Any l-state improved quasi-exact analytical solutions of the spatially dependent mass Klein-Gordon equation for the scalar and vector Hulthen potentials
aa r X i v : . [ qu a n t - ph ] J a n Any l -state improved quasi-exact analytical solutions of thespatially dependent mass Klein-Gordon equation for the scalarand vector Hulth´en potentials Sameer M. Ikhdair ∗ and Ramazan Sever † Department of Physics, Near East University, Nicosia, North Cyprus, Turkey Department of Physics, Middle East Technical University, 06800, Ankara,Turkey (Dated: November 6, 2018)
Abstract
We present a new approximation scheme for the centrifugal term to obtain a quasi-exact ana-lytical bound state solutions within the framework of the position-dependent effective mass radialKlein-Gordon equation with the scalar and vector Hulth´en potentials in any arbitrary D dimensionand orbital angular momentum quantum numbers l. The Nikiforov-Uvarov (NU) method is used inthe calculations. The relativistic real energy levels and corresponding eigenfunctions for the boundstates with different screening parameters have been given in a closed form. It is found that thesolutions in the case of constant mass and in the case of s -wave ( l = 0) are identical with the onesobtained in literature.Keywords: Bound states, approximation schemes, Hulth´en potential, Klein-Goron equation,position- dependent mass distributions, NU method PACS numbers: 03.65.-w; 02.30.Gp; 03.65.Ge; 34.20.Cf ∗ E-mail: [email protected] † E-mail: [email protected] . INTRODUCTION The bound and scattering states of the s - and l -waves for any interaction system haveraised a great interest in non-relativistic as well as in relativistic quantum mechanics [1-3].The exact solution of the wave equation is very important since the wavefunction containsall the necessary information regarding the quantum system under consideration. A numberof methods have been used to solve the wave equations exactly or quasi-exactly for non-zeroangular momentum quantum number ( l = 0) by means of a given potential. The boundstate eigenvalues were solved numerically [4,5] and quasi-analytically using variational [4,6],perturbation [7], shifted 1 /N expansion [8,9], NU [10,11], SUSYQM [12-14] and AIM [15]methods.The Hulth´en potential [10,12,13,15,16] is one of the important short–range potentials inphysics and it has been applied to a number of areas such as nuclear and particle physics[17], atomic physics [18,19], molecular physics [20,21] and chemical physics [22]. Therefore,it would be interesting and important to solve the relativistic equation for this potential for l = 0 case since it has been extensively used to describe the bound and continuum states ofthe interaction systems. Recently, the exact solutions for the bound and scattering statesof the s -wave Schr¨odinger [16,23], Klein-Gordon [1-3] and Dirac equation [24,25] with thescalar and vector Hulth´en potentials are investigated.Relativistic effects with the scalar plus vector Hulth´en-type potential [1,2] in three- and D dimensions and harmonic oscillator-type potential [26,27] have been also discussed in theliterature. The bound-states of the Dirac and Klein-Gordon equations with the Coulomb-likescalar plus vector potentials have been studied in arbitrary dimension [28-32]. Furthermore,the exact results for the scattering states of the Klein-Gordon equation with Coulomb-likescalar plus vector potentials have been investigated in an arbitrary dimension [33]. Thisequation has been exactly solved for a larger class of linear, exponential and linear plusCoulomb potentials to determine the bound state energy spectrum using two semiclassicalmethods with the following relationship between the scalar and vector potentials: V ( r ) = V + βS ( r ) , S ( r ) > V ( r ) where V and β being arbitrary constants [34]. In particular,inserting the constants V = 0 and β = ± V ( r ) = ± S ( r ).Also, the position-dependent mass solutions of the nonrelativistic and relativistic systems2ave received much attention recently. Many authors have used different methods to studythe partially exactly solvable and exactly solvable