of them can be solved by using Asymptotic Iteration Method (AIM. This work is limited to spin symmetry case. The relativistic energy euation and orbital uantum number euation l D have been obtained using Asymptotic Iteration Method. The upper radial wave function euations and angular wave function euations are also obtained by using this method. The relativistic energy levels are numerically calculated using Matlab and the increase of radial uantum number n causes the increase of bound state relativistic energy level in both dimensions D=5and D=3.The bound state relativistic energy level decreases with increasing of both deformation parameter and orbital uantum number n l.. Introduction Dirac euation as relativistic wave euation was formulated by P. A. M Dirac in 98; the exact solution of Dirac euation for some uantum potentials plays a fundamental role in relativistic uantum mechanics []. In order to investigate nuclear shell model spin symmetry and pseudospin symmetry solutions of Dirac euations have been an important field of study in nuclear physics. The concept of spin symmetry and pseudospin symmetry limit with nuclear shell model has been used widely in explaining a number of phenomena in nuclear physics and related field []. In nuclear physics spin symmetry and pseudospin symmetry concepts have been used to study the aspect of deformed and super deformation nuclei.theconceptofspinsymmetryhasbeenappliedtothe level of meson and antinucleon [3]. Pseudospin symmetry has been observed in deformed nuclei and can be enhanced in heavy proton-rich nuclei [4]. Solutions of Dirac euation for some potentials under limit case of spin symmetry and pseudospin symmetry have been investigated intensively whether in three- [5 6] two- or one- [7 3] dimensional space some D-dimensional spherical symmetric spacetimes [4 5]. However The D-dimensional Dirac euation with (D -dimensional separable noncentral potential has not been investigated yet; therefore it may be worthy to investigate Dirac euation in 5 dimensions with separable 4-dimensional noncentral potential in this study. In recent years some researchers have studied solution of Dirac euation with uantum potentials with different application and methods. These investigations include Eckart potential and trigonometric Manning-Rosen potential

2 Advances in High Energy Physics using Asymptotic Iteration Method (AIM [5] -deformed hyperbolic Pöschl-Teller potential and trigonometric Scarf II noncentral potential using Nikiforov-Uvarov method [6] -deformed trigonometric Scarf potential with -deformed Trigonometric Tensor Coupling Potential for Spin and Pseudospin 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] pseudospin symmetric solution of the Morse potential for any κ state using AIM [] Scalar Vector and Tensor Cornell Interaction using Ansatz method [] Scalar and Vector Generalized Isotonic Oscillators and Cornell Tensor Interaction using Ansatz method [3] Mie-type potentials for energy dependent pseudoharmonic potential via SUSYQM [6] trigonometric Scarf potential in D-dimension for spin and pseudospin symmetry using Nikiforov-Uvarov (NU method [5] Coulombic potential and its thermodynamics properties in D-dimensional space using NU method [7] and hyperbolic tangent potential and its application in material properties in D-dimensional space [8]. Asymptotic Iteration Methods (AIM have small deviation for determination of eigenenergies and eigenfunctions of Dirac euation. The separable D-dimensional uantum potentials are not studied yet by some researchers. In this paper we use Asymptotic Iteration Method (AIM to solve the Dirac euation under influence of separable D-dimensional