J. Nig. Soc. Phys. Sci. 5 (2023) 917 Journal of the Nigerian Society of Physical Sciences Dirac Equation for Energy-Dependent Potential With Energy-dependent Tensor Interaction C. A. Onatea, M. O. Oluwayemib,∗, I. B. Okonc aDepartment of Physics, Kogi State University, Anyigba, Nigeria b Department of Physical Sciences, Landmark University, Omu-Aran, Nigeria. cPhysics Department, University of Uyo, Uyo, Nigeria. Abstract The relativistic symmetries of the Dirac equation were investigated with an energy-dependent tensor potential interaction for two different energy- dependent potentials under parametric Nikiforov-Uvarov method and supersymmetric quantum mechanics and shape-invariance method. It is observed that the energy-dependent tensor interaction has stronger removal effect of the energy degeneracies in both the spin and pseudospin symmetries than the non-energy-dependent tensor interaction. DOI:10.46481/jnsps.2022.917 Keywords: Bound state, Dirac equation, Eigen solutions, Wave equation, potential function. Article History : Received: 05 July 2022 Received in revised form: 31 August 2022 Accepted for publication: 01 September 2022 Published: 14 January 2023 c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: Oluwatimilehin Oluwadare 1. Introduction The relativistic spin-1/2 particles in quantum mechanics is usually described by Dirac equation. The subject has drawn attention of physicist in the theoretical domain over the years. The theoretical physicists under the Dirac equation have anal- ysed the characteristic features of deformed nuclei, effective shell models [1-6], etc for the pseudospin symmetry (PS) for different physical potential models. The identical bands and mesons were also studied in details for the spin symmetry (SS). The theoretical reports of these studies showed that energy dou- blets were produced under the SS and PS for different levels. Very recently, tensor potential interaction was introduced to the ∗Corresponding author tel. no: +2348032652678 Email address: oluwayemimatthew@gmail.com (M. O. Oluwayemi ) Dirac equation to reduce the energy degeneracy. The reduc- tion of the degeneracy depends on the applied tensor potential. The Coulomb tensor potential for instance, reduced some de- generacies leaving some doublets unbroken. The application of Yukawa tensor potential also breaks some degeneracy doublets and produced another form of degeneracies. Onate et al. in their recent study applied Hellmann tensor potential and they found out that the whole degeneracies were broken even when the tensor strength is taken as small as 0.2. Owing to the ap- plication of Dirac equation, different authors have studied the Dirac equation in diverse areas using different traditional tech- niques [7-13]. However, it is very clear that the Dirac equation under SS and PS for energy-dependent potential (EDP) has not received attention to the best of our knowledge. Hence, the call for this study. Motivated by the application of relativistic wave equations particularly the Dirac equation, this study wants 1 C. A. Onate et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 917 2 to examine the effect of EDP potential on the eigenvalues of the SS and PS. This study will consider two sets of potentials using two different traditional methodologies. The two poten- tial models are modified Mie-type-constant EDP and Kratzer EDP respectively in the presence of an energy-dependent ten- sor (EDT) interaction via parametric Nikiforov-Uvarov method and supersymmetry quantum mechanics and shape invariance method. The major aim of this study is to determine the pro- duction of energy degenerate doublets by EDP and its removal by EDT interaction. The modified Mie-type-constant EDP and the Kratzer EDP respectively are given as V (r, E) = λ1(1 + a1 E) r2 + λ2(1 + a2 E) r−2 + λ3(1 + a3 E) (1) V (r, E) = λ4(1 + a4 E) r2 − λ5(1 + a5 E) r . (2) Here, λi(i = 1, 2, . . . ) are potential strengths and ai(i = 1, 2, . . . ) are potential parameters. Since this paper aim at determining the production and removal of energy degenerate state in the presence and absence of energy-dependent tensor interaction, we propose a Coulomb-constant energy-dependent tensor in- teraction of the form U(r, E) = H1(1 + b1 E) r + H2(1 + b2 E)r (3) 2. Dirac Equation (SS and PS) The Dirac equation with spin-1/2 particles