Schr¨odinger, Klein-Gordon and Diracequations in the presence of variable mass having a suitable mass distributions functionin 1 D, D and/or any dimension D cases for different potentials, such as the exponential-type potentials [35], the Coulomb potential [36], the Lorentz scalar interactions [37], thehyperbolic-type potentials [38], the Morse potential [39], the P¨oschl-Teller potential [40], theCoulomb and harmonic potentials [41], the modified Kratzer-type, rotationally correctedMorse potentials [42], Mie-type and pseudoharmonic potentials [43]. Recently, the pointcanonical transformation (PCT) has also been employed to solve the D -dimensional position-dependent effective mass Schr¨odinger equation for some molecular potentials to get the exactbound state solutions including the energy spectrum and corresponding wave functions [41-43].A new method to obtain the exactly solvable PT-symmetric potential potentials withinthe framework of the variable mass Dirac equation with the vector potential coupling schemein (1 + 1) dimensions [38]. Three PT-symmetric potentials are produced which are PT-symmetric harmonic oscillator-like potential, PT-symmetric of linear plus inversely linearpotential and PT-symmetric kink-like potential. The SUSYQM formalism and functionanalysis method are use to obtain the real energy levels and corresponding spinor componentsfor the bound states. Further, the position-dependent effective mass Dirac equation withthe PT-symmetric hyperbolic cosecant potential can be mapped into the Schr¨odinger-likeequation with the exactly solvable modified P¨oschl-Teller potential [38]. The real relativisticenergy levels and corresponding spinor wavefunctions for the bound states have been givenin a closed form.The Nikiforov-Uvarov (NU) method [44] and other methods have also been used to solvethe D -dimensional Schr¨odinger equation [45] and relativistic D -dimensional Klein-Gordon[46], Dirac [47] and spinless Salpeter equations [48].In strong coupling cases, it is crucial to understand relativistic effects on a moving par-ticle in a potential field. In a non-relativistic case, Schr¨odinger equation with the Hulth´enpotential [10,12,13,15] was solved using the usual existing approximation, r ≈ α e αr ( e αr − for the centrifugal potential which was found to be consistent with the results of othermethods [4,8,13,15]. Unfortunately, this approximation is valid only for small values of thescreening parameter α, but the agreement becomes poor in the high-screening region [10 , . particles, using a more general approximation scheme, r ≈ α e − γαr (1 − e − αr ) where γ is a dimensionless parameter ( γ = 0 , α and the di-mensionless parameter γ are taken as α = 0 . γ =1, respectively, which is simply thecase of the usual approximation [10,12,13,15]. Also, Jia and collaborators [50] have recentlyproposed an alternative approximation scheme, r ≈ α (cid:16) ωe αr − + e αr − (cid:17) where ω is a di-mensionless parameter ( ω = 1 . , for the centrifugal potential to solve the Schr¨odingerequation with the Hulth´en potential. Taking ω = 1 , their approximation can be reducedinto the usual approximation [10,12,13,15]. However, the accuracy of their numerical results[50] is found to be in poor agreement with the other numerical methods like integrationand variational methods [4,5]. In order to improve the accuracy of the used approxima-tion, we propose and apply an alternative shifted approximation scheme to approximate thecentrifugal term given by [51,52]1 r = lim α → α (cid:20) c + e αr ( e αr − (cid:21) , (1)where c is a shifting dimensionless parameter. The approximation scheme (1) emerged asa quite successful formalism to study the Schr¨odinger equation with the Manning-Rosen,hyperbolic and Hulth´en potentials in calculating the energy eigenvalues within the frame-work of the NU method [51-53]. The accuracy of the results are significantly improvedover all other existing literature approximation schemes and analytical methods [13,15,50].With extremely high accuracy, we have obtained the numerical energy eigenvalues as withthose obtained by the numerical integration [4,5,53], variational [4] methods and also by aMATHEMATICA package programmed by Lucha and Sch¨oberl [54].The purpose of this work is to employ the approximation scheme given in (1) to solvethe position-dependent mass radial Klein-Gordon equation with any orbital angular quan-tum number l for the scalar and vector Hulth´en potentials in D -dimensions. This offers asimple, accurate and efficient scheme for the exponential-type potential models in quantum4echanics.Our paper is organized as follows. In section 2, we review the NU method. In section 3,we present a brief a derivation to find the shifting parameter c . Then, the analytical solutionof the position-dependent mass Klein-Gordon equation with the scalar and vector Hulth´enpotentials is obtained for any l -state by means of the N-U method. Section 4 contains thesummary and conclusions. II. NU
The NU method is breifly outlined here and the details can be found in [44]. This methodis proposed to solve the second-order differential equation of the hypergeometric type: ψ ′′ n ( z ) + e τ ( z ) σ ( z ) ψ ′ n ( z ) + e σ ( z ) σ ( z ) ψ n ( z ) = 0 , (2)where σ ( z ) and e σ ( z ) are polynomials, at most, of second-degree, and e τ ( s ) is a first-degreepolynomial. In order to find a particular solution for Eq. (2), let us decompose the wave-function ψ n ( z ) as follows: ψ n ( z ) = φ n ( z ) y n ( z ) . (3)We can reduce Eq. (2) into the form σ ( z ) y ′′ n ( z ) + τ ( z ) y ′ n ( z ) + λy n ( z ) = 0 , (4)with τ ( z ) = e τ ( z ) + 2 π ( z ) , τ ′ ( z ) < , (5)where τ ′ ( z ) = dτ ( z ) dz is the derivative. Also, λ is a constant given in the form λ = λ n = − nτ ′ ( z ) − n ( n − σ ′′ ( z ) , n = 0 , , , · · · , (6)where λ = k + π ′ ( z ) . (7)The y n ( z ) can be written in terms of the Rodrigues relation y n ( z ) = B n ρ ( z ) d n dz n [ σ n ( z ) ρ ( z )] , (8)where B n is the normalization constant and the weight function ρ ( z ) satisfies σ ( z ) ρ ′ ( z ) + ( σ ′ ( z ) − τ ( z )) ρ ( z ) = 0 . (9)5he other wavefunction in the solution is defined by σ ( z ) φ ′ ( z ) − π ( z ) φ ( z ) = 0 . (10)Further, to find the weight function in Eq. (8) we need to obtain the following polynomial: π ( z ) = 12 [ σ ′ ( z ) − e τ ( z )] ± (cid:26)
14 [ σ ′ ( z ) − e τ ( z )] − e σ ( z ) + kσ ( z ) (cid:27) . (11)The expression under the square root sign in Eq. (11) can be arranged as the square of apolynomial. This is possible only if its discriminant is zero. In this regard, an equation for k is being obtained. After solving such an equation, the determined values of k are includedin the NU method. III. BOUND-STATE SOLUTIONSA. An Impoved Shifted Approximation Scheme
The approximation is based on the expansion of the centrifugal term in a series of expo-nentials depending on the intermolecular distance r. Therefore, instead of using the usualexisting approximation in literature, let us, instead, take the following exponential-typepotential to approximate the centrifugal potential,1 r ≈ α (cid:2) c + v ( r ) + v ( r ) (cid:3) , v ( r ) = e αr e αr − , r ≈ α (cid:20) c + 1 e αr − e αr − (cid:21) . (12)In the low-screening region, 0 . ≤ αr ≤ . α ) , Eq. (12)is a very well approximation to the centrifugal potential and the Schr¨odinger equation forsuch an approximation can be easily solved analytically. In Fig. 1, we give a plot of thevariation of the centrifugal potential and its approximation given in Eq. (12) versus αr.
Itshows that the approximation (12) and 1 /r are similar and coincide in both high-screeningas well as in the low-screening regions.Changing the r coordinate to x by shifting the parameters as x = ( r − r ) /r to avoidsingularities [55] , we obtains1 r (1 + x ) − = α " c + 1 e γ (1+ x ) − e γ (1+ x ) − , γ = αr , (13)6nd expanding Eq. (13) around r = r ( x = 0) , we obtain the following expansion:1 − x + O ( x ) = γ (cid:18) c + 1 e γ − e γ − (cid:19) − γ (cid:18) e γ − e γ − + 2( e γ − (cid:19) x + O ( x ) , (14)and consequently γ (cid:20) c + 1 e γ − e γ − (cid:21) = 1 ,γ (cid:18) e γ − e γ − + 2( e γ − (cid:19) = 2 . (15)By solving Eqs. (14) and (15) for the dimensionless parameter c , we obtain c = 1 γ − e γ − − e γ − = 0 . , (16)where e = 2 . γ = 0 . . Therefore, the centrifugal potential takes the formlim α → α " γ − e γ − − e γ − + e − αr − e − αr + (cid:18) e − αr − e − αr (cid:19) = 1 r . (17)Let us remark at the end of this analysis that the approximation used in many papers inliterature [10,12,13,15] is a special case of Eq. (12) if c is set to zero. B. A Quasi-Exactly Energy Eigenvalues and Eigenfunctions