uantum potentials. The relativistic energy levels can be obtained from calculation of relativistic energy euation using Matlab R3a. In Section we present basic theory of Dirac euation in hyperspherical coordinate with Ddimensional separable uantum potential. In this section deformed uantum potential is also included which is proposed by Dutra in 5 [9]. In Section 3 we present Asymptotic Iteration Method. Result and discussion are included in Section 4 and in Section 5 we present the special case in 3- dimensional space. In the last section we present conclusion.. Dirac Euation with Separable -Deformed Quantum Potential in the Hyperspherical Coordinates For single particle Dirac euation with vector potential V(r and scalar potential S(r in the hyperspherical coordinate can be expressed as follows (in the unit h=c= [3 4]: { α p+ β(m+s( r}ψ( r ={E V( r}ψ( r where p EandM are D-dimensional momentum operator total relativistic energy and relativistic mass of the particle respectively: ( α i =( σ i ( σ i β =( (3 where σ i are Pauli s matrices and is the unit matrix. Here we use relations between Pauli s matrices as σ i σ j + σ j σ i =δ ij. (4 The wave function of Dirac spinor can be classified in two forms upper spinor χ( rand lower spinor φ( ras follows [5 6]: ψ( r=( χ( r φ( r F nk (r r (D / Yl l...l D ( x =θ θ...θ D (5 =( i G. nk (r r (D / Y l l ( x =θ... l D θ...θ D By substituting ( (3 and (5 into ( we get σ pφ ( r = (M E+(V ( r +S( r χ ( r (6 σ pχ ( r = (M+E (V ( r S( r φ ( r. Exact spin symmetry limit is characterized with V( r = S( rbutforspinsymmetrylimitv(r S(r = Δ which is constant and for pseudospin symmetry limit V(r + S(r = Σ which is also constant. For spin symmetry limit we have dδ( r /dr = since Δ( r = C s = constant; therefore (6 can be rewritten as σ pφ ( r = {(M E +Σ( r} χ ( r (7 φ ( r = σ p (M + E C s χ ( If (8 is substituted into (7 we get p χ ( r r. (8 ={(M+E C s } [ {(M E + (Σ ( r}] χ ( r where ( σ p ( σ p = p. ( Using momentum operator definition in uantum mechanics where p = i D the hyperspherical Laplacian D is given as [5] D = r D r (rd r +(L D r. ( The eigenvalue of L D is l D (l D +D and the angular momentum operator is expressed as [5] L D = { sin D (sin D θ θ D θ D D θ D ( L D sin }. θ D (9

3 Advances in High Energy Physics 3 By inserting ( into (9 we get r D r (rd r χ ( =. r+(l D r χ( r (M E χ( r {(M+E C s }Σ( r χ ( r (3 The separable variable potential used in this study is -deformed hyperbolic Rosen-Morse potential plus deformed noncentral Scarf trigonometric potential in hyperspherical coordinate space. The effective potential can be written as V Σ (r =[{ cosh (αr +V tanh (αr} + r { V(θ sin θ sin θ 3 sin θ V(θ 3 sin θ 4 +V(θ 4 }] with angular potentials V(θ i taken as V(θ = b +a (a sin θ V (θ = b +a (a sin θ V (θ 3 = b 3 +a 3 (a 3 sin θ 3 V (θ 4 = b 4 +a 4 (a 4 sin θ 4 V(θ sin θ 3 sin θ 4 (4 b (a /cos θ sin θ (5 b (a /cos θ sin θ (6 b 3 (a 3 /cos θ 3 sin θ (7 3 b 4 (a 4 /cos θ 4 sin θ. (8 4 By substituting (4 into (3 and using variable separation method we get the radial part and the angular part of Dirac euations in hyperspherical coordinate with D=5... The Radial Part. The D-dimensional Dirac euation with -deformed hyperbolic Rosen-Morse potential plus deformed trigonometric Scarf noncentral potentials can be resolved into the form of radial part and angular part euations. The radial part of D-dimensional Dirac euation in this case can be expressed as d F nk (r dr (l D + (D / (l D + (D 3/ r F nk (r V +[{ cosh (αr V tanh (αr} {M+E} {M+E C s } {M E}]F nk (r = with λ 4 =λ D =l D (l D +D. (9.. The Angular Part. The angular part of 5-dimensional Dirac euation obtained from (3-(4 can be resolved into four parts and for C s =weget P (θ P (θ ( +{λ (M+E V(θ } = ( θ P (θ { P (sin θ (θ sin θ θ } θ +{λ λ sin θ (M+E V(θ } = P 3 (θ 3 sin (sin P θ 3 (θ 3 θ 3 θ 3 3 θ 3 +{λ 3 λ sin θ 3 (M+E V(θ 3 } = P 4 (θ 4 sin 3 (sin 3 P θ 4 (θ 4 θ 4 θ 4 4 θ 4 +{λ 4 λ 3 sin θ 4 {M+E} V(θ 4 } =. ( ( (3 Euations ( (3 are the angular part of Dirac euation for θ until θ 4 respectively.thed-dimensional relativistic wave functions and orbital uantum numbers are obtained from those euations. 