Effects of Kaluza-Klein theory and potential on a generalized Klein-Gordon oscillator in the cosmic string space-time
aa r X i v : . [ phy s i c s . g e n - ph ] J un Effects of Kaluza-Klein theory and potential on ageneralized Klein-Gordon oscillator in the cosmicstring space-time
Faizuddin Ahmed Ajmal College of Arts and Science, Dhubri-783324, Assam, India
Abstract
In this paper, we solve a generalized Klein-Gordon oscillator in thecosmic string space-time with a scalar potential of Cornell-type withinthe Kaluza-Klein theory and obtain the relativistic energy eigenvaluesand eigenfunctions. We extend this analysis by replacing the Cornell-type with Coulomb-type potential in the magnetic cosmic string space-time and analyze a relativistic analogue of the Aharonov-Bohm effectfor bound states. keywords:
Klein-Gordon oscillator, topological defects, Kaluza-Kleintheory, special functions, Aharonov-Bohm effect.
PACS Number(s):
A unified formulation of Einstein’s theory of gravitation and theory of electro-magnetism in four-dimensional space-time was first proposed by Kaluza [1]by assuming a pure gravitational theory in five-dimensional space-time. Theso called cylinder condition was later explained by Klein when the extra di-mension was compactified on a circle S with a microscopic radius [2], wherethe spatial dimension becomes five-dimensional. The idea behind introducing [email protected] ; faiz4U.enter@rediffmail.com − d version of Minkowski space-times [55]. Klein-Gordon oscillator was studied by several authors, such as,in the cosmic string space-time with an external fields [56], with Coulomb-type potential by two ways : (i) modifying the mass term m → m + S ( r )257], and (ii) via the minimal coupling [58] in addition to a linear scalar po-tential, in the background space-time generated by a cosmic string [59], inthe G¨odel-type space-times under the influence of gravitational fields pro-duced by topological defects [60], in the Som-Raychaudhuri space-time witha disclination parameter [61], in non-commutative (NC) phase space [62],in (1 + 2)-dimensional G¨urses space-time background [63], and in (1 + 2)-dimensional G¨urses space-time background subject to a Coulomb-type po-tential [64]. The relativistic quantum effects on oscillator field with a linearconfining potential was investigated in [65].We consider a generalization of the oscillator as described in Refs. [34, 64]for the Klein-Gordon. This generalization is introduced through a generalizedmomentum operator where the radial coordinate r is replaced by a generalfunction f ( r ). To author best knowledge, such a new coupling was firstintroduced by K. Bakke et al. in Ref. [41] and led to a generalization ofthe Tan-Inkson model of a two-dimensional quantum ring for systems whoseenergy levels depend on the coupling’s control parameters. Based on this, ageneralized Dirac oscillator in the cosmic string space-time was studied byF. Deng et al. in Ref. [66] where the four-momentum p µ is replaced withits alternative p µ + m ω β f µ ( x µ ). In the literature, f µ ( x µ ) has chosen similarto potentials encountered in quantum mechanics (Cornell-type, exponential-type, singular, Morse-type, Yukawa-like etc.). A generalized Dirac oscillatorin (2+1)-dimensional world was studied in [67]. Very recently, the generalizedK-G oscillator in the cosmic string space-time in [68], and non-inertial effectson a generalized DKP oscillator in the