Position-dependent-mass; Cylindrical coordinates, separability, exact solvability, and PT-symmetry
aa r X i v : . [ qu a n t - ph ] J u l Position-dependent-mass; Cylindrical coordinates,separability, exact solvability, and
P T -symmetry
Omar MustafaDepartment of Physics, Eastern Mediterranean University,G Magusa, North Cyprus, Mersin 10,TurkeyE-mail: [email protected]: +90 392 630 1314,Fax: +90 392 3651604November 27, 2018
Abstract
The kinetic energy operator with position-dependent-mass in cylin-drical coordinates is obtained. The separability of the correspondingSchr¨odinger equation is discussed within radial cylindrical mass settings.Azimuthal symmetry is assumed and spectral signatures of various z -dependent interaction potentials (Hermitian and non-Hermitian PT -symmetric)are reported.PACS codes: 03.65.Ge, 03.65.CaKeywords: Position-dependent-mass, cylindrical setting, separability,exact solvability, PT -symmetry. Introduction
The von Roos Hamiltonian for position-dependent-mass (PDM) quantum par-ticles is known to be associated with an ordering ambiguity problem manifestedby the non-unique representation of the kinetic energy operator [1]. In suchHamiltonian H = − ~ h m ( ~r ) γ ~ ∇ m ( ~r ) β · ~ ∇ m ( ~r ) α + m ( ~r ) α ~ ∇ m ( ~r ) β · ~ ∇ m ( ~r ) γ i + V ( ~r ) , (1)an obvious profile change in the effective potential is introduced when the para-metric values of the ambiguity parameters ( α, β, γ ) are changed (within the vonRoos constraint α + β + γ = − α = γ (cf., e.g., ref.[2] and the related references cited therein) . This wouldeffectively reduce the domain of the acceptable parametric values of the ambigu-ity parameters. In fact, the PDM Hamiltonian (1) is known to be a descriptivemodel for many physical problems (like but not limited to, many-body problem,electronic properties of semiconductors, etc.) [1-30]. It is, moreover, a math-ematically challenging and a useful model that enriches the class of exactlysolvable quantum mechanical systems.In the literature, nevertheless, one may find many suggestion on the am-biguity parametric values. For example, Gora and William have suggested β = γ = 0 , α = −
1, Ben Daniel and Duke α = γ = 0 , β = −
1, Zhu andKroemer α = γ = − / , β = 0, Li and Kuhn β = γ = − / , α = 0, andMustafa and Mazharimousavi α = γ = − / , β = − / m ( ~r ) = m ◦ M ( ρ, ϕ, z ) = M ( ρ, ϕ, z ) = M ( ρ ) = 1 /ρ ) and azimuthal sym-metrization is sought through the assumption that V ( ~r ) = V ( ρ, ϕ, z ) = ρ h ˜ V ( ρ ) + ˜ V ( z ) i . (2)Of course, this constitutes only a one feasible separability of the system (otherseparability options may occur as well), as justified in section 2. In section 3,within the radial cylindrical settings, we consider two examples of fundamentalnature. The radial cylindrical ”Coulombic” ˜ V ( ρ ) = − /ρ and the ”harmonicoscillator” ˜ V ( ρ ) = a ρ /
4. The spectral signatures of different ˜ V ( z ) settingson the Coulombic and harmonic oscillator spectra are reported for impenetrablewalls at z = 0 and z = L , for a Morse [31], for a non-Hermitian PT -symmetrizedScarf II [28,32,33], and for a non-Hermitian PT -symmetrized Samsonov [28,34]interaction models. Where, P denotes parity and T mimics the time reflection(cf., e.g., Ref.[28] and references cited therein on this issue). Our concludingremarks are in section 4. 3 Cylindrical coordinates and separability
