Journal of Physics A: Mathematical and Theoretical PAPER Generalized relativistic harmonic oscillator in minimal length quantum mechanics To cite this article: L B Castro and A E Obispo 2017 J. Phys. A: Math. Theor. 50 285202   View the article online for updates and enhancements. Related content An exact solution of the one-dimensional Dirac oscillator in the presence of minimal lengths Khireddine Nouicer - Effect of minimal lengths on electron magnetism Khireddine Nouicer - Casimir effect in the presence of minimal lengths Kh Nouicer - Recent citations On the Duffin–Kemmer–Petiau equation with linear potential in the presence of a minimal length Yassine Chargui - Quantum gauge freedom in the Lorentz violating background Mushtaq B. Shah and Prince A. Ganai - This content was downloaded from IP address 186.217.236.60 on 05/06/2019 at 20:52 https://doi.org/10.1088/1751-8121/aa70f1 http://iopscience.iop.org/article/10.1088/0305-4470/39/18/025 http://iopscience.iop.org/article/10.1088/0305-4470/39/18/025 http://iopscience.iop.org/article/10.1088/0305-4470/39/18/025 http://iopscience.iop.org/article/10.1088/1751-8113/40/9/017 http://iopscience.iop.org/article/10.1088/1751-8113/40/9/017 http://iopscience.iop.org/article/10.1088/0305-4470/38/46/009 http://iopscience.iop.org/article/10.1088/0305-4470/38/46/009 http://dx.doi.org/10.1016/j.physleta.2018.02.008 http://dx.doi.org/10.1016/j.physleta.2018.02.008 http://dx.doi.org/10.1016/j.physleta.2018.02.008 http://dx.doi.org/10.1142/S0219887818500093 http://dx.doi.org/10.1142/S0219887818500093 https://oasc-eu1.247realmedia.com/5c/iopscience.iop.org/590002079/Middle/IOPP/IOPs-Mid-JPA-pdf/IOPs-Mid-JPA-pdf.jpg/1? 1 Journal of Physics A: Mathematical and Theoretical Generalized relativistic harmonic oscillator in minimal length quantum mechanics L B Castro1 and A E Obispo1,2 1 Departamento de Física, Universidade Federal do Maranhão (UFMA), Campus Universitário do Bacanga, 65080-805, São Luís, MA, Brazil 2 Departamento de Física—IGCE, Universidade Estadual Paulista (UNESP), Campus de Rio Claro, 13506-900, Rio Claro, SP, Brazil E-mail: luis.castro@pq.cnpq.br, lrb.castro@ufma.br and signaux.fonce@gmail.com Received 26 December 2016, revised 11 April 2017 Accepted for publication 4 May 2017 Published 19 June 2017 Abstract We solve the generalized relativistic harmonic oscillator in 1 + 1 dimensions in the presence of a minimal length. Using the momentum space representation, we explore all the possible signs of the potentials and discuss their bound-state solutions for fermions and antifermions. Furthermore, we also find an isolated solution from the Sturm–Liouville scheme. All cases already analyzed in the literature are obtained as particular cases. Keywords: generalized uncertainty principle, dirac equation, relativistic oscillator 1. Introduction The concept of a minimal measurable length scale, which is expected to be of the order of the Planck length, given by lp = 1.62 × 10−33 cm, has emerged as a condition required for a consistent formulation of quantum theory of gravity [1]. The existence of such minimal length, which arise when quantum fluctuations of the gravitational field (at Planck scale) are taken into account, is a common feature among most of the theories of quantum gravity such as string theory [2], loop quantum gravity [3], quantum cosmology [4–6], noncommutative field theories [7–10], and black hole physics [11–16]. One of the interesting implications of introducing this minimal length is the modification of the standard commutation rela- tion between position and momentum, which is transformed into a generalized relation that includes an additional quadratic term in