cosmic string space-time in [69] wasstudied.The relativistic quantum dynamics of a scalar particle of mass m witha scalar potential S ( r ) [70, 71] is described by the following Klein-Gordonequation: (cid:20) √− g ∂ µ ( √− g g µν ∂ ν ) − ( m + S ) (cid:21) Ψ = 0 , (1)with g is the determinant of metric tensor with g µν its inverse. To couple3lein-Gordon field with oscillator [52, 53], following change in the momentumoperator is considered as in [72, 56]: ~p → ~p + i m ω ~r, (2)where ω is the oscillatory frequency of the particle and ~r = r ˆ r where, r beingdistance from the particle to the string. To generalized the Klein-Gordonoscillator, we adopted the idea considered in Refs. [34, 64, 66, 68, 69] byreplacing r → f ( r ) as X µ = (0 , f ( r ) , , , . (3)So we can write ~p → ~p + i m ω f ( r ) ˆ r , and we have, p → ( ~p + i m ω f ( r ) ˆ r )( ~p − i m ω f ( r ) ˆ r ). Therefore, the generalized Klein-Gordon oscillator equation: (cid:20) √− g ( ∂ µ + m ω X µ ) {√− g g µν ( ∂ ν − m ω X ν ) } − ( m + S ) (cid:21) Ψ = 0 , (4)where X µ is given by Eq. (3).Various potentials have been used to investigate the bound state solu-tions to the relativistic wave-equations. Among them, much attention hasgiven on Coulomb-type potential. This kind of potential has widely used tostudy various physical phenomena, such as in, the propagation of gravita-tional waves [73], the confinement of quark models [74], molecular models[75], position-dependent mass systems [76, 77, 78], and relativistic quantummechanics [58, 59, 57]. The Coulomb-type potential is given by S ( r ) = η c r . (5)where η c is Coulombic confining parameter.Another potential that we are interest here is the Cornell-type potential.The Cornell potential, which consists of a linear potential plus a Coulombpotential, is a particular case of the quark-antiquark interaction, one moreharmonic type term [79]. The Coulomb potential is responsible by the inter-action at small distances and the linear potential leads to the confinement.4ecently, the Cornell potential has been studied in the ground state of threequarks [80]. However, this type of potential is worked on spherical symmetry;in cylindrical symmetry, which is our case, this type of potential is knownas Cornell-type potential [60]. This type of interaction has been studied in[60, 65, 70, 81]. Given this, let us consider this type of potential S ( r ) = η L r + η c r , (6)where η L , η c are the confining potential parameters.The aim of the present work is to analyze a relativistic analogue of theAharonov-Bohm effect for bound states [19, 20, 21] for a relativistic scalarparticle with potential in the context of Kaluza-Klein theory. First, we studya relativistic scalar particle by solving the generalized Klein-Gordon oscillatorwith a Cornell-type potential in the five-dimensional cosmic string space-time. Secondly, by using the Kaluza-Klein theory [1, 2, 3] a magnetic fluxthrough the line-element of the cosmic string space-time is introduced, andthus write the generalized Klein-Gordon oscillator in the five-dimensionalspace-time. In the later case, a Coulomb-type potential by modifying themass term m → m + S ( r ) is introduced which was not study earlier. Then,we show that the relativistic bound states solutions can be achieved, wherethe relativistic energy eigenvalues depend on the geometric quantum phase[21]. Due to this dependence of the relativistic energy eigenvalue on thegeometric