Let us consider the kinetic energy operator of the PDM Hamiltonian in (1)and a PDM function of the form m ( ~r ) = m ◦ M ( ρ, ϕ, z ) = M ( ρ, ϕ, z ) (where ~ = m ◦ = 1 units are to be used hereinafter). Moreover, we consider thefollowing substitutions ~A = αM ( ρ, ϕ, z ) α − h ˆ ρ ∂ ρ + ˆ ϕρ ∂ ϕ + ˆ z ∂ z i M ( ρ, ϕ, z ) ,~B = βM ( ρ, ϕ, z ) β − h ˆ ρ ∂ ρ + ˆ ϕρ ∂ ϕ + ˆ z ∂ z i M ( ρ, ϕ, z ) ,~C = γM ( ρ, ϕ, z ) γ − h ˆ ρ ∂ ρ + ˆ ϕρ ∂ ϕ + ˆ z ∂ z i M ( ρ, ϕ, z ) , (3)to imply ~ ∇ M ( ρ, ϕ, z ) α = ~A + M ( ρ, ϕ, z ) α ~ ∇ ~ ∇ M ( ρ, ϕ, z ) β = ~B + M ( ρ, ϕ, z ) β ~ ∇ ~ ∇ M ( ρ, ϕ, z ) γ = ~C + M ( ρ, ϕ, z ) γ ~ ∇ . (4)Using the above identities, one (with M ( ρ, ϕ, z ) ≡ M for simplicity of notations)may rewrite M α ~ ∇ M β · ~ ∇ M γ = M α (cid:16) ~B · ~C (cid:17) + M α + γ (cid:16) ~B · ~ ∇ (cid:17) + M α + β h ~C · ~ ∇ + ~ ∇ · ~C i + M − ~ ∇ , (5)and M γ ~ ∇ M β · ~ ∇ M α = M γ (cid:16) ~B · ~A (cid:17) + M α + γ (cid:16) ~B · ~ ∇ (cid:17) + M γ + β h ~A · ~ ∇ + ~ ∇ · ~A i + M − ~ ∇ . (6)4et use now consider a class of the mass functions defined as M ( ρ, ϕ, z ) = g ( ρ ) f ( ϕ ) k ( z ) = ⇒ ∂ ρ M = M ρ = g ρ ( ρ ) f ( ϕ ) k ( z ) ⇒ ∂ ρ M = M ρρ = g ρρ ( ρ ) f ( ϕ ) k ( z ) . (7)Which would, in effect, imply that the PDM Schr¨odinger equation [ H − E ] Ψ ( ρ, ϕ, z ) =0 for Hamiltonian (1) be written as (cid:26) ∂ ρ + (cid:18) ρ − M ρ M (cid:19) ∂ ρ + 1 ρ (cid:18) ∂ ϕ − M ϕ M ∂ ϕ (cid:19) + (cid:18) ∂ z − M z M ∂ z (cid:19)(cid:27) Ψ ( ρ, ϕ, z )= { M V ( ρ, ϕ, z ) − M E − M W ( ρ, ϕ, z ) } Ψ ( ρ, ϕ, z ) , (8)where2 M W ( ρ, ϕ, z ) = ζM " M ρ + M ϕ ρ + M z − ( β + 1) M (cid:20) M ρ ρ + M ρρ + M ϕϕ ρ + M zz (cid:21) = ζ "(cid:18) g ′ ( ρ ) g ( ρ ) (cid:19) + 1 ρ (cid:18) f ′ ( ϕ ) f ( ϕ ) (cid:19) + (cid:18) k ′ ( z ) k ( z ) (cid:19) − ( β + 1) " g ′ ( ρ ) ρg ( ρ ) + g ′′ ( ρ ) g ( ρ ) + 1 ρ f ′′ ( ϕ ) f ( ϕ ) + k ′′ ( z ) k ( z ) . (9) ζ = α ( α −
1) + γ ( γ − − β ( β + 1) . (10)At this point, one should notice that the choice of the mass function in (7)is inspired by the appearance of terms like ( M ρ /M ), ( M ϕ /M ), and ( M z /M ) asmultiplicities of the first-order derivatives in (8). This would, in fact, make theseparability of (8) highly feasible and far less complicated. Moreover, followingthe traditional general wave function assumption,Ψ ( ρ, ϕ, z ) = R ( ρ ) Φ ( ϕ ) Z ( z ) ; ρ ∈ (0 , ∞ ) , ϕ ∈ (0 , π ) , z ∈ ( −∞ , ∞ ) (11)5o ease coordinates separability of (8), we obtain0 = 2 g ( ρ ) f ( ϕ ) k ( z ) [ E − V ( ρ, ϕ, z )]+ (cid:20) R ′′ ( ρ ) R ( ρ ) − (cid:18) g ′ ( ρ ) g ( ρ ) − ρ (cid:19) R ′ ( ρ ) R ( ρ )+ ζ (cid:18) g ′ ( ρ ) g ( ρ ) (cid:19) − ( β + 1)2 g ′ ( ρ ) ρg ( ρ ) + g ′′ ( ρ ) g ( ρ ) ! + " Z ′′ ( z ) Z ( z ) − k ′ ( z ) k ( z ) Z ′ ( z ) Z ( z ) + ζ (cid:18) k ′ ( z ) k ( z ) (cid:19) − ( β + 1)2 k ′′ ( z ) k ( z ) + 1 ρ " Φ ′′ ( ϕ )Φ ( ϕ ) − f ′ ( ϕ ) f ( ϕ ) Φ ′ ( ϕ )Φ ( ϕ ) + ζ (cid:18) f ′ ( ϕ ) f ( ϕ ) (cid:19) − ( β + 1)2 f ′′ ( ϕ ) f ( ϕ ) (12)It is obvious