3. Review of Asymptotic Iteration Method (AIM Asymptotic Iteration Method (AIM is an alternative method which has accuracy and high efficiency to determine eigenenergies and eigenfunctions for analytically solvable hyperbolic-like potential. Asymptotic Iteration Method is also giving solution for exactly solvable problem [7]. AIM is used to solve the second-order homogeneous linear euation as follows [5 3]: y n (x =λ (x y n (x +S (x y n (x (4 where λ (x =and prime symbol refers to derivation along x. Theotherparametern is interpreted as radial uantum number. Variables λ (x and S (x are variables that can be differentiated along x.togetthesolutionof(4wehaveto differentiate (4 along xandthenweget where y n (x =λ (x y n (x +S (x y n (x (5 λ (x =λ (x +S (x +λ (x S (x =S (x +S (x λ (x. (6 Asymptotic Iteration Method (AIM can be applied exactly in the different problem if the wave function has been known and fulfills boundary conditions zero (and infinity (.

4 4 Advances in High Energy Physics Euation ( can be simply iterated until (k+and(k+ k=3...andthenweget y n k+ (x =λ k (x y n (x +S k (x y n (x (7 y n k+ (x =λ k (x y n (x +S k (x y n (x (8 λ k (x =λ k (x +S k (x +λ (x λ k (x (9 S k (x =S k (x +S (x λ k (x (3 which is called as recurrence relation. Eigenvalue can be found using euation given as Δ k (z =λ k (z s k (z λ k (z s k (z = (3 where k = 3... is the iteration number and n r is representation of radial uantum number. Euation ( is the second-order homogeneous linear euation which can be solved by comparing it with the second-order linear euation as follows []: where y (z =( azn+ t+ bzn+ z y (z wz N y (z bzn+ (σ n = Γ (σ+n Γ (σ σ= t+n+3 N+ ρ= (t + b+a. (N+ b (3 (33 The wave function of ( is the solution of (3 which is given as [] y n (z = ( n C (N+ n (σ n F ( n ρ + n; σ; bz N+. (34 The -deformed hyperbolic and trigonometric functions are used as one of the parameters in the modified Rosen- Morse potential and noncentral Scarf trigonometric potentials were defined by Arai [4] some years ago as follows: sinh αr = eαr e αr (35a cosh αr = eαr +e αr (35b tanh αr = sinh αr cosh αr cosh αr sinh αr =. (35d (35c Deformation with-parameter in the hyperbolic function can be extended into trigonometric function. Definition of trigonometric function can be arranged by the same way as in the hyperbolic function introduced by Suparmi et al. [7] as follows: sin ar = eiar e iar cos ar = eiar +e iar cos ar + sin ar = tan ar = sin ar cos ar sec ar = cos ar d sin ar =acos dr ar d cos ar = asin dr ar d tan ar =asec dr ar. (36 By a convenient translation of spatial variable one can transform the deformed potentials into the form of nondeformed potentials or vice-versa. In analogy to the translation of spatial variable for hyperbolic function introduced by Dutra [9] we propose the translation of spatial variable for hyperbolic and trigonometric function as follows: r r+ ln α r r ln α r r+ ln iα r r ln iα. And then by inserting ( into (8 and (9 we have sinh αr sinh αr; cosh αr cosh αr; sin αr sin αr; cos αr cos αr. (37a (37b (38 Thetranslationofspatialvariablein(38canbeusedto map the energy and wave function of nondeformed potential toward deformed potential of Scarf potential.