momentum, namely, [X, P] = i� ( 1 + βP2 ) , where β = β0lp/�2, β0is a dimensionless constant. This generalized commutation relation leads to the modification of the Heisenberg uncertainty principle to a generalized uncertainty prin- ciple (GUP). This generalization, on the other hand, would imply in the modification of the properties of the quantum system under consideration, namely the eigenfunctions and the L B Castro and A E Obispo Generalized relativistic harmonic oscillator in minimal length quantum mechanics Printed in the UK 285202 JPHAC5 © 2017 IOP Publishing Ltd 50 J. Phys. A: Math. Theor. JPA 1751-8121 10.1088/1751-8121/aa70f1 Paper 28 1 10 Journal of Physics A: Mathematical and Theoretical IOP 2017 1751-8121/17/285202+10$33.00 © 2017 IOP Publishing Ltd Printed in the UK J. Phys. A: Math. Theor. 50 (2017) 285202 (10pp) https://doi.org/10.1088/1751-8121/aa70f1 https://orcid.org/0000-0002-7017-1791 mailto:luis.castro@pq.cnpq.br mailto:lrb.castro@ufma.br mailto:signaux.fonce@gmail.com http://crossmark.crossref.org/dialog/?doi=10.1088/1751-8121/aa70f1&domain=pdf&date_stamp=2017-06-19 publisher-id doi https://doi.org/10.1088/1751-8121/aa70f1 2 eigenvalues. This is the main reason why recent works on so-called minimal length quantum mechanics (MLQM) have undergone a significant growth [17–47]. The Dirac equation in 3 + 1 dimensions with a mixture of spherically symmetric scalar, vector and anomalous magnetic-like (tensor) interactions can be reduced to the 1 + 1 Dirac equation  with a mixture of scalar (Vs), time-component vector (Vt) and pseudoscalar (Vp) couplings when the fermion is limited to move in just one direction [48]. In this restricted motion the scalar and vector interactions preserve their Lorentz structures while the anoma- lous magnetic-like interaction becomes pseudoscalar. So, in the context of 1 + 1 dimensions the potential composed by Vs, Vt and Vp is the most general combination of Lorentz structures because there are only four linearly independent 2 × 2 matrices. In a recent published work, Hassanabadi et al [33] solved the minimal length Dirac equation with harmonic oscillator potential (scalar and vector interactions) and the energies and solutions are showed in quite a simple and systematic manner. To the best of our knowledge, no one has reported on the solution of the Dirac equation  with a generalized relativistic harmonic oscillator in 1 + 1 dimensions in the presence of a minimal length, and we believe that this problem deserves to be explored. In this context, the purpose of this work is to investigate the effects of GUP when brought into problem of relativistic fermions moving in 1 + 1 dimensions when a vector, scalar and pseudoscalar potentials are applied. We consider the effect of GUP on the definition of momentum operator and then we obtain a generalized Dirac equation and solve exactly the corresponding eigenvalue problem for the case of a generalized relativistic harmonic oscil- lator. This problem is mapped into a Schrödinger-like equation  embedded in a symmetric Pöschl–Teller potential. We explore all the possible signs of the potentials, thus paying atten- tion to bound states of fermions and antifermions. Finally, we show that our results obtained for the energy spectrum and the eigenfunctions are a generalization to those obtained in [19] (Dirac oscillator) and [33] (mixed vector–scalar harmonic oscillator). Also, in the limit of the ordinary quantum mechanics (β → 0) we are able to reproduce the case of generalized rela- tivistic harmonic oscillator [49]. 