that separability is granted through a variety of choices. Thesimplest of which may be sought in an obviously ”manifested-by-equation (12)”general identity of the form2 M V ( ρ, ϕ, z ) = 2 g ( ρ ) f ( ϕ ) k ( z ) V ( ρ, ϕ, z ) = ˜ V ( ρ ) + ˜ V ( z ) + 1 ρ ˜ V ( ϕ ) . (13)In this case, we may avoid any specifications on the forms of g ( ρ ), f ( ϕ ), and k ( z ) rather than being mathematically and quantum mechanically ”very well”defined. However, the energy term, 2 g ( ρ ) f ( ϕ ) k ( z ) E , in (12) suggests threefeasible separabilities for f ( ϕ ) = 1 = k ( z ), k ( z ) = 1 = g ( ρ ), and f ( ϕ ) = 1 = g ( ρ ). We focus on one of these cases in the sequel.Let us consider the position-dependent-mass function to be only an explicitfunction of ρ . Namely, we choose f ( ϕ ) = 1 = k ( z ) and g ( ρ ) = ρ − so that M ( ρ, ϕ, z ) = M ( ρ ) = ρ − . Under these settings, equation(12) collapses into asimple separable form0 = (cid:20) Φ ′′ ( ϕ )Φ ( ϕ ) + 2 E − ˜ V ( ϕ ) + 2 ( ζ − β − (cid:21) + ρ (cid:20) R ′′ ( ρ ) R ( ρ ) + 3 ρ R ′ ( ρ ) R ( ρ ) − ˜ V ( ρ ) + Z ′′ ( z ) Z ( z ) − ˜ V ( z ) (cid:21) . (14)6quation (14) with azimuthal symmetry (i.e., ˜ V ( ϕ ) = 0) would immediatelyimply that (cid:20) Φ ′′ ( ϕ )Φ ( ϕ ) + 2 E + 2 ( ζ − β − (cid:21) = K ϕ , (15)and " R ′′ ( ρ ) R ( ρ ) + 3 ρ R ′ ( ρ ) R ( ρ ) + K ϕ ρ − ˜ V ( ρ ) + (cid:20) Z ′′ ( z ) Z ( z ) − ˜ V ( z ) (cid:21) = 0 . (16)In due course, the solution of (15) reads Φ ( ϕ ) = exp ( imϕ ) where m = 0 , ± , ± , · · · is the magnetic quantum number and Φ ( ϕ ) satisfies the single valued conditionΦ ( ϕ ) = Φ ( ϕ + 2 π ). Moreover, we obtain K ϕ = 2 E + 2 ( ζ − β − − m . (17)Consequently, one may cast Z ′′ ( z ) Z ( z ) − ˜ V ( z ) = − K z (18)and R ′′ ( ρ ) R ( ρ ) + 3 ρ R ′ ( ρ ) R ( ρ ) + K ϕ ρ − ˜ V ( ρ ) = K z (19)In the following section, we consider ˜ V ( ρ ) to represent a ”Coulombic” and a”harmonic oscillator” and find the spectral signatures of different ˜ V ( z ) poten-tials of (18) on the over all spectra. 7 Two examples; the radial cylindrical Coulom-bic and the harmonic-oscillator
A priori, we remove the first-order derivative in the radial cylindrical part of(19) and redefine R ( ρ ) = ρ − / U ( ρ ) , (20)to obtain − U ′′ ( ρ ) + " / − K ϕ ρ + ˜ V ( ρ ) U ( ρ ) = − K z U ( ρ ) . (21)In fact, this 1D radial cylindrical Schr¨odinger equation provides an effectivetool to study the effect of different ˜ V ( z ) settings of (18) on the spectra of twointeresting models of fundamental nature. The Coulombic and the harmonicoscillator [30]. Of course, such effects could be tested for other models.Let us take a Coulombic radial cylindrical model ˜ V ( ρ ) = − /ρ . In this case,equation (21) would read − U ′′ ( ρ ) + (cid:20) ℓ − / ρ − ρ (cid:21) U ( ρ ) = − K z U ( ρ ) , (22)where ℓ = (cid:0) − K ϕ (cid:1) / , K z = ( n ρ + ℓ + 1) − , and n ρ = 0 , , , · · · is the radialquantum number. Hence, K z = 1 / (cid:16) n ρ + q − K ϕ + 1 (cid:17) and E = (cid:18) m + 32 (cid:19) − ( ζ − β ) − (cid:18) K z − n ρ − (cid:19) , (23)where K z is to be determined through the solution of (18) under different ˜ V ( z )settings.Next, we consider the radial cylindrical