quantum phase, we calculate the persistent currents [36, 37] thatarise in the relativistic system.This paper comprises as follow : In section 2 , we study a generalizedKlein-Gordon oscillator in the cosmic string background within the Kaluza-Klein theory with a Cornell-type scalar potential; in section 3 , a generalizedKlein-Gordon oscillator in the magnetic cosmic string in the Kaluza-Kleintheory subject to a Coulomb-type scalar potential and obtain the energyeigenvalues and eigenfunctions; and the conclusion one in section 4 .5 Generalized Klein-Gordon oscillator in cos-mic string space-time with a Cornell-typepotential in Kaluza-Klein theory
The purpose of this section is to study the Klein-Gordon equation in cosmicstring space-time with the use of Kaluza-Klein theory with interactions. Thefirst study of the topological defects within the Kaluza-Klein theory wascarried out in [16]. The metric corresponding to this geometry can be writtenas, ds = − dt + dr + α r dφ + dz + dx , (7)where t is the time coordinate, x is the coordinate associated with the fifthadditional dimension and ( r, φ, z ) are cylindrical coordinates. These coor-dinates assume the ranges −∞ < ( t, z ) < ∞ , 0 ≤ r < ∞ , 0 ≤ φ ≤ π ,0 < x < π a , where a is the radius of the compact dimension x . The α parameter characterizing the cosmic string, and in terms of mass density µ given by α = 1 − µ [82]. The cosmology and gravitation imposes limits tothe range of the α parameter which is restricted to α < − ∂ ∂t + 1 r (cid:18) ∂∂r + m ω f ( r ) (cid:19) (cid:18) r ∂∂r − m ω r f ( r ) (cid:19) + 1 α r ∂ ∂φ + ∂ ∂z + ∂∂x − ( m + S ( r )) ] Ψ( t, r, φ, z, x ) = 0 , [ − ∂ ∂t + ∂ ∂r + 1 r ∂∂r − m ω (cid:18) f ′ ( r ) + f ( r ) r (cid:19) − m ω f ( r ) + 1 α r ∂ ∂φ + ∂ ∂z + ∂∂x − ( m + S ( r )) ] Ψ( t, r, φ, z, x ) = 0 . (8)Since the metric is independent of t, φ, z, x . One can choose the followingansatz for the function ΨΨ( t, r, φ, z, x ) = e i ( − E t + l φ + k z + q x ) ψ ( r ) , (9)6here E is the total energy, l = 0 , ± , ± .. , and k, q are constants.Substituting the above ansatz into the Eq. (8), we get the followingequation for ψ ( r ) :[ d dr + 1 r ddr + E − k − q − l α r − m ω (cid:18) f ′ ( r ) + f ( r ) r (cid:19) − m ω f ( r ) − ( m + S ( r )) ] ψ ( r ) = 0 . (10)We choose the function f ( r ) a Cornell-type given by [34, 64, 66, 69] f ( r ) = a r + br , a, b > . (11)Substituting the function (11) and Cornell potential (6) into the Eq. (9),we obtain the following equation: (cid:20) d dr + 1 r ddr + λ − Ω r − j r − m η c r − m η L r (cid:21) ψ ( r ) = 0 , (12)where λ = E − k − q − m − m ω a − m ω a b − η L η c , Ω = q m ω a + η L ,j = r l α + m ω b + η c . (13)Transforming ρ = √ Ω r into the equation (12), we get (cid:20) d dρ + 1 ρ ddρ + ζ − ρ − j ρ − ηρ − θ ρ (cid:21) ψ ( ρ ) = 0 , (14)where ζ = λ Ω , η = 2 m η c √ Ω , θ = 2 m η L Ω . (15)Let us impose that ψ ( ρ ) → ρ → ρ → ∞ . Suppose thepossible solution to the Eq. (14) is ψ ( ρ ) = ρ j e − ( ρ + θ ) ρ H ( ρ ) . (16)7ubstituting the solution Eq. (16) into the Eq. (14), we obtain H ′′ ( ρ ) + (cid:20) γρ − θ − ρ (cid:21) H ′ ( ρ ) + (cid:20) − βρ + Θ (cid:21) H ( ρ ) = 0 , (17)where γ = 1 + 2 j, Θ = ζ + θ − j ) ,β = η + θ j ) . (18)Equation (17) is the biconfluent Heun’s differential equation [26, 27, 29, 30,31, 32, 33, 34, 58, 59, 60, 61, 64, 65, 70, 83, 84, 85, 86, 87] and H ( ρ ) is theHeun polynomials.The above equation (17) can be solved by the Frobenius