5 Advances in High Energy Physics 5 4. Result and Discussion 4.. Radial Part. The radial part of Dirac euation in hyperspherical space when C s =canbeexpressedas d F nk (r dr (l D + (D / (l D + (D 3/ r F nk (r V +[{ cosh (αr V tanh (αr} {M+E} {M E }] F nk (r =. (39 Euation (39 cannot be solved directly when l D = ; in this condition we use approximation to solve centrifugal term with Pekeris approximation. Because we have used deformed uantum potential the Pekeris approximation in this condition can be expressed as [3] r = e αr re (c +c +e αr +c ( e αr +e αr (4 with c = ( +exp ( αr e 8αr ( e αr e +exp ( αr e (4a (3+αr e c = (exp (αr e +(3( +exp ( αr e αr e (3+αr e ( +exp ( αr e αr e c =(exp (αr e + ( +exp ( αr e (3 αr e +αr e 4αr e +exp ( αr e. (4b (4c By inserting (38 into (37a and (37b and changing from exponential form to hyperbolic form we get d F nk (r dr (ω(c c + c +{M E } F nk (r {ω( c c +V {M+E}} tanh αrf nk (r +[{ ωc 4 +V {M+E}} cosh (αr]f nk (r = (4 with (l D + (D / (l D + (D 3/ =ω (43 r e where r e is euilibrium distance that can be derived by potential parameter. Let us assume α E = (ω(c c + c +{M E } α ρ={ω( c + c +V {M+E}} α ] (] + ={ ωc 4 +V {M+E}}. Euations (44 are inserted into (4 so we obtain Let d F nk (r dr α {ρ tanhαr ] (] + sech (αr}f nk (r = α E F nk (r. By using (45-(46 we get z ( z F nk (z z (44 (45 tanh αr = z. (46 + ( z F nk (z z {ρ tanh αr ] (] + sech αr} F 4z ( z nk (z E = 4z ( z F nk (z. (47 By substituting (46 and sech αr = 4z( z/ into (47 we get z ( z F nk (z z ] (] + +{ =. + ( z F nk (z z ( ρ E 4z E ( ρ 4 ( z } F nk (z (48 Euation (48 has two regular singular points for z=and z=so the general solution from (48 is F nκ (z = z δ ( z γ f n (z. Let 4δ =ρ E and 4γ = ρ E and then substitute them into (48 to get z ( z f n (z +{(δ + z (δ + γ + } f n (z ] (] + +{ (δ+γ(δ+γ+}f n (z =. (49

6 6 Advances in High Energy Physics Euation (49 is the second differential euation that can be manipulated into the form as in (: f n (z = {z (δ + γ + } (δ + f n z ( z (z + λ (z = {(δ+γ(δ+γ+ ](] + /} f z ( z n (z (5 z (δ + γ + (δ + (5 z ( z {(δ+γ(δ+γ+ ](] + /} S (z =. (5 z ( z Let A=(δ+γ(δ+γ+ ](] + / and using (9-(3 together with (5 and (5 we get λ (z (δ + (γ + ={ z + ( z }+{A z + A ( z } S (z (δ + (γ + +{ + z ( z } (53 From (57 we get that the relativistic energy euation of this system is (M E =α [ [ 4( ε n +/4 n / +( ε n + ] 4 n ] ρ ω(c c + c. (58 By comparing (3 and (5 and using (34 we obtain f n (z as f n (z = ( n C ( n (δ + n F ( n δ + γ + n + δ + z. (59 From (59 we determine unnormalized radial wave function with D = 5 for various n as shown in Table 6 here C is normalization constant. 