The D -dimensional time-independent radial position-dependent mass Klein-Gordon equa-tion with scalar and vector potentials S ( r ) and V ( r ) , respectively, r = | r | , and position-dependent mass m ( r ) describing a spin-zero particle takes the general form [3,46] ∇ D ψ ( l D − = l ) l ··· l D − ( x ) + 1 ~ c n [ E nl − V ( r )] − (cid:2) m ( r ) c + S ( r ) (cid:3) o ψ ( l D − = l ) l ··· l D − ( x ) = 0 , ∇ D = D X j =1 ∂ ∂x j , ψ ( l D − = l ) l ··· l D − ( x ) = R l ( r ) Y ( l ) l ··· l D − ( θ , θ , · · · , θ D − ) , (18)where E nl denotes the Klein-Gordon energy and ∇ D denotes the D -dimensional Laplacian.Further, x is a D -dimensional position vector. Let us decompose the radial wavefunction R l ( r ) as follows: R l ( r ) = r − ( D − / g ( r ) , (19)7e, then, reduce Eq. (18) into the following D -dimensional radial position-dependent effec-tive mass Schr¨odinger-like equation d g ( r ) dr + 1 ~ c (cid:26) [ E nl − V ( r )] − (cid:2) m ( r ) c + S ( r ) (cid:3) − ( D + 2 l − D + 2 l − ~ c r (cid:27) g ( r ) = 0 . (20)Further, taking the vector and scalar potentials as the Hulth´en potentials V ( r ) = − V e − αr − e − αr , S ( r ) = − S e − αr − e − αr , α = r − , (21)and choosing the following mass function m ( r ) = m + m e − αr − e − αr , (22)we can rewrite Eq. (20) as g ′′ ( r ) + 1 ~ c (cid:26) m c ( S − m c ) + E nl V ] e − αr − e − αr + h V − ( S − m c ) i e − αr − ~ c α ( D + 2 l − D + 2 l − e − αr (1 − e − αr ) g ( r )= 1 ~ c h(cid:0) m c (cid:1) − E nl + ∆ E l i g ( r ) , g (0) = 0 , (23)with the shift energy ∆ E l = ~ c α ( D + 2 l − D + 2 l − c / . On account of the wavefunction g ( r ) satisfying the standard bound-state condition (real values), i.e., g ( r → ∞ ) → . If we rewrite Eq. (23) by using a new variable of the form z = e − αr ( r ∈ [0 , ∞ ) , z ∈ [1 , , we get d g ( z ) dz + 1 − zz (1 − z ) dg ( z ) dz + 1[ z (1 − z )] × (cid:8) − ε nl + ( β − β − γ + 2 ε nl ) s − ( β + β + β − β + ε nl ) s (cid:9) g ( z ) = 0 , (24)where the following definitions of parameters ε nl = q ( m c ) − E nl + ∆ E l Q , β = 2 ( m c S + E nl V ) Q , β = S − V Q ,β = m c ( m c − S ) Q , β = 2 m m c Q , γ = ( D + 2 l − D + 2 l − , Q = ~ cα, (25)8re used. For bound-state solutions, we require that V ≤ ( S − m c ) and E nl ≤ p ( m c ) + ∆ E l . In order to solve Eq. (24) by means of the N-U method, we shouldcompare it with Eq. (2). The following values for parameters are found e τ ( z ) = 1 − z, σ ( z ) = z − z , e σ ( z ) = − ε nl +( β − β − γ +2 ε nl ) s − ( β + β + β − β + ε nl ) s . (26)If we insert these values of parameters into Eq. (11), with σ ′ ( z ) = 1 − z, the followinglinear function is obtained π ( z ) = − z ± q [1 + 4( β + β + β − β + ε nl − k )] z + [4( k − β + β + γ − ε nl )] z + 4 ε nl . (27)The determinant of the square root must be set equal to zero, that is, ∆ = ( k − β + β + γ − ε nl ) − ε nl [1 + 4( β + β + β − β + ε nl − k )] = 0 . Thus, the constant k found to be k = β − β − γ ± ε nl p β + β + γ ) . (28)In this regard, we can find four possible functions for π ( z ) as π ( s ) = − z ± ε nl ∓ (cid:2) ε nl − √ b (cid:3) z for k = d + ε nl √ b,ε nl ∓ (cid:2) ε nl + √ b (cid:3) z for k = d − ε nl √ b. (29)where b = β + β + γ and d = β − β − γ. Thus, taking the following values k = β − β − γ − ε nl p β + β + γ ) , (30)and π ( z ) = − z ε nl − (cid:20) ε nl + 12 p β + β + γ ) (cid:21) z, (31)they give τ ( z ) = 1 + 2 ε nl − (cid:20) ε nl + 12 p β + β + γ ) (cid:21) z,τ ′ ( s ) = − (cid:20) ε nl + 12 p β + β + γ ) (cid:21) < . (32)Eqs. (30)-(32) together with the assignments given in Eq. (26), the following expressionsfor λ are obtained λ n = λ = n + h ε nl + p β + β + γ ) i n, ( n = 0 , , , · · · ) , (33) λ = β − β − γ −