method. Weconsider the power series solution around the origin [88] H ( ρ ) = ∞ X i =0 c i ρ i (19)Substituting the above power series solution into the Eq. (17), we obtain thefollowing recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 j ) [ { β + θ ( n + 1) } c n +1 − (Θ − n ) c n ] . (20)And the various coefficients are c = (cid:18) ηγ − θ (cid:19) c ,c = 14 (1 + j ) [( β + θ ) c − Θ c ] . (21)The quantum theory requires that the wave function Ψ must be nor-malized. The bound state solutions ψ ( ρ ) can be obtained because there isno divergence of the wave function at ρ → ρ → ∞ . Since we have8ritten the function H ( ρ ) as a power series expansion around the originin Eq. (19). Thereby, bound state solutions can be achieved by impos-ing that the power series expansion (19) becomes a polynomial of degree n .Through the recurrence relation (20), we can see that the power series ex-pansion (19) becomes a polynomial of degree n by imposing two conditions[26, 27, 29, 30, 31, 32, 33, 34, 58, 59, 60, 61, 64, 65, 70, 83, 84, 85]:Θ = 2 n ( n = 1 , , ... ) ,c n +1 = 0 (22)By analyzing the condition Θ = 2 n , we get expression of the energyeigenvalues E n,l : λ Ω + θ − j ) = 2 n ⇒ E n,l = k + q + m + 2 Ω n + 1 + r l α + m ω b + η c ! +2 m ω a b + 2 m ω a + 2 η L η c − m η L Ω . (23)We plot graphs of the above energy eigenvalues w. r. t. different param-eters. In fig. 1, the energy eigenvalues E , against the parameter η c . In fig.2, the energy eigenvalues E , against the parameter η L . In fig. 3, the energyeigenvalues E , against the parameter M . In fig. 4, the energy eigenvalues E , against the parameter ω . In fig. 5, the energy eigenvalues E , againstthe parameter Ω.Now we impose additional condition c n +1 = 0 to find the individual energylevels and corresponding wave functions one by one as done in [89, 90]. Asexample, for n = 1, we have Θ = 2 and c = 0 which implies c = 2 β + θ c ⇒ (cid:18) η j − θ (cid:19) = 2 β + θ Ω ,l − η j ) Ω ,l − η θ ( 1 + j j ) Ω ,l − θ j ) = 0 (24)9 constraint on the parameter Ω ,l . The relation given in Eq. (24) gives thepossible values of the parameter Ω ,l that permit us to construct first degreepolynomial to H(x) for n = 1. Note that its values changes for each quantumnumber n and l , so we have labeled Ω → Ω n,l . Besides, since this parameter isdetermined by the frequency, hence, the frequency ω ,l is so adjusted that theEq. (24) can be satisfied, where we have simplified our notation by labeling: ω ,l = 1 m a q Ω ,l − η L . (25)It is noteworthy that a third-degree algebraic equation (24) has at least onereal solution and it is exactly this solution that gives us the allowed valuesof the frequency for the lowest state of the system, which we do not writebecause its expression is very long. We can note, from Eq. (24) that thepossible values of the frequency depend on the quantum numbers and thepotential parameter. In addition, for each relativistic energy level, we have adifferent relation of the magnetic field associated to the Cornell-type potentialand quantum numbers of the system { l, n } . For this reason, we have labeledthe parameters Ω and ω in Eqs. (24) and (25).Therefore, the ground state energy level and corresponding wave-functionfor n = 1 are given by E ,l = k + q + m + 2 Ω ,l r l α + m ω b + η c ! +2 m ω ,l a b + 2 m ω ,l a + 2 η L η c − m η L Ω ,l ,ψ ,l = ρ q l α + m ω ,l b + η c e − m ηL Ω 321 ,l + ρ ρ ( c + c ρ ) , (26)where c = 