under the poten- tials S (r) and V (r) as attractive scalar potential and repulsive vector potential is of the form [14, 15, 16] [Cα ·ρ + β ( MC2 + S (r) + V (r) − E ] ψnκ(r) = 0, (4) with E and M as energy and particle mass, ρ = −i~∇ defines momentum operator with α and β as 4 × 4 Dirac matrices, i.e. α = ( 0 σ1 σ1 0 ) , β = ( I 0 0 I ) (5) and σ1 = ( 0 1 1 0 ) ,σ2 = ( 0 −i i 0 ) ,σ3 = ( 1 0 0 −1 ) . (6) Here, I represents the 2×2 matrix identity and, σi are the Pauli 3-vector spin matrices. In the nuclei spherical symmetry, the angular momentum operator J and spin-orbit matrix operator κ = −β(σL + 1) commute with the Dirac Hamiltonian, where L is the total orbital angular momentum operator. The spinor wave functions can be classify following the radial quantum number n and the spin-orbit quantum number κ and can be ex- pressed according to the Pauli-Dirac representation [14, 15, 16]. ψnκ(r) = ( fnκ(r) gnκ(r) ) = 1 r  Fnκ(~r)Y`jmκ(θ,ϕ)iGnκ(~r)Y ¯̀jm(−κ)(θ,ϕ)  (7) where the upper and lower spinor components Fnκ(r) and Gnκ(r) are the real square-integral radial wave functions. Y`jmk(θ,φ) and Y ¯̀jm−k(θ,φ) are the spin spherical harmonic functions coupled to the total angular momentum j and its projection m on the z axis for κ(κ+1) = `(`+1) and κ(κ−1) = i(`+1). The quantum number κ is related to the quantum number ` for spin and Pseudospin symmetries as κ =  −(` + 1) = −( j + 12 ), (s1/2, p3/2, etc), j = ` + 1 2 , aligned spin (κ < 0) +`( j + 12 ), ( p1/2, d3/2, etc), j = `− 1 2 , unaligned spin (κ > 0) (8) The quasi-degenerate doublet structure can be expressed in terms of pseudospin angular momentum s̃ = 1/2 and pseudo- orbital angular momentum ˜̀ which is defined as κ =  − ˜̀ = (− j + 12 ), (s1/2, p3/2, etc), j = ˜̀ − 12 , aligned spin (κ < 0) +( ˜̀ + 1) = ( j + 12 ), (d3/2, f5/2, etc), j = ˜̀ + 12 , unaligned spin (κ > 0) (9) where κ = ±1,±2, . . . . Upon direct substitution of equation (7) into equation (4), we can obtain two radial coupled Dirac equation for the two symmetry components as follows:( d dr + κ r ) Fnκ (r) = [ MC2 + Enκ − ∆(r) ] Gnκ(r) (10) ( d dr − κ r ) Gnκ (r) = [ MC2 − Enκ + ∑ (r) ] Fnκ(r). (11) For the spin symmetry, ∆(r) = Cs = constant. Then, we obtain a second-order differential equation for upper-spinor component as[ − d2 dr2 + κ (κ + 1) r2 + 1 ~2C2 (MC2 + Enκ − Cs) ∑ (r) ] Fnκ(r) = 1 ~2C2 [ (E2nk − M 2C4 + Cs) ( MC2 − Enκ )] Fnκ(r), (12) and the lower-spinor component is given by Gnκ(r) = 1 MC2 + Enκ − Cs ( d dr + κ r ) Fnκ(r) (13) It is only the real positive energy states that exist when Cs = 0. However, under the pseudospin symmetry, Σ(r) = C p = con- stant, one can have from equation (10) a second-order differen- tial equation for the lower-spinor component as [14, 15][ − d2 dr2 + κ (κ− 1) r2 − 1 ~2C2 (MC2 − Enκ + C ps)∆(r) ] Gnκ(r) = 1 ~2C2 [ E2nκ − M 2C4 − C ps)(MC 2 − Enκ) ] Gnκ(r), (14) and the upper-spinor component Fnκ(r) as Fnκ(r) = 1 MC2 − Enκ + C ps ( d dr + κ r ) Gnκ(r) (15) It is only real negative energy states that exist when C p = 0. If we now include tensor interaction, then we obtain an equa- tion in each case for both spin and pseudospin symmetries as 2 C. A. Onate et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 917 3 follows[ d2 dr2 − κ (κ + 1) r2 + 2κ r U(r) − dU(r) dr − U 2(r) + d∆(r) dr M + Enκ − ∆(r) ( d dr + κ r − U(r) ) Fnκ(r) = [ (M + Enκ − ∆(r)) ( M − Enκ + ∑ (r) )] Fnκ(r) (16) [ d2 dr2 − κ (κ− 1) r2 + 2κ r U(r) + dU(r) dr − U 2(r) + d ∑ (r) dr M − Enκ + ∑ (r) ( d dr − κ r + U(r) ) Gnκ(r) = [ (M + Enκ − ∆(r)) ( M − Enκ + ∑ (r) )] Gnκ(r). (17) 2.1. Parametric Nikiforov-Uvarov Method (PNUM) The PNUM is one of the analytical techniques of mathe- matical physics that solves second-order differential equations in quantum mechanics. This method has a general form of the Schrödinger-like equation [17, 18, 19, 20][ d2 dr2 + c1 − c2 s s(1 − c3 s) d d s + −ξ1 s2 + ξ2 s − ξ3 s2(1 − c3 s) 2 ] ψ(s) = 0 (18) According to the PNUM, the eigenvalues and eigenfunctions can be obtain following the condition [17, 18, 19, 20, 21] c2n − (2n + 1)c5 + [ n2 − n + 2c8 ] c3 + √ c8 (√ c9 + 2nc3 + c3 ) + √ c9 ( 2n + 1 + √ c8 ) = −c7 (19) ψ(s) = sc12 (1 − s) −c12− c13 c3 P (c10−1, c11 c3 c10−1) n (1 − 2c3 s) (20) The values of the parametric constants in equations (19) and (20) are obtained as follows: c4 = 0.5(1 − c1), c5 = 0.5(c2 − 2c3), c6 = c 2 5 + ξ1, c7 = 2c4c5 − ξ2, c8 = c 2 4 + ξ3, c9 = c3(c7 + c3c8) + c6, c10 = 1 + 2 ( c4 + √ c8 ) , c11 = c2 − 2c5 + 2c3 √ c8 + 2 √ c9, c12 = c4 + √ c8, c13 = c5 − c3 √ c8 − √ c9. (21) 3. Solutions of Dirac Equation 3.1. SS Limit The SS limit occurs when d∆(r)/dr = 0 and ∆(r) = Cz with Σ(r) = V (r, E). Plugging equation (1) and equation (3) into equation (16), we have d2 Fnκ(r) dr2 + [ χ1s r2 + χ2s r 2 + χ3s ] Fnκ(r) = 0 (22) where χ1s = H1(1 + b1 Enκ) −λ1(1 + a1 Enκ)βs + 2κH1(1 + b1 Enκ) − H21 (1 + b1 Enκ) 2 − κ(κ + 1), (23) χ2s = −λ2(1 + a2 Enκ)βs − H 2 2 (1 + b2 Enκ) 2, (24) χ3s = H2(1 + b2 Enκ) [2κ− 1 − 2H1(1 + b1 Enκ)] −βs [λ3(1 + a3 Enκ) + M − Enκ] , (25) βs = M + Enκ − Cs. (26) Using a transformation of the form y = r2 in equation (22), we obtain d2 Fnκ(y) dy2 + 1 2y dFnκ(y) dy + χ2s y 2 + χ3s y + χ 1 s y2 Fnκ(y) = 0. (27) Comparing equation (27) with (18), we obtain the values of the parametric constants in equation (21) as follows c1 = 0.5, c2 = c3 = 0, c4 = 0.25, c5 = 0, c6 = −0.25χ 2 s, c7 = −0.24χ 3 s, c8 = 0.25(0.25 −χ 1 s ), c9 = −0.25χ 2 s, c10 = 1 + √ 0.25 −χ1s, c11 = √ −χ2s, c12 = 0.5 ( 0.5 + √ (0.25 −χ1s ) ) , c13 = − √ −0.25χ2s. (28) Plugging equation (28) into equation (19) and equation (20), respectively, gives λ1(1 + a1 Enκ)βs 4 + H1(1 + b1 Enκ) 2 [ H1(1 + b1 Enκ) − 1 2 − κ ] +( 1 + 2κ 4 )2 + βs [λ3 (1+a3 Enκ)+M−Enκ] 4 + H2 (1+b2 Enκ) 2 [ H1(1 + b1 Enκ) − κ− 1 2 ] + ℵs 1 + 2n + √ H22 (1 + b2 Enκ) 2 + λ2(1 + a2 Enκ)βs  2 = 0. (29) Fnκ(y) = y 0.5(0.5+ √ 0.25−χ1s e− √ −0.25χ2s y L √ 0.25−χ1s n ( √ −χ2s y ) (30) ℵs = ( n + 1 2 ) √ H22 (1 + b2 Enκ) 2 + λ2(1 + a2 Enκ)βs. (31) 3.2. PS limit The pseudospin symmetry limit occurs when dΣ(r)/dr = 0 and Σ(r) = C p. In this symmetry limit, the potential is taken as ∆(r) = V (r, E). Now, substituting equations (1) and (3) into equation (17) and by using the same transformation as before, 3 C. A. Onate et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 917 4 we have the parametric constants as c1 = 0.5, c2 = c3 = 0, c4 = 0.25, c5 = 0, c6 = 0.25 [ H22 (1 + b2 Enκ) 2 −λ2(1 + a2 Enκ)βp ] , c7 = 0.5H2(1 + b2 Enκ) [H1(1 + b1 Enκ) − 0.5 − κ] + 0.25 [M + Enκ −λ3(1 + a3 Enκ)] βp, c8 = 0.5H1(1 + b1 Enκ) [0.5H1(1 + b1 Enκ) + 0.5 − κ] + (0.25 − 0.5κ)2 −λ1(1 + a1 Enκ)βp, c9 = 0.25 [ H22 (1 + b2 Enκ) 2 −λ2(1 + a2 Enκ)βp ] , c10 = 1+√ 1 + H1(1 + b1 Enκ) [ H1(bE ) + 1 − 2κ ] + κ(κ− 1) −λ1(aE )βp, c11 = √ H22 (1 + b2 Enκ) 2 −λ2(1 + a2 Enκ)βp, c12 = 0.25+√ 0.5H1(bE ) [ 0.5H1(bE ) + 0.5 − κ ] + (0.25 − 0.5κ)2 −λ1(aE )βp, c13 = −0.5 √ H22 (1 + b2 Enκ) 2 −λ2(1 + a2 Enκ)βp, (32) where bE = 1 + b1 Enκ, aE = 1 + a1 Enκ Substituting equation (32) into equations (19) and (20), the energy for PS limit and its corresponding wave function are given as H1(1 + b1 Enκ) 2 [ H1(1 + b1 Enκ) + 1 2 − κ ] + ( 1 − 2κ 4 )2 − λ1(1 + a1 Enκ)βs 4 + βp [M−λ3 (1+a3 Enκ)+Enκ] 4 + H2 (1+b2 Enκ) 2 [ H1(1 + b1 Enκ) − κ− 1 2 ] + ℵp 1 + 2n + √ H22 (1 + b2 Enκ) 2 −λ2(1 + a2 Enκ)βp  2 = 0. (33) Fnκ(y) = y 0.25+ηp1 e0.5ηp2 y L ηp3 n (η4y) . (34) ℵp = ( n + 1 2 ) √ H22 (1 + b2 Enκ) 2 −λ2(1 + a2 Enκ)βp (35) ηp1 = √ 0.5H1(bE ) [ 0.5H1(bE ) + 0.5 − κ ] + Γ1, (36) where Γ1 = (0.25 − 0.5κ) 2 −λ1(1 + a1 Enκ)βp. ηp2 = √ H22 (1 + b2 Enκ) 2 −λ2(1 + a2 Enκ)βp, (37) ηp3 = √ 1 + H1(bE ) [ H1(bE ) + 1 − 2κ ] + κ(κ− 1) − Γ2, (38) where Γ2 = λ1(1 + a1 Enκ)βp. ηp4 = √ H22 (1 + b2 Enκ) 2 −λ2(1 + a2 Enκ)βp (39) 3.3. Solutions of the SS and PS via Supersymmetric Approach In this section, we obtain the solutions of the spin and pseu- dospin symmetry limits for Kratzer energy-dependent potential via SUSYQM. This method involves the proposition of super- potential function which is the solution of the non-linear Riccati equation. 3.3.1. Solution of the SS limit To obtain the solution of the spin symmetry limit of the Dirac equation with Kratzer energy-dependent potential, we substitute equations (2) and (3) into equation (16) to have a second-order differential equation of the form d2 Fnκ(r) dr2 =[ χs1 r2 − λ5(1 + a5 Enκ)βs r + H22 (1 + b2 Enκ) 2r2 + χs2 ] Fnκ(r) (40) where we have defined the following for mathematical simplic- ity χs1 = κ(κ + 1) + λ4(1 + a4 Enκ)βs+ H1(1 + b1 Enκ) [H1(1 + b1 Enκ) − 2κ− 1] (41) χs2 = H2(1 + b2 Enκ) [2H1(1 + b1 Enκ) − 2κ + 1] + (M − Enκ)βs (42) For a non-energy-dependent potential in the absence of tensor interaction, the energy eigenvalues in equation (40) purely de- pends on the