2. The generalized uncertainty principle We consider the following one-dimensional deformed commutation relation [X, P] = i� ( 1 + βP2) , (1) where 0 � β � 1. The limits β → 0 and β → 1 correspond to the ordinary quantum mechan- ics (OQM) and the extreme quantum gravity (EQG), respectively. This deformed commuta- tion relation leads to the following generalized uncertainty principle (GUP) ∆X∆P � � 2 [ 1 + β(∆P)2] . (2) The peculiarity of (2) is that it implies the existence of a non-zero minimal uncertainty in posi- tion (minimal length). The minimization of (2) with respect to ∆P gives (∆X)min = � √ β. (3) Following [17], we consider the following simple one-dimensional realization of the position and momentum operators obeying the relation (1): X = i� ( 1 + βp2) ∂ ∂p , P = p. (4) L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 3 It is important to note that the scalar product in this case is not the usual one, but is defined as 〈 f | g〉 = ∫ +∞ −∞ dp (1 + βp2) f ∗( p)g( p). (5) 3. The Dirac equation in 1 + 1dimensions The 1 + 1 dimensional time-independent Dirac equation for a fermion of rest mass m under the action of vector (Vt), scalar (Vs) and pseudoscalar (Vp) potentials can be written, in terms of the combinations Σ = Vt + Vs and ∆ = Vt − Vs , as Hψ = Eψ, (6) with H = cσ1P + σ3mc2 + I + σ3 2 Σ+ I − σ3 2 ∆+ σ2Vp, (7) where E is the energy of the fermion and P is the momentum operator. The matrices σ1, σ2 and σ3 denote the Pauli matrices, and I denotes the 2 × 2 unit matrix. The positive definite function |ψ|2 = ψ†ψ, satisfying a continuity equation, is interpreted as a position probability density and its norm is a constant of motion. This interpretation is completely satisfactory for single-particle states [50]. 3.1. Equations of motion and isolated solution If we now write the spinor ψ in terms of its components ψT = ( f , g), the Dirac equation gives rise to two coupled first-order equations for the upper, f, and the lower, g, components of the spinor: [cP − iVp(X)] g = [ E − mc2 − Σ(X) ] f , (8) [cP + iVp(X)] f = [ E + mc2 −∆(X) ] g. (9) In terms of f and g the spinor is normalized as ∫ +∞ −∞ dP 1 + βP2 (| f |2 + |g|2) = 1, (10) so that f and g are square integrable functions. For ∆ = 0 with E �= −mc2, the Dirac equation becomes g = (cP + iVp) f E + mc2 , (11) c2P2f − ic[Vp, P] f + [ (E + mc2)Σ + V2 p ] f = (E2 − m2c4) f , (12) and for Σ = 0 with E �= mc2, the Dirac equation becomes f = (cP − iVp)g E − mc2 , (13) c2P2g + ic[Vp, P]g + [ (E − mc2)∆ + V2 p ] g = (E2 − m2c4)g. (14) L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 4 Either for ∆ = 0 with E �= −mc2 or Σ = 0 with E �= mc2 the solution of the relativistic prob- lem is mapped into a Sturm–Liouville problem in such a way that solution can be found by solving a Schrödinger-like problem. The solutions for ∆ = 0 with E = −mc2 and Σ = 0 with E = mc2 (isolated solution) can be obtained directly from the original first-order equations (8) and (9). They are (cP − iVp(X))g = [ −2mc2 − Σ(X) ] f , (15) (cP + iVp(X)) f = 0, (16) for ∆ = 0 with E = −mc2, and (cP − iVp(X))g = 0, (17) (cP + iVp(X)) f = [ 2mc2 −∆(X) ] g, (18) for Σ = 0 with E = mc2. It is worthwhile to note that this sort of isolated solution cannot describe scattering states and is subject to the normalization condition (10). Because f and g are normalizable functions, the possible isolated solution presupposes Vp(X) �= 0. 3.2. The nonrelativistic limit In the nonrelativistic limit (with small potential energies compared to mc2 and E ∼ mc2), the equation (12) becomes P2 2m f + ( V2 p 2mc2 − i [Vp, P] 2mc +Σ ) f = E f , (19) where E = E − mc2 is the nonrelativistic energy and f obeys a Schrödinger equation with a effective potential expressed in terms of the original potentials Σ and Vp. Note that, in this approximation, Σ preserves its original structures itself. However, Vp provides two terms pro- portional to i [Vp, P] and V2 p , which do not have a nonrelativistic counterpart. Therefore, we can say that Vp is a potential intrinsically relativistic. Furthermore, the term V2 p in (19) allows us to infer that even a potential unbounded from below could be a confining potential. 