1Ω ,l m η c (cid:16) q l α + m ω ,l b + η c (cid:17) − m η L Ω ,l c . (27)10hen, by substituting the real solution of Eq. (25) into the Eqs. (26)-(27)it is possible to obtain the allowed values of the relativistic energy for theradial mode n = 1 of a position dependent mass system. We can see thatthe lowest energy state defined by the real solution of the algebraic equationgiven in Eq. (25) plus the expression given in Eq. (26) is defined by theradial mode n = 1, instead of n = 0. This effect arises due to the presenceof the Cornell-type potential in the system.For α →
1, the relativistic energy eigenvalue (24) becomes E n,l = k + q + m + 2 Ω (cid:16) n + 1 + p l + m ω b + η c (cid:17) +2 m ω a b + 2 m ω a + 2 η L η c − m η L Ω . (28)Equation (28) is the relativistic energy eigenvalue of a scalar particles viathe generalized Klein-Gordon oscillator subject to a Cornell-type potentialin the Minkowski space-time in the Kaluza-Klein theory.We discuss bellow a very special case of the above relativistic system. Case A : Considering η L = 0, that is, only Coulomb-type potential S ( r ) = η c r . We want to investigate the effect of Coulomb-type potential on a scalarparticle in the background of cosmic string space-time in the Kaluza-Kleintheory. In that case, the radial wave-equation Eq. (12) becomes (cid:20) d dr + 1 r ddr + λ − m ω a r − j r − m η c r (cid:21) ψ ( r ) = 0 , (29)where λ = E − k − q − m − m ω a − m ω a b (30)Transforming ρ = √ m ω a r into the Eq. (29), we get (cid:20) d dρ + 1 ρ ddρ + λ m ω a − ρ − j ρ − m η c √ m ω a ρ (cid:21) ψ ( ρ ) = 0 . (31)11uppose the possible solution to Eq. (31) is ψ ( ρ ) = ρ j E − ρ H ( ρ ) . (32)Substituting the solution Eq. (32) into the Eq. (31), we obtain H ′′ ( ρ ) + (cid:20) jρ − ρ (cid:21) H ′ ( ρ ) + (cid:20) − ˜ ηρ + λ m ω a − j ) (cid:21) H ( ρ ) , (33)where ˜ η = m η c √ m ω a . Equation (33) is the Heun’s differential equation [26, 27,29, 30, 31, 32, 33, 34, 58, 59, 60, 61, 64, 65, 70, 83, 84, 85, 86, 87] with H ( ρ )is the Heun polynomial.Substituting the power series solution Eq. (19) into the Eq. (33), weobtain the following recurrence relation for coefficients c n +2 = 1( n + 2)( n + 2 + 2 j ) (cid:20) ˜ η c n +1 − { λ m ω a − j ) − n } c n (cid:21) (34)The power series solution becomes a polynomial of degree n provided [26, 27,29, 30, 31, 32, 33, 34, 58, 59, 60, 61, 64, 65, 70, 83, 84, 85] λ m ω a − j ) = 2 n ( n = 1 , , ... ) c n +1 = 0 . (35)Using the first condition, one will get the following energy eigenvalues ofthe relativistic system : E n,l = ±{ k + q + m + 2 m ω a n + 2 + r l α + m ω b + η c ! +2 m ω a b } . (36)The ground state energy levels and corresponding wave-function for n = 1are given by E ,l = ±{ k + q + m + 2 m ω ,l a r l α + m ω b + η c ! +2 m ω a b } ,ψ ,l ( ρ ) = ρ q l α + m ω ,l b + η c e − ρ ( c + c ρ ) , (37)12here c = 2 m η c √ m ω ,l a (cid:16) q l α + m ω ,l b + η c (cid:17) =
21 + 2 q l α + m ω ,l b + η c c ,ω ,l = 2 m η c a (cid:16) q l α + m ω ,l b + η c (cid:17) . (38)a constraint on the frequency parameter ω ,l . Case B :We consider another case corresponds to a → b → η L = 0, thatis, a scalar quantum particle in the cosmic string background subject to aCoulomb-type scalar potential within the Kaluza-Klein theory. In that case,from Eq. (12) we obtain the following equation: ψ ′′ ( r ) + 1 r ψ ′ ( r ) + [˜ λ − ˜ j r − m η c r ] ψ ( r ) = 0 . (39)Equation (39) can be written as ψ ′′ ( r ) + 1 r