quantum number n and the spin-orbit coupling term κ. This is physically related to energy as Enκ = (n,κ(κ+1)). This shows that for κ , 0, the states are degenerate. To solve equation (40) using SUSY approach [22, 23, 24, 25], we can write F0κ(r) = ex p ( − ∫ W(r)dr ) , (43) where W(r) is a superpotential which determines the solution of equation (40). To proceed to the next level, it is necessary to propose a superpotential function [22, 23, 24, 25, 26]. In this work, our superpotential function is proposed as W(r) = δ0 −δ1r −1 (44) where δ0 and δ1 are two different constants. For equation (44) to determine the solution of equation (40), the following condition must be satisfied W 2(r) − dW(r) dr = χs1 r2 − λ5(1 + a5 Enκ)βs r + H22 (1 + b2 Enκ) 2r2 + χs2 (45) Substituting equation (44) into equation (45), we easily deter- mine the values of the two constants in equation (44) as δ20 = χ s 2 (46) δ1 = 1 2 ± √ 1 + 4 [ χs1 + H 2 2 (1 + b2 Enκ) 2 ] 2 (47) 4 C. A. Onate et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 917 5 δ0 = λ5(1 + a5 Enκ)βsδ−11 2 (48) To test the correctness of the superpotential function, we construct the partner potentials of the supersymmetric quan- tum mechanics and examine the shape invariance condition [27, 28, 29]. Our partner potentials are obtain in the follow- ing forms V+(r) = W 2(r) + dW(r) dr = δ20 − 2δ0δ1 r + δ1(δ1 − 1) r2 (49) V−(r) = W 2(r) − dW(r) dr = δ20 − 2δ0δ1 r + δ1(δ1 + 1) r2 . (50) Equations (49) and (50) satisfied the shape-invariance condition and so, the following relationship exist V+(r, a0) = V−(r, a1) + R(a1) (51) In equation (51), a0 = δ1 and the Hamiltonian is shape- invariant. However, a1 = f (a0) = a0 + 1, which simply means that the potentials V±(r) are the same apart from a constant and the residual term R(a1) is independent of the variable r. In this case, a1 is uniquely determined from an old set of parameter a0. From the mapping, we now established that δ1 → δ1 + 1 using the negative partner potential. In terms of the newly introduced parameters, we express the residual term as R(a1) = λ25(1 + a5 Enκ) 2β2s 4a20 − λ25(1 + a5 Enκ) 2β2s 4a21 (52) R(a2) = λ25(1 + a5 Enκ) 2β2s 4a21 − λ25(1 + a5 Enκ) 2β2s 4a22 (53) R(a3) = λ25(1 + a5 Enκ) 2β2s 4a22 − λ25(1 + a5 Enκ) 2β2s 4a23 (54) The energy of the system is obtained using the above summa- tion given by Σnk=1R(ak) which is generalized as R(an) = λ25(1 + a5 Enκ) 2β2s 4a2n−1 − λ25(1 + a5 Enκ) 2β2s 4a2n (55) In view of the negative partner potential, the complete energy equation for the spin symmetry is given as H2(1 + b2 Enκ) [1 − 2κ + H2(1 + b2 Enκ)] + (M − Enκ)βs = λ25 [ (1 + a5 Enκ)βs ]2 4(δ1 + n) 2 (56) 3.3.2. Solution of the PS limit To obtain the solution of the pseudospin symmetry limit, we substitute equations (2) and (3) into equation (17) to have d2Gnκ(r) dr2 = χ p 1 + λ5(1 + a5 Enκ)βpr + χ p 2 r 2 + H22 (1 + b2 Enκ) 2r4 r2 Gnκ(r) (57) Table 1: Energies in the SS limit for modified Mie-type-constant EDP with λ1 = λ2 = λ3 = 0.5, a1 = 3, a2 = 2, a3 = 1, b1 = b2 = 0.1, M = 5 f m−1 and Cs = 1 f m−1. ` κ n (`, j) H1,2 = 0 H1,2 = 0.1 H1,2 = 0.2, 0.1 H1,2 = 0.1, 0.2 0 -1 0 0s1/2 1.942623900 1.958361278 1.959096533 1.972020529 0 -1 1 1s1/2 2.941736416 2.957567872 2.958305747 2.971859998 0 -1 2 2s1/2 4.060873315 4.076229954 4.076927679 4.090458326 0 -1 3 3s1/2 5.254527544 5.269270999 5.269915044 5.283193841 1 -2 0 0p3/2 1.841748059 1.861208075 1.850731135 1.889789645 1 -2 1 1p3/2 2.850509762 2.869662294 2.859097611 2.898566721 1 -2 2 2p3/2 3.981140297 3.999437112 3.989244626 4.027491284 1 -2 3 3p3/2 5.185188130 5.202574765 5.192859711 5.229500168 2 -3 0 0d5/2 1.644091503 1.670587775 1.650604087 1.715582858 2 -3 1 1d5/2 2.667778395 2.692515093 2.671366252 2.737514242 2 -3 2 2d5/2 3.820578636 3.843285779 3.822525569 3.886288212 2 -3 3 3d5/2 5.045509610 5.066514491 5.046609652 5.107241634 3 -4 0 0f7/2 1.362484903 1.400794769 1.374384444 1.463630855 3 -4 1 1f7/2 2.394333292 2.428432855 2.397977358 2.491841820 3 -4 2 2f7/2 3.577245062 3.606844862 3.576008806 3.666695125 3 -4 3 3f7/2 4.833507250 4.859747029 4.829865691 4.915639885 1 1 0 0p1/2 1.841748059 1.852413864 1.877371357 1.836808899 1 1 1 1p1/2 2.850509762 2.861449107 2.885908697 2.847162481 1 1 2 2p1/2 3.981140297 3.991828450 4.015024293 3.978925935 1 1 3 3p1/2 5.185188130 5.195485760 5.217335080 5.183790476 2 2 0 0d3/2 1.644091503 