4. The generalized relativistic harmonic oscillator Let us consider Σ = k1X2, ∆ = 0, Vp = k2X. (20) Note that, by making the changes Σ → ∆, m → −m, Vp → −Vp, f → g and g → f in equa- tion (8) (or equation (12)), we obtain equation (9) (or equation (14)). This symmetry also is present in the isolated solution and can be clearly seen from the two equation pairs (15)–(18). This symmetry provides a simple mechanism by which one can go from the results from the case Σ = k1X2, ∆ = 0, Vp = k2X to the case when ∆ = k1X2, Σ = 0, Vp = k2X by just changing the sign of m and of k2 in the relevant expressions. The configuration for the potentials (20) was chosen conveniently so that one obtains Schrödinger-like equations with the harmonic oscillator potential. We can note that, for the case ∆ = 0 (Vt = Vs) and a confining vector potential (∼ X2), the scalar potential Vs is also a confining potential. In this case, the Σ potential, which appears in the Schrödinger-like L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 5 equation for the component f (equation (12)) becomes proportional to X2 (confining potential). For Σ = 0 (Vt = −Vs) and a confining vector potential (∼ X2), the scalar potential is not a confining potential (∼ −X2). In this case, the ∆ potential, which appears in the Schrödinger- like equation for the component g (equation (14)) also becomes proportional to X2 (confin- ing potential). In both cases for a linear pseudoscalar potential, we always have a confining potential in the Schrödinger-like equation for f and g components. The linear pseudoscalar potential is associated with a well known system, the Dirac oscillator [51]. The Dirac oscil- lator is a natural model for studying properties of physical systems, it is an exactly solvable model; several research have been developed in the context of this theoretical framework in recent years [48]. 4.1. Isolated solution The isolated solution with E = −mc2 is obtained from equations (15) and (16). Substituting (20) in equations (15) and (16) and using the operator relation (4), we obtain dg( p) dp + cp �k2 (1 + βp2) g( p) = Q( p) f ( p), (21) and df ( p) dp − cp �k2 (1 + βp2) f ( p) = 0, (22) respectively. It is important to mention that the expression for Q( p ) is irrelevant to our analy- sis. The general solutions for (21) and (22) are given by f ( p) = N+ ( 1 + βp2) c 2β�k2 , (23) g( p) = ( 1 + βp2)− c 2β�k2 [N+I( p) + N−] , (24) where N+ and N− are normalization constants, and I( p) = ∫ Q( p) ( 1 + βp2) c β�k2 dp. (25) Observing (23) and (24), we can conclude that it is impossible to have both nonzero comp- onents simultaneously as physically acceptable solution. A normalizable solution for k2 > 0 and β �= 0 is possible if N+ = 0. Thus, ψ( p) = N(1 + βp2) − c 2β�k2 ( 0 1 ) , (26) where N = √ 1 δ , (27) with δ = ∫ +∞ −∞ dp (1 + βp2) ( β�k2+c β�k2 ) . (28) L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 6 4.2. Solution for ∆ = 0 and E �= −mc2 For ∆ = 0 and E �= −mc2, the equation (12) takes the form ( c2 + �cβk2 ) P2f (P) + κ2X2f (P) = ε2f (P), (29) where κ2 = ( E + mc2) k1 + k2 2, (30) ε2 = E2 − m2c4 − �ck2, (31) and the equation (11) becomes g(P) = cP + ik2X E + mc2 f (P). (32) Using the operator relation (4), the equation (29) gets d2f ( p) dp2 + 2βp 1 + βp2 df ( p) dp + η2 (1 + βp2)2 [ ε2 − (c2 + �cβk2) p2] f ( p) = 0, (33) where