ψ ′ ( r ) + 1 r ( − ξ r + ξ r − ξ ) ψ ( r ) = 0 , (40)where ξ = − ˜ λ = − ( E − k − q − m ) , ξ = − m η c , ξ = ˜ j = l α + η c . (41)Compairing the Eq (40) with Equation (A.1) in appendix A, we get α = 1 , α = 0 , α = 0 , α = 0 , α = 0 , α = ξ ,α = − ξ , α = ξ , α = ξ , α = 1 + 2 p ξ ,α = 2 p ξ , α = p ξ , α = − p ξ . (42)13he energy eigenvalues using Eqs. (41)-(42) into the Eq. (A.8) in ap-pendix A is given by E n,l = ± m vuut − η c ( n + q l α + η c + ) + k m + q m , (43)where n = 0 , , , .. is the quantum number associated with the radial modes, l = 0 , ± , ± , . are the quantum number associated with the angular momen-tum operator, k and q are arbitrary constants. Equation (43) correspondsto the relativistic energy eigenvalues of a free-scalar particle subject to aCoulomb-type scalar potential in the background of cosmic string within theKaluza-Klein theory.The corresponding radial wave-function is given by ψ n,l ( r ) = | N | r ˜ j e − r L (˜ j ) n ( r )= | N | r q l α + η c e − r L ( q l α + η c ) n ( r ) . (44)Here | N | is the normalization constant and L ( q l α + η c ) n ( r ) is the generalizedLaguerre polynomial.For α →
1, the relativistic energy eigenvalues Eq. (43) becomes E n,l = ± m s − η c ( n + p l + η c + ) + k m + q m . (45)Equation (45) correspond to the relativistic energy eigenvalue of a scalarparticle subject to a Coulomb-type scalar potential in the Minkowski space-time within the Kaluza-Klein theory. Let us consider the quantum dynamics of a particle moving in the magneticcosmic string background. In the Kaluza-Klein theory [1, 2, 18], the corre-14ponding metrics with Aharonov-Bohm magnetic flux Φ passing along thesymmetry axis of the string assumes the following form ds = − dt + dr + α r dφ + dz + ( dx + Φ2 π dφ ) (46)with cylindrical coordinates are used. The quantum dynamics is describedby the equation (4) with the following change in the inverse matrix tensor g µν , g µν = − α r − Φ2 π α r − Φ2 π α r Φ π α r . (47)By considering the line element (46) into the Eq. (4), we obtain the followingdifferential equation :[ − ∂ t + ∂ r + 1 r ∂ r + 1 α r ( ∂ φ − Φ2 π ∂ x ) + ∂ z + ∂ x − m ω (cid:18) f ′ ( r ) + f ( r ) r (cid:19) − m ω f ( r ) − ( m + S ( r )) ] Ψ = 0 . (48)Since the space-time is independent of t, φ, z, x , substituting the ansatz (9)into the Eq. (48), we get the following equation : ψ ′′ ( r ) + 1 r ψ ′ ( r ) + [ E − k − q − l eff r − m ω (cid:18) f ′ ( r ) + f ( r ) r (cid:19) − m ω f ( r ) − ( m + S ( r )) ] ψ ( r ) = 0 , (49)where the effective angular quantum number l eff = 1 α ( l − q Φ2 π ) . (50)Substituting the function (11) into the Eq. (49) and using Coulomb-typepotential (5), the radial wave-equation becomes (cid:20) d dr + 1 r ddr + λ − m ω a r − χ r − m η c r (cid:21) ψ ( r ) = 0 , (51)15here λ = E − k − q − m − m ω a − m ω a b,χ = s ( l − q Φ2 π ) α + m ω b + η c . (52)Transforming ρ = √ m ω a r into the equation (51), we get (cid:20) d dρ + 1 ρ ddρ + λ m ω a − ρ − χ ρ − ˜ ηρ (cid:21) ψ ( ρ ) = 0 , (53)where ˜ η = m η c √ m ω a .Suppose the possible solution to Eq. (53) is ψ ( ρ ) = ρ χ e − ρ H ( ρ ) (54)Substituting the solution Eq. (54) into the Eq. (53), we obtain H ′′ ( ρ ) + (cid:20) χρ − ρ (cid:21) H ′ ( ρ ) + (cid:20) − ˜ ηρ + λ m ω a − χ ) (cid:21) H ( ρ ) . (55)Equation (55) is the second order Heun’s differential equation [26, 27, 29, 30,31, 32, 33, 34, 58, 59, 60, 61, 64, 65, 70, 83, 84, 85, 86, 87] with H ( ρ ) is