12 (1 + 2 ε nl ) h p β + β + γ ) i , (34)9here n is the radial quantum number. By defining δ = 12 (cid:16) p β + β + γ ) (cid:17) , (35)where β + β = δ − δ − γ. With the aid of Eq. (35), we can easily obtain the energyeigenvalue equation of the Hulth´en potential by solving Eqs. (33) and (34): ε ( D ) nl = ( β − β − γ − n ) − (2 n + 1) δ n + δ )= 4 [ β − β − n − (2 n + 1) δ ] − ( D + 2 l − D + 2 l − n + δ )= 2 h m c e S + E ± nl V i + e S − V Q ( n + δ ) − n + δ , ( n = 0 , , , · · · ) , (36)where e S = S − m c is the modified scalar potential. Solving the last equation for theenergy eigenvalues E ± nl , we obtain E ± nl = V − e S (cid:16) e S + 2 m c (cid:17) V + κ nl ± κ nl vuuut ξ − − e S (cid:16) e S + 2 m c (cid:17) V + κ nl ,ξ = (2 m c ) + ~ c α ( D + 2 l − D + 2 l − c V + κ nl ,κ nl = ~ cα (2 n + 1) + r (cid:16) e S − V (cid:17) + ( ~ cα ) ( D + 2 l − , (37)where n = 0 , , , · · · and l = 0 , , , · · · signify the usual radial and angular momentumquantum numbers, respectively, and( ~ cα ) ( D + 2 l − + 4 e S ≥ V , ξ ≥ − e S (cid:16) e S + 2 m c (cid:17) V + κ nl , (38)are constraints over the strength of the potential coupling parameters. In the above equation,let us remark that it is not difficult to conclude that all bound-states appear in pairs, twoenergy solutions are valid for the particle E p = E + nl and the second one corresponds to theanti-particle energy E a = E − nl in the Hulth´en field. When we take the scalar and vectorpotentials as e S = 0 (i.e., S = m c ) and V = 0, the energy equation (37) becomes E ± nl = V ± κ nl s (2 m c ) + ~ c α ( D + 2 l − D + 2 l − c V + κ nl − , m c ) + 4 ~ c α ( D + 2 l − D + 2 l − c ≥ V + κ nl ,κ nl = ~ cα (2 n + 1) + q ( ~ cα ) ( D + 2 l − − V , D ≥ , (39)with the following constraints on the coupling parameter of the vector potential:( ~ cα ) ( D + 2 l − ≥ V , (40)must be fulfilled for real eigenvalues.Therefore, having solved the D -dimensional position-dependent mass Klein-Gordon equa-tion for scalar and vector usual Hulth´en potentials, we should make some useful remarks.(i) For s -wave ( l = 0), the exact energy eigenvalues of the 1 D Klein-Gordon equationbecomes E ± n = V − e S (cid:16) e S + 2 m c (cid:17) V + κ n ± κ n vuuut m c V + κ n − − e S (cid:16) e S + 2 m c (cid:17) V + κ n ,κ n = ~ cα (2 n + 1) + q ( ~ cα ) + 4( e S − V ) , (41)In order that at least one level might exist, it is necessary that the inequalities ~ c α + 4 e S ≥ V , m c V + κ n ≥ − e S (cid:16) e S + 2 m c (cid:17) V + κ n , (42)are fulfilled. In the case e S = 0 , V = 0 , the energy spectrum (in units where ~ = c = 1): E ± n = V ± (cid:20) α (2 n + 1) + q α − V (cid:21) vuut m V + h α (2 n + 1) + p α − V i − , (43)with the following constraints on the potential coupling constant:16 m ≥ V + (cid:20)q α − V + (2 n + 1) (cid:21) , α ≥ V , (44)are fulfilled for bound state solutions. We notice that the result given in Eq. (43) is identicalto Eq. (31) of Ref. [56]. As can be seen from Eq. (43), there are only two lower-lying states( n = 0 ,
1) for a Klein-Gordon particle of rest mass m = 1 and screening parameter α = 1with vector coupling strength V ≤ / . As an example, one may calculate the ground stateenergy for the vector coupling strength V = α/ E ± = V " ± s m V − . (45)11urther, in the case of pure scalar potential ( V = 0 , S = m c ) , the energy spectrum E ± n = ± s m c − ( ~ cα ) ( n + 1) , m c ≥ ( ~ cα ) ( n + 1) . (46)Since the Klein-Gordon equation is independent of the sign of E n for scalar potentials, thewavefunctions become the same for both energy values. If the range parameter α is chosento be α = 1 /λ c , where λ c = ~ /m c = 1 /m denotes the Compton wavelength of the Klein-Gordon particle. It can be seen easily that while S → m c in ground state ( n = 0) , allenergy eigenvalues tend to the value E ≈ . m . (ii) For D = 3 , the mixed scalar and vector Hulth´en potentials, the energy eigenvaluesfor l = 0 are given by E ± nl = V − e S (cid:16) e S + 2 m c (cid:17) V + e κ nl ± e κ nl vuuute ξ − − e S (cid:16) e S + 2 m c (cid:17) V + e κ nl , e ξ = ( m c ) + ~ c α l ( l + 1) c V + e κ nl , e κ nl = ~ cα (2 n + 1) + r ( ~ cα ) (2 l + 1) + 4 (cid:16) e S − V (cid:17) . (47)Further, in order that at least one real eigenvalue might exist, it is necessary that theinequality ( ~ cα ) (2 l + 1) + 4 e S ≥ V , e ξ ≥ − e S (cid:16) e S + 2 m c (cid:17) V + e κ nl , (48)must be fulfilled. For the case where e S = 0 in the spatial-dependent mass ( S = 0 , in theconstant mass case) [46], the energy eigenvalues turn out to be E ± nl = V ± η nl s ( m c ) + ~ c α l ( l + 1) c V + η nl − ,η nl = ~ cα (2 n + 1) + q ( ~ cα ) (2 l + 1) − V , ~ cα (2 l + 1) ≥ V , (49)with the following constraint over the potential parameters:(4 m c ) + 16 ~ c α l ( l + 1) c ≥ V + (cid:20) ~ cα (2 n + 1) + q ( ~ cα ) (2 l + 1) − V (cid:21) . (50)12iii) When D = 3 and l = 0 , the centrifugal term ( D +2 l − D +2 l − r = 0 and consequently theapproximation term ( D +2 l − D +2 l − α h c + e − αr (1 − e − αr ) i = 0 , too. Thus, the energy eigenval-ues turn to become p ( m c ) − E ± n = 2 h m c e S + E ± n V i + e S − V ~ cα ( n + δ ) − ~ cα (cid:18) n + δ (cid:19) ,δ = 12 " ~ cα ) r ( ~ cα ) + 4 (cid:16) e S − V (cid:17) , ( n = 0 , , , , · · · ) (51)which gives E ± n = V − e S (cid:16) e S + 2 m c (cid:17) V + ξ n ± ς n vuuut ( m c ) V + ς n − − e S (cid:16) e S + 2 m c (cid:17) V + ς n ,ς n = ~ cα (2 n + 1) + r ( ~ cα ) + 4 (cid:16) e S − V (cid:17) , ( ~ cα ) + 4 e S ≥ V , (4 m c ) ≥ (cid:0) V + ς n (cid:1) − e S (cid:16) e S + 2 m c (cid:17) V + ς n (52)(iv) For equal scalar and vector usual Hulth´en potential (i.e., S = V ), Eq. (36) with the aidof Eq. (25) can be reduced to the relativistic energy equation (in the conventional atomicunits ~ = c = 1): s m + ( D + 2 l − D + 2 l − c r − E R = 2 r V [ m + E R − m ] + r ( m − m ) m n + δ ) − n + δ r ,δ = 12 (cid:20) q ( D + 2 l − + (2 r m c ) − r V m c (cid:21) , ( n = l = 0 , , , , · · · ) , (53)which is Eq. (22) of Ref. [58] if the perturbed mass m = 0 and shifting parameter c = 0 . (v) We discuss non-relativistic limit of the energy equation (53). When V = S , Eq.(23) reduces to a Schr¨odinger-like equation for the potential 2 V ( r ) . In other words, the non-relativistic limit is the Schr¨odinger equation for the potential − V e − r/r / (cid:2) − e − r/r (cid:3) , r = α − . After making the parameter replacements m + E R → m and m − E R → − E NR inEq. (53)[58], it reduces into the non-relativistic energy equation of Refs. [10,12,13,15,57,59]: E NR = α ( D + 2 l − D + 2 l − c m − m α " (2 V − m )(2 m − m ) − α ( n + δ ) ( n + δ ) , = 12 (cid:20) α q α ( D + 2 l − + (2 m c ) − V m c (cid:21) , ( n = l = 0 , , , , · · · ) (54)which is Eq. (23) of Ref. [57] when c and m are set to zero. It is noted that Eq. (54)is identical to Eq. (59) of Ref. [56] for s -wave in 1 D when the potential is 2 V ( r ) , when α becomes pure imaginary, i.e., α → iα and when we set m = 1 , m = 0 and c = 0 . Equation (54) can be reduced to the constant mass ( m = 0) case in the three-dimensionalSchr¨odinger equation: E NR = α m ( l ( l + 1) c − (cid:20) V m α ( n + l + 1) − n + l + 12 (cid:21) ) , which is identical to the expressions given in Refs. [50,52] when the vector potential istaken as 2 V ( r ) , c = 0 and ω = 1 in Ref. [50] . The numerical approximation to the energyeigenvalues in Ref. [50] for the last energy equation was found to be more efficient than theapproximation given in Eq. (19) of Ref. [50]. Taking V = Zαe as in [50], we obtain E NR = α m ( l ( l + 1) c − (cid:20) m Ze α ( n + l + 1) − ( n + l + 1)2 (cid:21) ) . For the s -wave ( l = 0) , the above energy spectrum is identical to the factorization method[23], SUSYQM [12,13] and NU [46] methods. Expanding the energy equation (53) underthe weak coupling conditions [( n + δ ) /m r ] ≪ V r / ( n + δ )] ≪ , retaining onlythe terms containing the power of (1 /m r ) and ( r V ) , we obtain the relativistic energyequation E R ≈ E NR + m + 2(2 m − m ) (cid:18) (2 V − m )2 α ( n + δ ) (cid:19) , (55)which is simply Eq. (24) of Ref. [57], where δ is given in Eq. (54). The first term is thenon-relativistic energy and third term is the relativistic approximation to energy.Now, let us find the wave function y n ( s ) , which is the polynomial solution ofhypergeometric-type