η2 = 1 �2κ2, and (32) becomes g( p) = − �k2 ( 1 + βp2 ) E + mc2 [ d dp − cp �k2 (1 + βp2) ] f ( p). (34) Implementing a change of variable defined by q = 1 � √ βκ arctan (√ βp ) , (35) we can rewrite the equation (33) as d2f (q) dq2 − c2 + �cβk2 β tan2 ( � √ βκq ) f (q) + ε2f (q) = 0. (36) where we recognize the effective potential as the exactly solvable symmetric Pöschl–Teller potential [52] with k2 > − c �β. The corresponding effective eigenenergy is given by ε2 �2βκ2 = n (n + 2λ) + λ, (37) where n is a non-negative integer and λ = 1 2 + 1 2�β √ �2β2(E + mc2)k1 + (�βk2 + 2c)2 (E + mc2)k1 + k2 2 . (38) Now, (31), (37) and (38) tell us that E2 = �2β [ (E + mc2)k1 + k2 2 ]( n2 + n + 1 2 ) + ( n + 1 2 ) � √ �2β2(E + mc2)k1 + (�βk2 + 2c)2 [ (E + mc2)k1 + k2 2 ]−1/2 + m2c4 + �ck2. (39) L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 7 The solutions of (39) determine the eigenvalues of our problem. This equation can be solved easily with a symbolic algebra program. The solution for the differential equation (36) becomes f (q) = A ( cos ( � √ βκq ))λ Cλ n ( sin ( � √ βκq )) , (40) where A is a normalization constant and Cλ n (z) are the Gegenbauer polynomials [53]. The lower component obtained from (34) is given by g(q) = − 2iAc ( cos ( � √ βκq ))λ+1 √ β (E + mc2) Cλ+1 n−1 ( sin ( � √ βκq )) . (41) Using the relation (35) the solutions (40) and (41) can be rewritten in function of the original variable p as f ( p) = A ( 1 + βp2)−λ/2 Cλ n ( p √ β√ 1 + βp2 ) , (42) and g( p) = − 2iAc ( 1 + βp2 )−λ−1 √ β (E + mc2) Cλ+1 n−1 ( p √ β√ 1 + βp2 ) , (43) respectively. The normalization condition (10) dictates that the normalization constant can be written as A = 2λβ1/4 √ 2π [ Γ(2λ+ n) n!(n + λ) (Γ(λ)) 2 + c2 β (E + mc2) 2 Γ(2λ+ n + 1) (n − 1)!(n + λ) (Γ(λ+ 1))2 ]−1/2 . (44) 5. Particular cases on minimal length quantum mechanics (MLQM) 5.1. Pure pseudoscalar linear potential (one-dimensional Dirac oscillator) For β �= 0 and k1 = 0 the expression (39) reduces to E2 − m2c4 = �2βk2 2 ( n2 + n + 1 2 ) + � ( n + 1 2 ) |�βk2 + 2c‖k2|+ �ck2. (45) One can readily envisage that two different classes of solutions can be distinguished depend- ing on the sign of k2. For k2 > 0, we obtain E = ± √ m2c4 + 2 (n + 1) �ck2 + β�2k2 2 (n + 1)2. (46) For − c �β < k2 < 0, we obtain E = ± √ m2c4 + 2n�c|k2|+ β�2k2 2n2. (47) L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 8 This last result is exactly the equation (37) of [19]. Note that n � 0 for k2 > 0 and n � 1 for − c �β < k2 < 0, because for the lower component is proportional to a Gegenbauer polynomial of degree n − 1. Our results (46) and (47), allow us to conclude that E± = ± √ m2c4 + 2 (n + 1) �c|k2|+ β�2k2 2 (n + 1)2, (48) with n = 0, 1, . . ., and it is independent of the sign of k2. Note that both particle (E+ ) and antiparticle (E−) energy levels are members of the spectrum. Let us consider the limit of the ordinary quantum mechanics (β → 0). In this limit (48) becomes E± = ± √ m2c4 + 2 (n + 1) �c|k2|, (49) which is in accordance with [49]. 5.2. Mixed vector–scalar harmonic oscillator potentials For β �= 0 and k2 = 0 the expression (39) reduces to E2 − m2 = ( n + 1 2 ) � √ �2β2(E + mc2)k1 + 4c2 [(E + mc2)k1] −1/2 + �2β ( n2 + n + 1 2 )[ (E + mc2)k1 ] . (50) This result is equivalent to equation (12) of [33]. The quantization condition expressed by (50) can be rewritten as (E − mc2) √ (E + mc2) sgn(k1) = sgn(k1) [ ( n + 1 2 ) � √ |k1| [�2β2(E + mc2)sgn(k1)|k1|+ 4c2] −1/2 + �2β ( n2 + n + 1 2 )√ (E + mc2) sgn(k1)|k1| ] . (51) This result implies that when k1 > 0 there are only bound states for fermions with E > mc2. On the other hand, for k1 < 0 there are only bound states for antifermions with E < −mc2. Therefore, the positive (negative) energies for fermions (antifermions) never sink into the Dirac sea of negative (positive) energies. This fact means that there is no channel for spontane- ous fermion–antifermion creation (Klein’s paradox). Now, let us consider the limit of the ordinary quantum mechanics (β → 0). In this limit (51) becomes (E − mc2) √ (E + mc2) sgn(k1) = sgn(k1) ( n + 1 2 ) �c √ 4|k1|, (52) which is in accordance with [49]. 