1.649346874 1.685727304 1.616913066 2 2 1 1d3/2 2.667778395 2.674654649 2.711126809 2.644310749 2 2 2 2d3/2 3.820578636 3.827991306 3.862736003 3.800294001 2 2 3 3d3/2 5.045509610 5.052997276 5.085717151 5.027654662 3 3 0 0f5/2 1.362484903 1.357450308 1.402754356 1.305640324 3 3 1 1f5/2 2.394333292 2.393986861 2.442017313 2.344787927 3 3 2 2f5/2 3.577245062 3.579492175 3.625897479 3.534961395 3 3 3 3f5/2 4.833507250 4.837006867 4.880792058 4.796634088 Table 2: Energies in the PS limit for modified Mie-type-constant EDP with λ1 = λ3 = 0.5, λ2 = −0.5, a1 = −3, a2 = −2, a3 = −1, b1 = b2 = 0.1, M = 5 f m−1 and Cs = 1 f m−1. ` κ n (`, j) H1,2 = 0 H1,2 = 0.1 H1,2 = 0.2, 0.1 H1,2 = 0.1, 0.2 1 -1 1 1s1/2 -1.758132998 -1.769902692 -1.779164454 -1.773124361 2 -2 1 1p3/2 -1.891711144 -1.914557485 -1.929357426 -1.923214633 3 -3 1 1d5/2 -2.100102133 -2.132843969 -2.153277701 -2.145586292 4 -4 1 1f7/2 -2.390428405 -2.431076159 -2.456965418 -2.446048454 1 -1 2 2s1/2 -2.424883237 -2.435163923 -2.443110934 -2.438169927 2 -2 2 2p3/2 -2.549727011 -2.569554464 -2.582073017 -2.577422600 3 -3 2 2d5/2 -2.740445644 -2.768710697 -2.785650672 -2.780425981 4 -4 2 2f7/2 -3.000000000 -3.035085686 -3.056131285 -3.049319087 1 2 1 0d3/2 -1.758132998 -1.735243670 -1.728251600 -1.719908479 2 3 1 0f5/2 -1.891711144 -1.858269052 -1.845486264 -1.837996526 3 4 1 0g7/2 -2.100102133 -2.341193985 -2.038745712 -2.316883988 4 5 1 0h9/2 -2.390428405 -2.341193985 -2.316253094 -2.316883988 1 2 2 1d3/2 -2.424883237 -2.404762492 -2.398620806 -2.391343780 2 3 2 1f5/2 -2.549727011 -2.956867588 -2.509496082 -2.502502503 3 4 2 1g7/2 -2.740445644 -2.956867588 -2.936575208 -2.682113098 4 5 2 1h9/2 -3.000000000 -2.956867588 -2.936575208 -2.934010648 where χ p 1 = κ(κ− 1) + H1(1 + b1 Enκ) [H1(1 + b1 Enκ) − 2κ + 1]− λ4(1 + a4 Enκ)βp (58) χ p 2 = H2(1 + b2 Enκ) [2H1(1 + b1 Enκ) − 2κ− 1] + (M + Enκ)(M − Enκ + C p) (59) To avoid repetition of works and algebra, we follow the same steps as in the spin symmetry and obtain the energy equation for the pseudospin spin symmetry as H2(1 + b2 Enκ) [2H1(1 + b1 Enκ) − 2κ− 1] + (M + Enκ)βp = [ −λ5(1 + a5 Enκ)βp 2(δp + n) ]2 (60) 5 C. A. Onate et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 917 6 δp = 1 2 + 1 2 √ 4χp1 + 1 + 4H 2 2 (1 + b2 Enκ) 2 (61) Table 3: Energies in the SS limit for modified Mie-type-constant EDP with λ1 = λ2 = λ3 = 0.5, a1 = 3, a2 = 2, a3 = 1, b1 = b2 = 0.1, M = 5 f m−1 and Cs = 1 f m−1 ` κ n (`, j) H1,2 = 0.2, 0 H1,2 = 0, 0.2 0 -1 0 0s 1/2 1.942079678 1.968171442 0 -1 1 1s 1/2 2.941066860 2.967805266 0 -1 2 2s 1/2 4.060044875 4.086322787 0 -1 3 3s 1/2 5.253529416 5.279016863 1 -2 0 0p3/2 1.818436560 1.897264650 1 -2 1 1p3/2 2.826966136 2.905851918 1 -2 2 2p3/2 3.958335296 4.034259115 1 -2 3 3p3/2 5.163329221 5.235688819 2 -3 0 0d5/2 1.601252028 1.732796555 2 -3 1 1d5/2 2.622584840 2.755428278 2 -3 2 2d5/2 3.776272981 3.903614939 2 -3 3 3d5/2 5.003004662 5.123603755 3 -4 0 0f7/2 1.306444952 1.487803464 3 -4 1 1f7/2 2.329887999 2.519191159 3 -4 2 2f7/2 3.512241944 3.694086997 3 -4 3 3f7/2 4.770643993 4.941943853 1 1 0 0p1/2 1.889417309 1.808576037 1 2 1 1p1/2 2.897131599 2.819215988 1 1 2 2p1/2 4.025207202 3.952158620 1 3 3 3p1/2 5.226530876 5.158310400 2 4 0 0d3/2 1.713977782 1.577278752 2 2 1 1d3/2 2.737952422 2.604208929 2 2 2 2d3/2 3.887416000 3.761839331 2 2 3 3d3/2 5.108364728 4.991200184 3 3 0 0f5/2 1.449429077 1.257493042 3 3 1 1f5/2 2.486841044 2.293004999 3 3 2 2f5/2 3.666840421 3.484650816 3 3 3 3f5/2 4.918103438 4.748960521 4. Discussion In Table 1, we presented the energy eigenvalues of the spin symmetry for equal and unequal values of H1 and H2. For H1 = H2 = 0, which is the solution without energy-dependent tensor interaction, the following degeneracies were produced: 0p3/2 = 0p1/2, 1p3/2=1p1/2, 3p3/2=3p1/2, 0d5/2, 0d3/2, 1d5/2=1d3/2, 2d5/2 = 2d3/2, 3d5/2=3d3/2, 0f7/2=0f5/2, 1f7/2=1f5/2. 