equation. We multiply Eq. (4) by the weight function ρ ( s ) so thatit can be rewritten in self-adjoint form [45,46][ ω ( s ) y ′ n ( s )] ′ + λρ ( s ) y n ( s ) = 0 . (56)The weight function ρ ( s ) that satisfies Eqs. (9) takes the following form ρ ( z ) = z ε nl (1 − z ) β , β = 2 δ − y nl ( z ) = B nl z − ε nl (1 − z ) − β d n dz n (cid:2) z n +2 ε nl (1 − z ) n + β (cid:3) = B nl P (2 ε nl ,β ) n (1 − z ) . (58)On the other hand, inserting the values of σ ( s ) , π ( s ) and τ ( s ) given in Eqs. (26), (31) and(32) into Eq. (10), we get the other part of the wave function φ ( s ) = z ε nl (1 − z ) δ . (59)Hence, the wave function g n ( z ) = φ n ( z ) y n ( z ) becomes g ( z ) = C nl z ε nl (1 − z ) δ P (2 ε nl ,β ) n (1 − z )= C nl z ε ( D ) nl (1 − z ) δ P (2 ε ( D ) nl ,β ) n (1 − z ) , z ∈ [1 , . (60)Finally, the radial wave functions of the Klein-Gordon equation are obtained as R l ( r ) = N nl r − ( D − / e − ε ( D ) nl αr (1 − e − αr ) δ P (2 ε ( D ) nl ,β ) n (1 − e − αr ) , (61)with ε ( D ) nl = 1 ~ cα r ( m c ) + ~ c α ( D + 2 l − D + 2 l − c − E nl ,β = 1 ~ cα r (cid:16) e S − V (cid:17) + ( ~ cα ) ( D + 2 l − , δ = 12 (1 + β ) , (62)where E nl is given in Eq. (37) and N nl is the radial normalization factor. The Jacobi polyno-mials P (2 ε ( D ) nl ,β ) n (1 − e − αr ) [60] in the last result can be written in terms of the hypergeometricfunction F ( − n, n + 2 ε ( D ) nl + β + 1 , ε ( D ) nl ; e − αr ) which gives the same result obtained in Ref.[57].(i) The exact radial wave functions for the s -wave Klein-Gordon equation in 1 D reducesto the following form (in ~ = c = 1) : R n ( x ) = C n e − √ m − E n x (1 − e − x/r ) (1+ a ) / P (2 r √ m − E n ,a ) n (1 − e − x/r ) ,a = r r (cid:16) e S − V (cid:17) , (63)where E n is given in Eq. (41). The last formula is identical to Eq. (35) of Ref. [56] whenthe modified scalar potential, e S , is set to zero.15ii) Choosing the atomic units h/ π = ~ = c = 1 , the exact radial wave functions for the s -wave Klein-Gordon equation in 3 D reduces to the following form: R n ( r ) = N n e − √ m + − E n r (1 − e − r/r ) (1+ a ) / P (2 r √ m − E n ,a ) n (1 − e − r/r ) ,P (2 r √ m − E n ,a ) n (1 − e − r/r ) = F ( − n, n + 2 r q m − E n + a + 1 , r q m − E n ; e − αr ) , (64)where E n and a are given in Eq. (52) and Eq. (63), respectively. The last formula is identicalto Eq. (22) of Ref. [57] when the perturbed mass m is set to zero.(iii) The quasi-exact radial wave functions for the l -wave Klein-Gordon equation in 3 D reduces to the following form (in ~ = c = 1) : R nl ( r ) = N nl e − r m + l ( l +1) c r − E nl r (1 − e − r/r ) (1+ a l ) / P (2 r r m + l ( l +1) c r − E nl ,a l ) n (1 − e − r/r ) ,P (2 r r m + l ( l +1) c r − E nl ,a l ) n (1 − e − r/r )= F ( − n, n + 2 r s m + l ( l + 1) c r − E nl + a l + 1 , r s m + l ( l + 1) c r − E nl ; e − αr ) ,a l = r (2 l + 1) + 4 r (cid:16) e S − V (cid:17) , (65)where E nl is given in Eq. (43) and α = r − . It is identical to Ref. [57] when m = 0. Theeigenfunctions in the constant mass case are written as R nl ( r ) = N nl e − r m + l ( l +1) c r − E nl r (1 − e − r/r ) (1+ a l ) / P (2 r r m + l ( l +1) c r − E nl ,b l ) n (1 − e − r/r ) ,b l = q (2 l + 1) + 4 r ( S − V ) . (66)At the end of these calculations, the total wave functions of the Klein-Gordon equation withposition-dependent mass for the scalar and vector Hulth´en potentials are ψ ( l D − = l ) l ··· l D − ( x ) = N nl r − ( D − / e − ε ( D ) nl αr (1 − e − αr ) δ P (2 ε ( D ) nl ,β ) n (1 − e − αr )1 √ π exp( ± il θ ) D − Y j =2 s (2 l j + j − n j !2Γ ( l j + l j − + j −