5.3. Generalized relativistic harmonic oscillator (ordinary quantum mechanics) Now, let us consider the limit of the ordinary quantum mechanics (β → 0). In this limit (39) reduces L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 9 E2 − m2c4 = (2n + 1) �c √ (E + mc2) k1 + k2 2 + �ck2. (53) This result is exactly the equation (44) of [49]. 6. Final remarks In this paper, we have studied the problem of relativistic fermions moving in 1 + 1 dimensions under the influence of mixture of a vector, scalar and pseudoscalar potentials (most general Lorentz structure) in the context of minimal length quantum mechanics. Using the momentum space representation and a convenient representation of the Dirac matrices, we solved the first order Dirac equation and found a isolated solution for ∆ = 0 with E = −mc2 for the case of a generalized relativistic harmonic oscillator. It is important to highlight that those solutions only exist for β �= 0 and are normalizable when k2 > 0. Normalizable solutions for Σ = 0 with E = mc2 can be easily obtained from the symmetries of the system, as mentioned above. The expression for the energy spectrum and the corresponding eigenstates for ∆ = 0 and E �= −mc2 were obtained exactly from the second–order differential equation for the Dirac spinor components after being mapped into a Sturm–Liouville problem (Schrödinger-like) with a symmetric Pöschl–Teller potential. For the general case, the quantization condition can be solved easily with a symbolic algebra program and the Dirac spinor were expressed in terms of the Gegenbauer polynomials. We discussed in detail all the possible signs of the potentials and determined which values of k1 and k2 allow there to be a spectrum of both fer- mion and antifermion bounded solutions simultaneously or just one of these type of solutions. Furthermore, a remarkable feature of this problem is that we were able to reproduce well– known particular cases of relativistic harmonic oscillator in the presence of a minimal length, as for instance: the cases of harmonic oscillator potential (scalar and vector couplings) [33] and the so–called Dirac oscillator [19]. Also, the results obtained in this work are consistent in the limit of the ordinary quantum mechanics (β → 0) with those found in [49] for the general- ized relativistic harmonic oscillator. Acknowledgments This work was supported in part by means of funds provided by CNPq (grants 455719/2014-4 and 304105/2014-7). Angel E Obispo thanks CAPES for support through a scholarship under the CAPES/PNPD program. Angel E Obispo also thanks CNPq (grant 312838/2016-6) and Secti/FAPEMA (grant FAPEMA DCR-02853/16), for financial support. ORCID L B Castro https://orcid.org/0000-0002-7017-1791 References [1] Hossenfelder S 2013 Living Rev. Relativ. 