4.. Solution of Angular Part. In this study the four angular parts of Dirac euations are presented in ( (3 so we have to solve each euation of angular Dirac euation using AIM. ={ A z + A ( z } +{ A z + A + (γ + }{ (δ + ( z z ( z }. The eigenvalue of (5 can be found by using (3: Δ =s λ s λ = ε =(δ+γ(δ+γ+ Δ =s λ s λ = ε =(δ+γ+(δ+γ+. ε n =(δ+γ+n(δ+γ+n+ (54 ( Euation of Angular Part for θ. We can solve ( by changing it into the form of the second-order hypergeometric-type differential euation that is similar to ( after we insert (5 in ( P (θ θ {M+E} { b +a (a sin θ P (θ +λ P (θ = + b (a /cos θ sin θ } andsubstitutecos θ = ( z into (6 so we get z ( z d P (z dz + ( z dp (z dz +{λ (6 where ε n is nth eigenvalue when n =... and if n is radial uantum number. By using (9-(3 and the last euation in (55 we have ε n = ] (] + ={ (l 4 +3/(l 4 +/ r e 4α c + V (M+E α }. The last euation in (55 can be rewritten as (56 ε n =(δ+γ+(n+ (δ+γ+(n+. (57 {M+E} [(b +a (a /+(b (a / /] 4z {M+E}[(b +a (a / (b (a / /] } 4( z P (z =. (6 Euation (6 has two regular singular points for z =and z = and then the solution of P (z is set as P (z =z δ s ( z γ s f nl (z. (6

7 Advances in High Energy Physics 7 Table : Relativistic energies with variation of and n l. D=5 V =6 V = a =a =a 3 =a 4 =b =b =b 3 =b 4 = r e (fm n n l =n l =n l3 =n l4 E (fm n l =n l =n l3 =n l4 n r e (fm E nl (fm If we replace P (z in (6 with (6 and simplify it by using appropriate variable substitution as follows: O s =λ (63a δ s (δ s =(M+E ( b +a (a γ s (γ s =(M+E ( b +a (a then (6 reduces to f n l (z b (a / + b (a / = {(δ s +γ s +z (δ s +/} f n z ( z l (z From (64 we get + {(δ s +γ s O s } f z ( z nl (z. (63b (63c (64 λ (z = {(δ s +γ s +z (δ s +/} ; (65 z ( z S (z = {(δ s +γ s O s } (66 z ( z λ (z ={ (δ s +/ z + (γ s +/ ( z } +{ I s I + s z ( z } +{ (δ s +/ + (γ s +/ } z ( z (67 Table : Relativistic energies with variation of radial uantum number n. D=5 V =6 V = α=5 (fm and r e =.67 (fm a =a =a 3 =a 4 =b =b =b 3 =b 4 = n n l =n l =n l3 =n l4 E n (fm Table 3: The relativistic energies with variation of potential parameters a ;a ;a 3 ;a 4 and b ;b ;b 3 ;b 4 with M = 5; C s = ; n r = ; n l =n l =n l3 =n l4 =; V =6; α =.5; =; V = ;and r e =.67. a b a b a 3 b 3 a 4 b 4 E ab (fm S (z ={ I s z + I s ( z }+{I s z + { (δ s +/ + (γ s +/ }. z ( z I s ( z } (68 By using (65 (68 and using relation of (3 we get (l nl =(δ s +γ s +n l (69

8 8 Advances in High Energy Physics where n l is orbital uantum number. The angular wave function of (64 is determined by using (34 and then we get solution for θ bysettingsubscripti with 3 and 4. So we get the solution for orbital uantum number for θ θ 3 and θ 4 respectively as follows: f nl = ( n l C ( n l (δ s + n l F ( n l δ s +γ s +n l δ s + z (7 (l n l =δ s +γ s +(n l for θ n l =... (7 where C is normalization constant and F (ab;c;z is hypergeometry function. From (7 we solve unnormalized angular wave function as function of z completely for variation of n l as shown in Table Euation of Angular Part for θ θ 3 andθ 4. Solutions for θ θ 3 andθ 4 are determined by the same way as the (l 3 n l =δ s3 +γ s3 +(n l for θ 3 n l =... (7 (l 4 n l =δ s4 +γ s4 +(n l 3 for θ 4 n l =... (73 where λ + (M+E((b δ s = +a (a / b (a // + / λ +/4+(M+E ((b3 δ s3 = +a 3 (a 3 / b 3 (a 3 // + / λ 3 /+(M+E ((b4 δ s4 = +a 4 (a 4 / b 4 (a 4 // + / λ + (M+E((b γ s = +a (a /+b (a // + / λ + (M+E((b3 