2) (sin θ j ) lj − nj P ( l j − n j +( j − / ,l j − n j +( j − / n j (cos θ j ) s (2 n D − + 2 m ′ + 1) n D − !2Γ ( n D − + 2 m ′ ) (sin θ D − ) l D − P ( m ′ ,m ′ ) n D − (cos θ D − ) , (67)16here ε ( D ) nl and β are given in Eq. (62) and E nl is given in Eq. (37) [46].To check the accuracy of the resulting analytical expressions. We give a few numericalreal eigenvalues for some selected values of the mass m and m and potential parameters S and V . In Tables 1 and 2, taking α = 1 and m = 1 , we present some numerical valuesfor the energy spectrum of the constant mass Klein-Gordon equations with the condition S = V for all possible real eigenvalues. To get more real energy eigenvalues in the constantmass case (e.g., m = 1 , m = 0), the vector parameter V of the Hulth´en potential shouldbe increased. As shown in Tables 1 and 2, when the parameter V = S = 1 , , , , , , we obtain N = 1,3,6,10,15,36 real energy eigenvalues, respectively. The numerical solutionof the position-dependent mass case with vector and scalar Hulth´en potential parameterssatisfying the conditions S = ± V and S > V are presented in Table 3. For example, inTable 3, when the Hulth´en potential parameter V = S = 1 , m = 5 and m = 0 , we obtain N = 46 real energy eigenvalues. Obviously, the number of real eigenvalues increases in thesolution of the position-dependent case than in the constant mass case where the condition S ≥ V must be fulfilled. IV. COCLUSIONS
In summary, we have proposed an alternative approximation scheme for the centrifu-gal potential similar to the non-relativistic case. This is because the usual approximation[10,13,15] for the centrifugal term is only valid for low-screening region, however, for thehigh screening region where α increases, the agreement between the old approximation andcentrifugal term decreases. Using this approximation scheme, the analytical solutions of theradial Klein-Gordon equation with position-dependent mass for scalar and vector Hulth´enpotentials can be approximately obtained for any dimension D and orbital angular mo-mentum quantum number l . It is found that the expressions for the eigenvalues and thecorresponding eigenfunctions become complicated and tedious since the eigenvalues are re-lated to the parameters m o , m , S , V , c and α. We have investigated the possibility toobtain the bound-state (real) energy spectra with some constraints to be imposed on theparameters and, further, the relationship between the strengths of vector V and scalar S coupling parameters. In one- and three-dimensions, the special cases for the angular momen-tum l = 0 , m = 0) and s -wave ( l = 0) , the results are reduced to exact solution of bound states of s -wave Klein-Gordon equationwith scalar and vector Hulth´en potentials. To test our results, we have also calculated theenergy eigenvalues of a particle and antiparticle for the constant mass limit as well as theposition-dependent mass case. The case of spatial-dependent mass with scalar potential S = m c is found to be equivalent to the constant mass with scalar potential S = 0 in apure vector case. Hence, the spectrum is found to be same. Acknowledgments
Work partially supported by the Scientific and Technological Research Council of Turkey(T ¨UB˙ITAK). We thank the kind referees for their positive and invaluable suggestions whichhave improved the paper greatly. 18
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050 to α = 0 .
250 in steps of 0 . . ABLE I: The energy spectrum of the scalar and vector Hulth´en potential for m = 1 and m = 0 .V = S n l E + nl a E − nl a E + nl [61,62] b E − nl [61,62] b . − . . − . − − − − . − . . − . . − . − − − − − − . − . . − . − − − − . − . . − . . − . − − . . − − − − − − . − . . − . . . − − − − − − . . . . − − − − − . − . − . − . . − . − − . − . − − . . − − . − . . − . . − . − − . − . − − − − − − . − . . − . . . − − − − − − . . . . a The present NU method. b The results from AIM and SUSY. ABLE II: The energy spectrum of the scalar and vector Hulth´en potential for m = 1 and m = 0 .V = S n l E + nl E − nl V = S n l E + nl E − nl − . − . − . − . − . − . − . − . . − . − . − . . − . . − . . . . − . − − . . − . − . . . . − . − . − . . − . − . − . . . . − . − − . − . . − . . . . − . . . . . − . − . − − . − . . − . . − . . . . . − − . . . . . − . − − . − . − − . . − . − . . . − . − . . − . − . − . . . − . − . . . . − . . . . − . . . . . . . . . − − ABLE III: The energy spectrum of the scalar and vector Hulth´en potential for m = 0 .m m V S n l E + E − m m V S n l E + E − .
01 2 2 1 0 0 . − . −
10 20 1 0 4 . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . − . . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . − −− −