16 (https:\\doi.org\10.12942/lrr-2013-2) [2] Konishi K, Paffuti G and Provero P 1990 Phys. Lett. B 234 276 [3] Garay L J 1995 Int. J. Mod. Phys. A 10 145 [4] Maggiore M 1993 Phys. Lett. B 304 65 [5] Ali A F, Faizal M and Khalil M M 2015 J. Cosmol. Astropart. Phys. JCAP9(2015)025 L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 https://orcid.org/0000-0002-7017-1791 https://orcid.org/0000-0002-7017-1791 https://doi.org/10.12942/lrr-2013-2 https://doi.org/10.1016/0370-2693(90)91927-4 https://doi.org/10.1016/0370-2693(90)91927-4 https://doi.org/10.1142/S0217751X95000085 https://doi.org/10.1142/S0217751X95000085 https://doi.org/10.1016/0370-2693(93)91401-8 https://doi.org/10.1016/0370-2693(93)91401-8 https://doi.org/10.1088/1475-7516/2015/09/025 https://doi.org/10.1088/1475-7516/2015/09/025 10 [6] Garattini R and Faizal M 2016 Nucl. Phys. B 905 313 [7] Bing-Sheng L, Tai-Hua H and Wei C 2014 Commun. Theor. Phys. 61 605 [8] Faizal M 2014 Int. J. Mod. Phys. A 29 1450106 [9] Faizal M and Majumder B 2015 Ann. Phys., NY 357 49 [10] Pramanik S, Moussa M, Faizal M and Ali A F 2015 Ann. Phys., NY 362 24 [11] Scardigli F 1999 Phys. Lett. B 452 39 [12] Bina A, Jalalzadeh S and Moslehi A 2010 Phys. Rev. D 81 023528 [13] Faizal M and Khalil M M 2015 Int. J. Mod. Phys. A 30 1550144 [14] Gangopadhyay S, Dutta A and Faizal M 2015 Eur. Phys. Lett. 112 20006 [15] Wang P, Yang H and Ying S 2016 Class. Quantum Grav. 33 025007 [16] Gangopadhyay S 2016 Int. J. Theor. Phys. 55 617 [17] Kempf A, Mangano G and Mann R B 1995 Phys. Rev. D 52 1108 [18] Nozari K and Karami M 2005 Mod. Phys. Lett. A 20 3095 [19] Nouicer K 2006 J. Phys. A: Math. Gen. 39 5125 [20] Das S and Vagenas E C 2008 Phys. Rev. Lett. 101 221301 [21] Merad M and Falek M 2009 Phys. Scr. 79 015010 [22] Jana T and Roy P 2009 Phys. Lett. A 373 1239 [23] Das S and Vagenas E C 2009 Can. J. Phys. 87 233 [24] Chargui Y, Trabelsi A and Chetouani L 2010 Phys. Lett. A 374 531 [25] Das S, Vagenas E C and Ali A F 2010 Phys. Lett. B 690 407 [26] Ali A F, Das S and Vagenas E C 2011 Phys. Rev. D 84 044013 [27] Hassanabadi H, Zarrinkamar S and Maghsoodi E 2012 Phys. Lett. B 718 678 [28] Taşkın F and Yaman Z 2012 Int. J. Theor. Phys. 51 3963 [29] Ghosh S and Roy P 2012 Phys. Lett. B 711 423 [30] Dey S, Fring A and Gouba L 2012 J. Phys. A: Math. Theor. 45 385302 [31] Betrouche M, Maamache M and Choi J R 2013 AdHEP 2013 383957 [32] Menculini L, Panella O and Roy P 2013 Phys. Rev. D 87 065017 [33] Hassanabadi H, Zarrinkamar S and Rajabi A 2013 Phys. Lett. B 718 1111 [34] Hassanabadi H, Zarrinkamar S and Maghsoodi E 2013 Eur. Phys. J. Plus 128 25 [35] Pedram P 2013 AdHEP 2013 853696 [36] Haouat S and Nouicer K 2014 Phys. Rev. D 89 105030 [37] Haouat S 2014 Phys. Lett. B 729 33 [38] Bouaziz D 2015 Ann. Phys., NY 355 269 [39] Hassanabadi H, Hooshmand P and Zarrinkamar S 2015 Few-Body Syst. 56 19 [40] Falek M, Merad M and Moumni M 2015 Found. Phys. 45 507 [41] Faizal M, Ali A F and Nassar A 2015 Int. J. Mod. Phys. A 30 1550183 [42] Dey S and Hussin V 2015 Phys. Rev. D 91 124017 [43] Faizal M and Kruglov S I 2016 Int. J. Mod. Phys. D 25 1650013 [44] Faizal M, Khalil M M and Das S 2016 Eur. Phys. J. C 76 30 [45] Deb S, Das S and Vagenas E C 2016 Phys. Lett. B 755 17 [46] Bernardo R C S and Esguerra J P H 2016 Ann. Phys., NY 373 521 [47] Faizal M 2016 Phys. Lett. B 757 244 [48] Strange  P 1998 Relativistic Quantum Mechanics with Applications in Condensed Matter and Atomic Physics (Cambridge: Cambridge University Press) [49] de Castro A S, Alberto P, Lisboa R and Malheiro M 2006 Phys. Rev. C 73 054309 [50] Thaller B 1992 The Dirac Equation (Berlin: Springer) [51] Moshinsky M and Szczepaniak A 1989 J. Phys. A: Math. Gen. 22 L817 [52] Castro L B and de Castro A S 2007 J. Phys. A: Math. Theor. 40 263 [53] Abramowitz M and Stegun I A 1965 Handbook of Mathematical Functions (Toronto: Dover) L B Castro and A E Obispo J. Phys. A: Math. Theor. 