γ s3 = +a 3 (a 3 /+b 3 (a 3 // + /4 + / λ 3 /+(M+E ((b4 γ s4 = +a 4 (a 4 /+b 4 (a 4 // + /. (74 (75 (76 (77 (78 (79 The unnormalized angular wave functions for θ θ 3 and θ 4 are in the same pattern as the angular wave function for θ so we obtain the last three angular wave functions by simple change of parameters in (7 with parameters in (74 (79. The relativistic energy levels are calculated numerically using Matlab program R3a. Table shows the relativistic energy levels as a function of deformation parameter ; the relativistic energy E decreases when deformation parameter increases. Here we apply the value of from. until.withstep..therelativisticenergylevelsasafunction of orbital uantum numbers n l are also shown in Table. The magnitude of energies E decreases when the orbital uantum numbers n l increase. The relativistic energy levels as a function of radial uantum number n are shown in Table. The relativistic energies E increase when radial uantum number n increases. The relativistic energies numerically also change as a function of potential parameters a i and b i where i = 3 4 are the components of ith noncentral potential (Table 3. The relativistic energies E increase when both potential parameters in each potential component increase. This suggests that the bounded energies become less bounded with increasing of potential parameters. The unnormalizedangularwavefunctionsarelistedintable4. The unnormalized radial wave functions are plotted by using (46 and (59 as shown in Figure. From Figures (a (c it is seen that the amplitude of the wave function increases when the orbital uantum number increases. This suggests that the probability of finding particles is larger for higher orbital uantum number n l. 5. Special Case for Three-Dimensional System In three-dimensional case D=3the bound state relativistic energy levels are calculated numerically from (58 (43 (44 (56 (69 (74 and (77 using Matlab software R3a and are presented in Tables 5 6 and 7. By using (44 and (6 the unnormalized radial wave functions are listed in Table 8. In Table 5 the relativistic energy levels decrease

9 Advances in High Energy Physics 9 Table 4: The unnormalized upper angular wave function for higher dimension D=5. n l Unnormalized angular wave function Q(θ =C ( cos θ δ s +cos θ ( Q(θ =C ( cos θ δ s +cos θ ( Q(θ 3 =C ( cos θ 3 Q(θ 4 =C ( cos θ 4 Q(θ =C ( cos θ δ s3 +cos θ ( 3 δ s4 +cos θ ( 4 Q(θ =C ( cos θ δ s +cos θ ( Q(θ 3 =C ( cos θ δ s3 3 +cos θ ( 3 γ s γ s γ s3 γ s4 Q(θ 4 =C ( cos θ δ s4 4 +cos θ ( 4 δ s γ +cos θ s ( ( (δ s + [+ ( (δ s +γ s +(( cos θ / ] (δ s +/ γ s ( (δ s + [+ ( (δ s +γ s +(( cos θ / ] (δ s +/ γ s3 ( (δ s3 + [+ ( (δ s3 +γ s3 +(( cos θ 3 / ] (δ s3 +/ γ s4 ( (δ s4 + [+ ( (δ s4 +γ s4 +(( cos θ 4 / ] (δ s4 +/ F (r (fm.. F (r (fm 5 5 r (fm r (fm (a (b 5 F (r (fm r (fm (c Figure : Unnormalized radial wave functions versus hyperspherical radius r with variation of radial uantum number where M=5fm C s = V =6 V = r e =.67 fm and n li = i = 3 4.(aForn=(bforn=and(cforn=.