harmonic oscillator model ˜ V ( ρ ) =8 ρ / E = (cid:18) m + 32 (cid:19) − ( ζ − β ) − (cid:18) K z a + 2 n ρ + 1 (cid:19) , (24)where, again, K z is to be determined through the solution of (18) under differ-ent ˜ V ( z ) settings in the sequel subsections. Nevertheless, it is obvious that theposition-dependent-mass spectral signature is documented through the ambigu-ity parameters appearance in the constant shift, (i.e., [ − ( ζ − β − z = 0 and z = L Lets us now consider that the above mentioned position-dependent-mass particleis trapped to move between two impenetrable walls at z = 0 and z = L . Wemay then take ˜ V ( z ) = < z < L ∞ ; elsewhere . (25)Consequently, equation (18) reads Z ′′ ( z ) + K z Z ( z ) = 0 , (26)where Z ( z ) satisfies the boundary conditions Z ( z = 0) = 0 = Z ( z = L ) andimplies that Z ( z ) = sin K z z ; K z = n z πL , n z = 1 , , , · · · . (27)Hence, K z = n z π /L and the quantum PDM particle here is quasi-free in the z -direction (i.e., ˜ V ( z ) = 0) but constrained to move between the two impenetrable9alls at z = 0 and z = L . The spectral signature of such z -dependent potentialsettings is clear, therefore. That is, a quantum particle endowed with a position-dependent-mass M ( ρ, ϕ, z ) = M ( ρ ) = ρ − and subjected to an interactionpotential of the form V ( ρ, ϕ, z ) = − ρ + ρ ˜ V ( z ) , (28)with ˜ V ( z ) defined in (25), would admit exact energy eigenvalues given by E n ρ ,m,n z = (cid:18) m + 32 (cid:19) − ( ζ − β ) − (cid:18) Ln z π − n ρ − (cid:19) , (29)On the other hand, a quantum particle endowed with a position-dependent-mass M ( ρ, ϕ, z ) = M ( ρ ) = ρ − subjected to an interaction potential of theform V ( ρ, ϕ, z ) = a ρ / ρ ˜ V ( z ) , (30)with ˜ V ( z ) defined in (25), would be accompanied by exact energy eigenvaluesof the form E n ρ ,m,n z = (cid:18) m + 32 (cid:19) − ( ζ − β ) − (cid:18) n z π aL + 2 n ρ + 1 (cid:19) . (31) ˜ V ( z ) Morse model
Consider a Morse type interaction ˜ V ( z ) = D (cid:0) e − ǫz − e − ǫz (cid:1) ; D >
0, in (18).We may then closely follow the methodical proposal of Chen [31] to obtain K z = √ Dǫ − ˜ n z − ! , ˜ n z = 0 , , , , · · · (32)where one should consider 2 m = ~ = 1, a → ǫ , E → K z and x → z of Chen[31] to match our settings in (18). Therefore, a PDM quantum particle endowed10ith M ( ρ, ϕ, z ) = M ( ρ ) = ρ − and subjected to an interaction potential of theform V ( ρ, ϕ, z ) = − ρ + Dρ (cid:0) e − ǫz − e − ǫz (cid:1) ; D > , (33)would admit exact energy eigenvalues given by E n ρ ,m, ˜ n z = (cid:18) m + 32 (cid:19) − ( ζ − β ) − q √ Dǫ − ˜ n z − − n ρ − . (34)Obviously, the condition (cid:16) √ D/ǫ − ˜ n z − (cid:17) > M ( ρ, ϕ, z ) = M ( ρ ) = ρ − subjected toan interaction potential V ( ρ, ϕ, z ) = a ρ / Dρ (cid:0) e − ǫz − e − ǫz (cid:1) ; D > , (35)would indulge the exact energy eigenvalues E n ρ ,m, ˜ n z = (cid:18) m + 32 (cid:19) − ( ζ − β ) − a " √ Dǫ − ˜ n z − + 2 n ρ + 1 ! . (36) PT -symmetrized ˜ V ( z ) spectral signatures We may now consider a PT -symmetrized ˜ V ( z ) Scarf II in (18) so that˜ V ( z ) = − A z − i A sinh z cosh z , (37)where the corresponding Hamiltonian is known to be a non-Hermitian PT -symmetric Hamiltonian that admits exact eigenvalues (cf., e.g., Mustafa and11azharimousavi { K z = − (cid:0) n z + − A (cid:1) ; n z = 0 , , , , · · · < A − , for A ≥ , − ; for A < , . (38)Hence, a quantum particle with M ( ρ, ϕ, z ) = M ( ρ ) = ρ − moving in V ( ρ, ϕ, z ) = − ρ − ρ (cid:20) A z + i A sinh z cosh z (cid:21) , (39)would encounter complex pairs of energy eigenvalues since K z = i (cid:0) n z + − A (cid:1) (i.e., E n ρ ,m,n z ∈ C ). Whereas, when the same PDM-particle is moving in V ( ρ, ϕ, z ) = a ρ / − ρ (cid:20) A z + i A sinh z cosh z (cid:21) , (40)it would admit exact and real energy eigenvalues as E = (cid:18) m + 32 (cid:19) − ( ζ − β ) − (cid:18) K z a + 2 n ρ + 1 (cid:19) , (41)with K z defined in (38). Of course, this should never be attributed to PT -symmetricity or non- PT -symmetricity of the original Hamiltonian (1) with theattendant complex non-Hermitian settings. It is very much related to the natureof separability we followed in this methodical proposal.One may wish to consider the PT -symmetric Samsonov [34,28] model˜ V ( z ) = − z + 2 i sin z ; z ∈ [ − π, π ] , (42)in (18). In this case Z ( − π ) = Z ( π ) = 0 and K z = n z / n z = 1 , , , · · · , (43)12ith a missing state n z = 2 (the reader may refer to Samsonov [34] on moredetails on this missing state). Hence, for a PDM quantum particle endowedwith M ( ρ, ϕ, z ) = M ( ρ ) = ρ − and subjected to an interaction potential of theform V ( ρ, ϕ, z ) = − ρ − z + 2 i sin z ; z ∈ [ − π, π ] , (44)the exact energy eigenvalues would read E n ρ ,m,n z = (cid:18) m + 32 (cid:19) − ( ζ − β ) − (cid:18) n z − n ρ − (cid:19) . (45)Whereas, for V ( ρ, ϕ, z ) = a ρ / − z + 2 i sin z ; z ∈ [ − π, π ] , (46)the exact energy eigenvalues would read E n ρ ,m,n z = (cid:18) m + 32 (cid:19) − ( ζ − β ) − (cid:18) n z a + 2 n ρ + 1 (cid:19) . (47)where ˜ n z = 1 , , , · · · . The kinetic energy operator in the PDM Hamiltonian (1) is a problem with manyaspects that are yet to be explored. In the current work, we tried to study thisproblem within the context of cylindrical coordinates ( ρ, ϕ, z ). In due course,the essentials related with the kinetic energy operator in (1) are reported. Theseparability of the Schr¨odinger equation is sought through a radial cylindricalposition dependent mass M ( ρ, ϕ, z ) = M ( ρ ) = 1 /ρ accompanied by an az-imuthally symmetrized interaction potential V ( ρ, ϕ, z ) = ρ h ˜ V ( ρ ) + ˜ V ( z ) i / V ( ϕ ) = 0. Such a combination is not a unique one and some other sep-arability settings could be sought. However, we have chosen to stick with theabove mentioned combination for it leads into a handy though rather construc-tive separable system of the one-dimensional Schr¨odinger equations (15), (18),and (19).Assuming azimuthal symmetrization of the problem at hand and within theradial settings, we consider two examples of fundamental nature. The radialcylindrical Coulombic ˜ V ( ρ ) = − /ρ and the radial cylindrical harmonic oscilla-tor ˜ V ( ρ ) = a ρ /