2f7/2=2f5/2 and 3f7/2=3f5/2/. These degeneracies are the usual degenera- cies obtained with non-energy-dependent potentials. However, for H1 = H2 = 0.1, which is a solution with energy-dependent tensor interaction, there are no degeneracies. This shows that the energy-dependent tensor potential has broken the energy degenerate doublets in the system. For a non-energy-dependent tensor potential, even at H = 0.5 and H = 1, there are still degenerate doublets. For H1 = 0.2, H2 = 0.1 and H1 = 0.1, H2 = 0.2, there are no degeneracy production. In Table 2, we presented energy eigenvalues of the pseudospin symmetry for equal and unequal values of the tensor strengths i.e. H1 Table 4: Energies in the PS limit for modified Mie-type-constant EDP with λ1 = λ3 = 0.05, λ2 = −0.05, a1 = 6, a2 = −4, a3 = 5, b1 = b2 = 0.1, M = 15 f m−1 and Cs = 5 f m−1. ` κ n (`, j) H1,2 = 0.2, 0 H1,2 = 0, 0.2 1 -1 1 1s 1/2 -1.775253455 -1.763815218 2 -2 1 1p3/2 -1.920130164 -1.908460034 3 -3 1 1d5/2 -2.140140406 -2.125315642 4 -4 1 1f7/2 -2.441826999 -2.420424044 1 -1 2 2s1/2 -2.439648344 -2.430168824 2 -2 2 2p3/2 -2.573814603 -2.564901362 3 -3 2 2d5/2 -2.773637600 -3.028396662 4 -4 2 2f7/2 -3.041719317 -3.028396662 1 2 1 0d3/2 -1.742804990 -1.726917318 2 3 1 0f5/2 -1.865069919 -1.850942038 3 4 1 0g7/2 -2.061722917 -2.053329000 4 5 1 0h9/2 -2.340384372 -2.342315770 1 2 2 1d3/2 -2.411514069 -2.397477706 2 3 2 1f5/2 -2.526953758 -2.513529136 3 4 2 1g7/2 -2.708412767 -2.698017234 4 5 2 1h9/2 -2.959189164 -2.954567108 Table 5: Energies in the SS limit for Kratzer EDP with λ4 = λ5 = 0.5, a1 = 4.0, a2 = 2, b1 = b2 = 0.1, M = 10 f m−1 and Cs = 5 f m−1 ` κ n (`, j) Kratzer potential Coulomb potential H1,2 = 0 H1,2 = 0.1 H1,2 = 0.1 H1,2 = 0.1, 0 0 -1 0 0s1/2 0.370204340 0.393187068 1.252040915 1.230390156 0 -1 1 1s1/2 0.945739430 0.963640344 2.192321102 2.167520674 0 -1 2 2s1/2 1.435407674 1.453860405 2.839097594 2.811476651 0 -1 3 3s1/2 1.899432950 1.919827326 3.304673071 3.274484128 1 -2 0 0p3/2 0.568272837 0.605461699 2.196388371 2.167520674 1 -2 1 1p3/2 1.067819409 1.102125131 2.847354128 2.811476651 1 -2 2 2p3/2 1.530395283 1.565774885 3.316370162 3.274484128 1 -2 3 3p3/2 1.979996399 2.018417622 3.663421017 3.616377889 2 -3 0 0d5/2 1.262754541 0.865450254 2.858804254 2.811476651 2 -3 1 1d5/2 0.820311004 1.309314656 3.330517856 3.274484128 2 -3 2 2d5/2 1.694588465 1.744582496 3.679846038 3.616377889 2 -3 3 3d5/2 2.125227188 2.180498632 3.943595506 3.873809868 3 -4 0 0f7/2 1.085509689 1.138674624 3.345044630 3.274484128 3 -4 1 1f7/2 1.493098313 1.550825160 3.696546742 3.616377889 3 -4 2 2f7/2 1.901400337 1.965254589 3.962133536 3.873809868 3 -4 3 3f7/2 2.315910246 2.387891090 4.166042378 4.070830840 1 1 0 0p1/2 0.568272837 0.542744504 1.976844362 1.975929806 1 3 1 1p1/2 1.067819409 1.048345302 2.667541872 2.668584697 1 1 2 2p1/2 1.530395283 1.513218985 3.163567616 3.166216009 1 5 3 3p1/2 1.979996399 1.963746819 3.529456299 3.533423105 2 6 0 0d3/2 0.820311004 0.786755216 2.654220464 2.668584697 2 2 1 1d3/2 1.262754541 1.231133014 3.147935985 3.166216009 2 2 2 2d3/2 1.694588465 1.663175926 3.511836761 3.533423105 2 2 3 3d3/2 2.125227188 2.092809055 3.785235098 3.809600325 3 3 0 0f5/2 1.085509689 1.044886371 3.132908744 3.166216009 3 3 1 1f5/2 1.493098313 1.451243785 3.494674460 3.533423105 3 3 2 2f5/2 1.901400337 1.857234266 3.766256136 3.809600325 3 3 3 3f5/2 2.315910246 2.268233378 3.973416820 4.020621390 and H2. For H1 = H2 = 0, the following degenerate dou- blets are obtain: 1s1/2 = 0d3/2, 1p3/2 = 0f5/2, 1d5/2 = 0g7/2, 1f7/2 = 0h9/2, 2s1/2 = 1d3/2, 2p3/2 = 1f5/2, 2d5/2 = 1g7/2 and 2f7/2 = 1h9/2. These degeneracies are also equal to the de- generacies produced for non EDP for non-tensor interaction. For H1 = H2 = 0.1, there are no degeneracies. Similarly, for H1 > H2 and H1 < H2, there are no degenerate doublets. This also shows that the inclusion of the EDT term breaks the whole degeneracies even at small values of the tensor strengths. In 6 C. A. Onate et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 917 7 Table 6: Energies in the PS limit for Kratzer EDP with λ4 = λ5 = 0.5, λ2 = 0, a1 = 4.0, a2 = 2, b1 = b2 = 0.1, M = 10 f m−1 and Cs = 5 f m−1 ` κ n (`, j) Kratzer potential Coulomb potential H1,2 = 0 H1,2 = 0.1 H1,2 = 0.1 H1,2 = 0.1, 0 1 -1 1 1s1/2 -9.096854737 -9.097243710 -3.957853178 -3.956166280 2 -2 1 1p3/2 -9.103820973 -9.104900620 -4.705516764 -4.702953988 3 -3 1 1d5/2 -9.114041578 -9.115790810 -5.338447790 -5.334597330 4 -4 1 1f7/2 -9.127260481 -9.129650770 -5.877946235 -5.872766190 1 -1 2 2s1/2 -9.165271209 -9.165625395 -4.704351342 -4.702953988 2 -2 2 2p3/2 -9.171400175 -9.172390830 -5.337077070 -5.334597330 3 -3 2 2d5/2 -9.180401886 -9.182010810 -5.876538030 -5.872766190 4 -4 2 2f7/2 -9.192060938 -9.194263445 -6.338969120 -6.333932324 1 2 1 0d3/2 -9.096854737 -9.095134090 -3.852996864 -3.855104284 2 3 1 0f5/2 -9.103820973 -9.101415860 -4.624203482 -4.628695316 3 4 1 0g7/2 -9.114041578 -9.110975965 -5.272927745 -5.279249573 4 5 1 0h9/2 -9.127260481 -9.123565500 -5.823169340 -5.831036969 1 2 2 1d3/2 -9.165271209 -9.163683540 -4.625917968 -4.628695316 2 3 2 1f5/2 -9.171400175 -9.169181270 -5.274472190 -5.279249573 3 4 2 1g7/2 -9.180401886 -9.177572380 -5.824661165 -5.831036969 4 5 2 1h9/2 -9.192060938 -9.188647735 -6.294424900 -6.302161924 Note: Hi, j = 0 means Hi = H j = 0. Hi, j = 0.1, 0.2 means Hi = 0.1, H j = 0.2. Figure 1: Energies in the SS limit against mass M for modified Mie-type- constant EDP with λ1 = λ2 = λ3 = 1, a1 = 3, a2 = 2, a3 = 1, b1 = b2 = 1, and Cs = 5 f m−1. Tables 3 and 4, we presented the energy for SS and PS respec- tively for Coulomb energy-dependent tensor potential ( H2 = 0) and constant energy-dependent tensor potential ( H1 = 0). In both cases, there are no degeneracies. To check the accuracy and correctness of the energy-dependent tensor potential, we also studied the solutions of the spin and pseudospin symme- tries with the same energy-dependent tensor potential with the Kratzer energy-dependent potential. The special cases of this potential was studied numerically. The results of the two sym- metries are given in Tables 5 and 6. In Table 5, the energy for spin symmetry is given for both Kratzer and Coulomb energy- dependent potentials. For Kratzer energy-dependent potential, the degeneracies obtained in Table 1 were equal obtained. For Coulomb energy-dependent potential ( λ4 = 0), it was consid- ered for Coulomb-constant energy-dependent tensor potential and Coulomb energy-dependent tensor potential ( H2 = 0). For H1 = H2 = 0.1, the same energy degeneracies obtained Figure 2: Energies in the PS limit against mass M for modified Mie-type- constant EDP with λ1 = λ2 = λ3 = 1, a1 = 3, a2 = 2, a3 = 1, b1 = b2 = 1, and Cs = 5 f m−1. in Table 1 were also obtained, but for H2 = 0, which re- duces the Coulomb-constant energy-dependent tensor potential to Coulomb energy-dependent tensor potential, a new set of en- ergy degeneracies were formed. The degeneracies formed are 1s1/2 = 0p3/2, 2s1/2 = 1p3/2 = 0d5/2, 3s1/2 = 2p3/2 = 1d5/2 = 0f7/2, 3p3/2 = 2d5/2 = 1f 7/2, 3d5/2 = 2f7/2, 1p1/2 = 0d3/2, 2p1/2 = 1d3/2 = 0f5/2, 3p1/2 = 2d3/2 = 1f5/2 and 3d3/2 = 2f5/2. These are new degeneracies different from the degeneracies obtained with ordinary Coulomb tensor potential. For the pseudospin symmetry in Table 6, the following degeneracies were formed with Coulomb energy-dependent tensor potential for Coulomb energy-dependent potential: 1d5/2 = 2p3/2, 1f7/2 = 2d5/2, 0f5/2 = 1d3/2, 0g7/2 = 1f5/2 and 0h9/2 = 1g7/2. For ordinary Coulomb tensor potential with H = 0.5 and H = 1, the degeneracies formed in each case are different from those formed in the present work. Figure 1 and Figure 2 showed the variation of the energy of SS and PS respectively with the mass M for the modified Mie-type potential. The energy of the SS increases with the mass while that of the PS decreases with the mass. 5. Conclusion In this work, we have employed two traditional techniques to solve Dirac equation with two energy-dependent potential and an energy-dependent tensor interaction without the use of any approximation scheme to the centrifugal term. The degen- eracies formed in our results without the application of tensor potential are exactly the degeneracies formed for non-energy- dependent potential. However, the degeneracies formed when the energy-dependent tensor potential was applied differ from the degeneracies formed with non-energy-dependent tensor po- tential. In the case of Coulomb energy-dependent potential with Coulomb energy-dependent tensor potential under spin symme- try, there are four degenerate doublets, three degenerate dou- blets and two degenerate doublets which cannot be formed in 7 C. A. Onate et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 917 8 the case of non-energy-dependent tensor potentials. For the combination of Coulomb-constant energy-dependent tensor po- tential, a small value of the tensor strengths can easily break the energy doublets which is not possible for ordinary tensor poten- tial. References [1] D. Troltenier, C. Bahri & J.P. Draayer, “Effects of pairing in the pseudo- SU(3) model”, Nucl. Phys. A 586 (1995) 75. [2] A.E. Stuchbery, “Magnetic