10 Advances in High Energy Physics Table 5: Relativistic energy with variation of and n l. V =6 V = α=5 (fm a =a =b =b = r e (fm n n l =n l E (fm n l =n l n r r e (fm E nl (fm Table6:Relativisticenergieswithvariationofn r and potential parameter. M=5 V =6 V = = α=5 (fm and r e =.67 (fm n n l =n l a =a =b =b E nnl n l (fm n n l =n l a =a =b =b E nnl n l (fm Table 7: Relativistic energies for M=fm C s = V =6andV = without presence of -deformed noncentral potentials. l n K (lj E ss nk (fm = r e =.67 E ss nk (fm =. r e =.7 E ss nk (fm =.4 r e =.334 p 3/ d 5/ f 7/ g 9/ p 3/ d 5/ f 7/ g 9/ p / d 3/ f 5/ g 7/ p / d 3/ f 5/ g 7/ n Radial wave function F (r = C ( tanh δ αr F (r = C ( tanh δ αr Table 8: Unnormalized upper radial wave function for dimension D=3. ( +tanh γ αr ( +tanh γ αr ( C (δ + ( + ( (δ + γ + (( tanh αr / (δ +!

11 Advances in High Energy Physics Table 9: The unnormalized upper angular wave function for dimension D=3. n l Unnormalized angular wave function Q(θ =C ( cos θ δ s +cos θ ( Q(θ =C ( cos θ δ s +cos θ ( γ s γ s Q(θ =C ( cos θ δ s +cos θ ( Q(θ =C ( cos θ δ s +cos θ ( γ s ( (δ s + [+ ( (δ s +γ s +(( cos θ / ] (δ s +/ γ s ( (δ s + [+ ( (δ s +γ s +(( cos θ / ] (δ s +/ with the increase of both deformation parameter and orbital uantum number n l wheren l is orbital uantum number for θ and n l is orbital uantum number for θ. From Table 7 we can conclude that the value of relativistic energies increases with the increase of the radial uantum number n and with either the presence or absence of angular potential parameter. The angular potential parameters a and b influence the relativistic energy level where the relativistic energy levels increase by increasing the angular potential parameter. The unnormalized upper radial wave functions are listed in Table 8 when the radial uantum numbers are n = and n = and C is the normalization constant. The unnormalized angular wave functions as functions of θ and θ are listed in Table 9. Without the presence of the noncentral potential degeneracy energy spectra occur in the spin doublets with uantum numbers (nlj = l+/ and (nlj = l / where n l andj are the radial the orbital and the total angular momentum uantum numbers respectively for example (np / np 3/ forl = but for difference values of j=/and 3/ (nd 3/ nd 5/ forl= with (j =3/and 5/ (nf 5/ nf 7/ forl=3with (j =5/ and 7/and(ng 7/ ng 9/ forl=4with (j =7/and 9/. The degeneracy energies can be removed with the presence of noncentral potential by changing l K where K= l and K=lfor K<and K>. 6. Conclusion In this study we have obtained the bound state solution of the 5-dimensional Dirac euation with separable -deformed uantum potential under condition of spin symmetry. The upper component of Dirac spinors and relativistic energy have been obtained using Asymptotic Iteration Method (AIM. The numerical result shows that the bound state relativistic energy level for spin symmetry case with both dimensions D=5and D=3increases with increasing of radial uantum number n and decreases with increasing of both deformation parameter and orbital uantum number n l. The degeneracy energy states occur in the absence of deformed noncentral potential. Competing Interests The authors declare that they have no competing interests. Acknowledgments This Research is partly supported by Higher Education Project Grant with contract no. 63/UN7/7./LT/6. References [] W. Greiner Relativistic Quantum Mechanics Springer New York NY USA 3rd edition. [] B. J. Falaye and K. J. Oyewumi Solutions of the Dirac euation with spin and pseudospin symmetry for the trigonometric Scarf potential in D-dimensions The African Review of Physics vol. 6 article 5. [3] S. M. Ikhdair Approximate solutions of the Dirac euation for the Rosen-Morse potential including the spin-orbit centrifugal term Mathematical Physics vol. 5 no. ArticleID 