4. They are indeed exactly solvable within the settings of (21)and admit exact energy eigenvalues documented in (23) and (24), respectively.Nevertheless, the appearance of K z and K z in (23) and (24), respectively, of-fered an opportunity to study their spectral signatures mandated by different˜ V ( z ) interaction models. Namely, the spectral signatures of ˜ V ( z ) for impen-etrable walls at z = 0 and z = L (27), for a Morse (32), for a non-Hermitian PT -symmetrized Scarf II (38), and for a non-Hermitian PT -symmetrized Sam-sonov [28,34] (43) are reported.To summarize, we have assumed azimuthal symmetry and used the radialcylindrical Coulomb and harmonic oscillator to obtain exact eigenvalues for anew set of interaction potentials (represented in their general form in (2) anddetailed in (28), (30), (33), (35), (39), (40), (44), and (46)). In fact, under suchazimuthal symmetrization and ˜ V ( z ) setting, this set of exactly-solvable modelsmay grow up as long as one can find exactly-solvable radially cylindrical models(hereby, exact-solvability may even include numerically exactly-solvable modelsas well). The recipe as how to collect the energy eigenvalues is clear in the abovemethodical proposal. Acknowledgement 1
I would like to thank both referees for their valuablecomments and suggestions. eferences [1] O von Roos, Phys. Rev. B 27 (1983) 7547.[2] O Mustafa, S.Habib Mazharimousavi, Int. J. Theor. Phys (2007) 1786.R. Koc, G. Sahinoglu, M. Koca, Eur. Phys. J. B 48 (2005) 583.[3] O Mustafa, S H Mazharimousavi, Phys. Lett.
A 373 (2009) 325.[4] A de Souza Dutra, C A S Almeida, Phys Lett.
A 275 (2000) 2[5] A Puente, M Casas, Comput. Mater Sci. (1994) 441[6] A R Plastino, M Casas, A Plastino, Phys. Lett. A281 (2001) 297.[7] A Schmidt, Phys. Lett.
A 353 (2006) 459.A Schmidt, J Phys A: Math. Theor.
42 ( ) A 337 (2005) 313.[9] I O Vakarchuk, J. Phys. A ; Math. Gen. (2005) 4727.[10] C Y Cai, Z Z Ren, G X Ju, Commun. Theor. Phys. (2005) 1019.[11] B Roy, P Roy, Phys. Lett. A 340 (2005) 70.[12] B Gonul, M Kocak, Chin. Phys. Lett. (2005) 2742.[13] S. Cruz y Cruz, J Negro, L. M. Nieto, Phys. Lett. A 369 (2007) 400.[14] S. Cruz y Cruz, O Rosas-Ortiz, J Phys A : Math. Theor. (2009) 185205[15] J Lekner, Am. J. Phys. (2007) 1151[16] C Quesne, V M Tkachuk, J. Phys. A : Math. Gen. (2004) 4267.[17] L Jiang, L Z Yi, C S Jia, Phys. Lett. A 345 (2005) 279.1518] O Mustafa, S H Mazharimousavi, Phys. Lett.
A 358 (2006) 259.[19] J I Diaz, J Negro, L M Nieto, O Rosas-Ortiz, J Phys A ; Math. Gen. (1999) 8447[20] A D Alhaidari, Phys. Rev. A 66 (2002) 042116.[21] O Mustafa, S H Mazharimousavi, J. Phys. A : Math. Gen. (2006) 10537.[22] B Bagchi, A Banerjee, C Quesne, V M Tkachuk, J. Phys. A ; Math. Gen. (2005) 2929.[23] J Yu, S H Dong, Phys. Lett. A 325 (2004) 194.[24] C Quesne, Ann. Phys. (2006) 1221.[25] T Tanaka, J. Phys. A ; Math. Gen. (2006) 219.[26] A de Souza Dutra, J. Phys. A ; Math. Gen. (2006) 203.A de Souza Dutra, J A Oliveira, J Phys A: Math. Theor.
42 ( ) (2006) 297.[28] O Mustafa, S H Mazharimousavi, Phys. Lett. A 357 (2006) 295.[29] O Mustafa, S H Mazharimousavi, J Phys A: Math. Theor.
41 ( ) (1993) 1327.[31] G. Chen, Phys. Lett. A 326 (2004) 55.[32] Z. Ahmed, Phys. Lett.
A 282 (2001) 343.[33] A. Khare, Phys. Lett.
A 288 (2001) 69.[34] B. F. Samsonov, P. Roy, J. Phys. A ; Math. Gen.38