properties of rotational states in the pseudo- Nilsson model”, Nucl. Phys. A 700 (2002) 83. [3] A. Bohr, I. Hamamoto & B.R, “Motteison, Pseudospin in rotating nuclear potentials”, Phys. Scr. 26 (1982) 267. [4] W. Nazarewicz, P.J. Twin, P. Fallon & J.D. Garrett, “Natural-parity states in superdeformed bands and pseudo SU(3) symmetry at extreme condi- tions”, Phys. Rev. Lett. 64 (1990) 1654. [5] A.E. Stuchbery, Magnetic behaviour in the pseudo-Nilsson model,” J. Phys. G. 25 (1999) 611. [6] D. Troltenier, W. Nazarewicz, Z. Szymanski & J.P. Draayer, “On the va- lidity of the pseudo-spin concept for axially symmetric deformed nuclei”, Nucl. Phys. A 567 (1994) 591. [7] B.J. Falaye, “Arbitrary `-state solutions of the hyperbolical potential by the asymptotic iteration method”, Few-Body Syst. 53 (2012) 557. [8] A. Soylu, O. Bayrak & I. Boztosun, “An approximate solution of Dirac- Hulthen problem with pseudospin and spin symmetry for any state”, J. Math. Phys. 48 (2007) 082302 [9] H. Hassanabadi, E. Maghsoodi, S. Zarrinkamar & H. Rahimov, “An ap- proximate solution of the Dirac equation for hyperbolic scalar and vector potentials and a Coulomb tensor interaction by SUSYQM”, Mod. Phys. Lett. A 26 (2011) 2703. [10] C.A. Onate & J.O. Ojonubah, “Relativistic and nonrelativistic solutions of the generalized Pöschl-Teller and hyperbolical potentials with some thermodynamic properties”, Int. J. Mod. Phys. E 24 (2015) 1550020. [11] S.M. Ikhdair & R. Sever, “Two approximation schemes to the bound states of the Dirac-Hulthen problem”, J. Phys. A: Math. Theor. 44 (2011) 355301. [12] M. Hamzavi, A.A. Rajabi & H. Hassanabadi, “Exact Solutions of Dirac Equation with Hartmann Potential by Nikiforov-Uvarov Method”, Int. J. Mod. Phys. A 26 (2010) 1363. [13] M. C. Onyeaju, A.N. Ikot, C.A. Onate, O. Ebomwonyi, M.E. Udoh & J.O.A. Idiodi, “Approximate bound-states solution of the Dirac equation with some thermodynamic properties for the deformed Hylleraas plus de- formed Woods-Saxon potential”, Eur. Phys. J. Plus 132 (2017) 302. [14] J.N. Ginocchio, “The relativistic foundations of pseudospin symmetry in nuclei”, Nucl. Phys. A 654 (1999) 663c. [15] J.N. Ginocchio¸ “A relativistic symmetry in nuclei”, Phys. Rep. 315 (1999) 231. [16] G.F. Wei & S.H. Dong, “Algebraic approach to pseudospin symmetry for the Dirac equation with scalar and vector modified Pöschl-Teller poten- tials”, European Physics Letter (EPL) 87 (2009) 40004. [17] C. Tezcan & R. Sever, “A general approach for the exact solution of the Schrödinger equation”, Int. J. Theor. Phys. 48 (2009) 337. [18] I. J. Njoku, C. P. Onyenegecha , C. J. Okereke, A. I. Opara, U. M. Ukewuihe, F. U. Nwaneho, “Approximate solutions of Schrodinger equation and thermodynamic properties with Hua potential”, Results in Physics 24 (2021) 104208. [19] R. Khordad, B. Mirhosseini, Linear and nonlinear optical properties in spherical quantum dots: Rosen-Morse potential. Opt. Spectroscopy 117 (2014) 440 [20] A. Ghanbari, R. Khordad, & F. Taghizadeh, “Influence of Coulomb term on thermal properties of fluorine”, Chem. Phys. Lett. 801 (2022) 139725. [21] J.O.A. Idiodi & C.A. Onate, “Entropy, Fisher Information and Variance with Frost-Musulin Potenial”, Commun. Theor. Phys. 66 (2016) 269. [22] F. Cooper, A. Khare & U. Sukhatme, “Supersymmetry and quantum me- chanics”, Phys. Rep. 251 (1995) 267. [23] F. Cooper, J.N. Ginocchio & A. Wipf, “Derivation of the S-Matrix using super symmetry”, Phys. Lett. A 129 (1988) 145. [24] F. Cooper, A. Khare & U. Sukhatme, “Relationship between supersym- metry and solvable potentials”, Phys. Rev. D 36 (1987) 2458. [25] A. Khare & U. Sukhatme, “New shape-invariant potentials in supersym- metric quantum mechanics”, J. Phys. A 26 (1993) L901. [26] E. Witten, “Dynamical breaking of supersymmetry”, Nucl. Phys. B 185 (1981) 513. [27] L.E. Gendenshtein, “Derivation of exact spectra of the Schrodinger equa- tion by means of supersymmetry”, Sov. Phys.: JETP Lett. 38 (1983) 356. [28] C.S. Jia, X.G. wang, X.K. Yao, P.C. Chen & W. Xiao, “A unified recur- rence operator method for obtaining normalized explicit wave functions for shape-invariant potentials”, J. Phys A: Math. Gen. 31 (1998) 4763. [29] C. A. Onate, G. O. Egharevba, & D. T. Bankole, “Eigensolution to Morse potential for Scandium and Nitrogen Monoiodides”, J. Nig. Soc. Phys. Sci. 3 (2021) 286. 8