355. [4] J. N. Ginocchio Pseudospin as a relativistic symmetry Physical Review Letters vol. 78 no. 3 pp [5] R. A. Sari A. Suparmi and C. Cari Solution of Dirac euation for Eckart potential and trigonometric Manning Rosen potential using asymptotic iteration method Chinese Physics B vol. 5 no. Article ID 3 5. [6] A. Kurniawan A. Suparmi and C. Cari Approximate analytical solution of the Dirac euation with -deformed hyperbolic Pöschl-Teller potential and trigonometric Scarf II non-central potential Chinese Physics B vol.4no.3articleid33 5. [7] A. Suparmi C. Cari U. A. Deta A. S. Husein and H. Yuliani Exact solution of Dirac euation for -deformed trigonometric Scarf potential with -deformed trigonometric tensor coupling potential for spin and pseudospin symmetries using Romanovski polynomial JournalofPhysics:ConferenceSeries vol.539no.articleid44. [8] M. Hamzavi and A. A. Rajabi Generalized nuclear Woods- Saxon potential under relativistic spin symmetry limit ISRN High Energy Physics vol. 3 Article ID pages 3. [9] M. Sameer Approximate relativistic bound states of a particle in Yukawa field with Coulomb tensor interaction Physica Scriptavol.87no.3ArticleID353. [] Suparmi and Cari Bound state solution of Dirac euation for generalized Pöschl-Teller plus trigomometric Pöschl-Teller non-central potential using SUSY uantum mechanics Journal of Mathematical and Fundamental Sciences vol.46no.3pp

12 Advances in High Energy Physics [] O. Bayrak and I. Boztosun The pseudospin symmetric solution of the Morse potential for any κ state Physics A vol. 4 no. 36 pp [] H. Hassanabadi E. Maghsoodi S. Zarrinkamar and H. Rahimov Dirac euation under scalar vector and tensor cornell interactions Advances in High Energy Physics vol. Article ID pages. [3] H. Hassanabadi E. Maghsoodi A. N. Ikot and S. Zarrinkamar Dirac euation under scalar and vector generalized isotonic oscillators and cornell tensor interaction Advances in High Energy Physics vol. 4 Article ID pages 4. [4] A. Lopez-Ortega The Dirac euation in D-dimensional spherically symmetric spacetimes [5] R. L. Hall and P. Zorin Refined comparison theorems for the Dirac euation in d dimensions Annalen der Physik vol. 57 no. 5-6 pp [6] A. N. Ikot H. Hassanabadi E. Maghsoodi and S. Zarrinkamar D-dimensional Dirac euation for energy-dependent pseudoharmonic and Mie-type potentials via SUSYQM Communications in Theoretical Physicsvol.6no.4pp [7] C. Cari A. Suparmi M. Yunianto and A. S. Husein Solution of D-dimensional Dirac euation for Coulombic potential using NU method and its thermodynamics properties AIP Conference Proceedingvol.7ArticleID396. [8] A. Suparmi C. Cari B. N. Pratiwi and U. A. Deta Solution of D dimensional Dirac euation for hyperbolic tangent potential using NU method and its application in material properties AIP Conference Proceedings vol. 7 Article ID 3 6. [9] A. S. Dutra Mapping deformed hyperbolic potentials into nondeformed ones Physics Letters A vol. 339 no. 3 5 pp [] T. Das Exact solutions of the Klein-Gordon euation for -deformed manning-rosen potential via asymptotic iteration method [] A. Soylu O. Bayrak and I. Boztosun An approximate solution of Dirac-Hulthén problem with pseudospin and spin symmetry for any κ state Mathematical Physics vol. 48 no. 8 Article ID [] A. Soylu O. Bayrak and I. Boztosun K state solutions of the Dirac euation for the Eckart potential with pseudo-spin and spin symmetry Physics A: Mathematical and Theoreticalvol.4ArticleID [3] G. Kocak O. Bayrak and I. Boztosun Supersymmetric solution of Schrödinger euation by using the asymptotic iteration method Annalen der Physikvol.54no.6-7pp [4] B.-H. Nam J.-Y. Moon E.-H. Park et al. Conserved gene structure and function of interleukin- in teleost fish Journal of Animal and Veterinary Advances vol. 3 no. pp

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