50 (2017) 285202 https://doi.org/10.1016/j.nuclphysb.2016.02.023 https://doi.org/10.1016/j.nuclphysb.2016.02.023 https://doi.org/10.1088/0253-6102/61/5/11 https://doi.org/10.1088/0253-6102/61/5/11 https://doi.org/10.1142/S0217751X14501061 https://doi.org/10.1142/S0217751X14501061 https://doi.org/10.1016/j.aop.2015.03.022 https://doi.org/10.1016/j.aop.2015.03.022 https://doi.org/10.1016/j.aop.2015.07.026 https://doi.org/10.1016/j.aop.2015.07.026 https://doi.org/10.1016/S0370-2693(99)00167-7 https://doi.org/10.1016/S0370-2693(99)00167-7 https://doi.org/10.1103/PhysRevD.81.023528 https://doi.org/10.1103/PhysRevD.81.023528 https://doi.org/10.1142/S0217751X15501444 https://doi.org/10.1142/S0217751X15501444 https://doi.org/10.1209/0295-5075/112/20006 https://doi.org/10.1209/0295-5075/112/20006 https://doi.org/10.1088/0264-9381/33/2/025007 https://doi.org/10.1088/0264-9381/33/2/025007 https://doi.org/10.1007/s10773-015-2699-7 https://doi.org/10.1007/s10773-015-2699-7 https://doi.org/10.1103/PhysRevD.52.1108 https://doi.org/10.1103/PhysRevD.52.1108 https://doi.org/10.1142/S0217732305018517 https://doi.org/10.1142/S0217732305018517 https://doi.org/10.1088/0305-4470/39/18/025 https://doi.org/10.1088/0305-4470/39/18/025 https://doi.org/10.1103/PhysRevLett.101.221301 https://doi.org/10.1103/PhysRevLett.101.221301 https://doi.org/10.1088/0031-8949/79/01/015010 https://doi.org/10.1088/0031-8949/79/01/015010 https://doi.org/10.1016/j.physleta.2009.02.007 https://doi.org/10.1016/j.physleta.2009.02.007 https://doi.org/10.1139/P08-105 https://doi.org/10.1139/P08-105 https://doi.org/10.1016/j.physleta.2009.11.028 https://doi.org/10.1016/j.physleta.2009.11.028 https://doi.org/10.1016/j.physletb.2010.05.052 https://doi.org/10.1016/j.physletb.2010.05.052 https://doi.org/10.1103/PhysRevD.84.044013 https://doi.org/10.1103/PhysRevD.84.044013 https://doi.org/10.1016/j.physletb.2012.11.005 https://doi.org/10.1016/j.physletb.2012.11.005 https://doi.org/10.1007/s10773-012-1288-2 https://doi.org/10.1007/s10773-012-1288-2 https://doi.org/10.1016/j.physletb.2012.04.033 https://doi.org/10.1016/j.physletb.2012.04.033 https://doi.org/10.1088/1751-8113/45/38/385302 https://doi.org/10.1088/1751-8113/45/38/385302 https://doi.org/10.1155/2013/383957 https://doi.org/10.1155/2013/383957 https://doi.org/10.1103/PhysRevD.87.065017 https://doi.org/10.1103/PhysRevD.87.065017 https://doi.org/10.1016/j.physletb.2012.11.044 https://doi.org/10.1016/j.physletb.2012.11.044 https://doi.org/10.1140/epjp/i2013-13025-1 https://doi.org/10.1140/epjp/i2013-13025-1 https://doi.org/10.1155/2013/853696 https://doi.org/10.1155/2013/853696 https://doi.org/10.1103/PhysRevD.89.105030 https://doi.org/10.1103/PhysRevD.89.105030 https://doi.org/10.1016/j.physletb.2013.12.060 https://doi.org/10.1016/j.physletb.2013.12.060 https://doi.org/10.1016/j.aop.2015.01.032 https://doi.org/10.1016/j.aop.2015.01.032 https://doi.org/10.1007/s00601-014-0910-7 https://doi.org/10.1007/s00601-014-0910-7 https://doi.org/10.1007/s10701-015-9880-y https://doi.org/10.1007/s10701-015-9880-y https://doi.org/10.1142/S0217751X15501833 https://doi.org/10.1142/S0217751X15501833 https://doi.org/10.1103/PhysRevD.91.124017 https://doi.org/10.1103/PhysRevD.91.124017 https://doi.org/10.1142/S0218271816500139 https://doi.org/10.1142/S0218271816500139 https://doi.org/10.1140/epjc/s10052-016-3884-4 https://doi.org/10.1140/epjc/s10052-016-3884-4 https://doi.org/10.1016/j.physletb.2016.01.059 https://doi.org/10.1016/j.physletb.2016.01.059 https://doi.org/10.1016/j.aop.2016.07.035 https://doi.org/10.1016/j.aop.2016.07.035 https://doi.org/10.1016/j.physletb.2016.03.074 https://doi.org/10.1016/j.physletb.2016.03.074 https://doi.org/10.1103/PhysRevC.73.054309 https://doi.org/10.1103/PhysRevC.73.054309 https://doi.org/10.1088/0305-4470/22/17/002 https://doi.org/10.1088/0305-4470/22/17/002 https://doi.org/10.1088/1751-8113/40/2/005 https://doi.org/10.1088/1751-8113/40/2/005