theHeun polynomial.Substituting the power series solution Eq. (19) into the Eq. (55), weobtain the following recurrence relation for the coefficients: c n +2 = 1( n + 2) ( n + 2 + 2 χ ) (cid:20) ˜ η c n +1 − (cid:26) λ m ω a − − χ − n (cid:27) c n (cid:21) . (56)The power series becomes a polynomial of degree n by imposing the followingconditions [26, 27, 29, 30, 31, 32, 33, 34, 58, 59, 60, 61, 64, 65, 70, 83, 84, 85] c n +1 = 0 , λ m ω a − − χ = 2 n ( n = 1 , , ... ) (57)By analyzing the second condition, we get the following energy eigenvalues E n,l : E n,l = k + q + m + 2 m ω a n + 2 + s ( l − q Φ2 π ) α + m ω b + η c +2 m ω a b. (58)16quation (58) is the energy eigenvalues of a generalized Klein-Gordon oscilla-tor in the magnetic cosmic string with a Coulomb-type scalar potential in theKaluza-Klein theory. Observed that the relativistic energy eigenvalues Eq.(58) depend on the Aharonov-Bohm geometric quantum phase [21]. Thus,we have that E n,l (Φ + Φ ) = E n,l ∓ τ (Φ) where, Φ = ± πq τ with τ = 0 , , .. .This dependence of the relativistic energy eigenvalue on the geometric quan-tum phase Φ gives rise to a relativistic analogue of the Aharonov-Bohm effectfor bound states [26, 29, 18, 19, 20, 21].We plot graphs of the above energy eigenvalues w. r. t. different param-eters. In fig. 6, the energy eigenvalues E , against the parameter η c . In fig.7, the energy eigenvalues E , against the parameter M . In fig. 8, the energyeigenvalues E , against the parameter ω . In fig. 9, the energy eigenvalues E , against the parameter Φ.The ground state energy levels and corresponding wave-function for n = 1are given by E ,l = k + q + m + 2 m ω ,l a s ( l − q Φ2 π ) α + m ω b + η c +2 m ω ,l a b ,ψ ,l ( ρ ) = ρ r ( l − q Φ2 π )2 α + m ω ,l b + η c e − ρ ( c + c ρ ) , (59)where c = 2 m η c √ m ω ,l a (1 + 2 q ( l − q Φ2 π ) α + m ω ,l b + η c )=
21 + 2 q ( l − q Φ2 π ) α + m ω ,l b + η c c .ω ,l = 2 m η c a (cid:18) q ( l − q Φ2 π ) α + m ω ,l b + η c (cid:19) (60)17 constraint on the physical parameter ω ,l .Equation Eq. (59) is the ground states energy eigenvalues and correspond-ing eigenfunctions of a generalized Klein-Gordon oscillator in the presence ofCoulomb-type scalar potential in a magnetic cosmic string space-time in theKaluza-Klein theory.For α →
1, the energy eigenvalues (58) becomes E n,l = k + m + q + 2 m ω a n + 2 + r ( l − q Φ2 π ) + m ω b + η c ! +2 m ω a b. (61)Equation (61) is the relativistic energy eigenvalue of the generalized Klein-Gordon oscillator field with a Coulomb-type scalar potential with a magneticflux in the Kaluza-Klein theory. Observed that the relativistic energy eigen-value Eq. (61) depend on the geometric quantum phase [21]. Thus, we havethat E n,l (Φ + Φ ) = E n,l ∓ τ (Φ) where, Φ = ± πq τ with τ = 0 , , .. . Thisdependence of the relativistic energy eigenvalue on the geometric quantumphase gives rise to an analogous effect to Aharonov-Bohm effect for boundstates [26, 29, 18, 19, 20, 21]. Case A :We discuss below a special case corresponds to b → a →
0, that is, ascalar quantum particle in a magnetic cosmic string background subject toa Coulomb-type scalar potential in the Kaluza-Klein theory. In that case,from Eq. (51) we obtain the following equation: ψ ′′ ( r ) + 1 r ψ ′ ( r ) + [˜ λ − ˜ χ r − m η c r ] ψ ( r ) = 0 , (62)where ˜ λ = E − k − q − m , ˜ χ = s ( l − q Φ2 π ) α + η c . (63)18he above Eq. (62) can be written as ψ ′′ ( r ) + 1 r ψ ′ ( r ) + 1 r (cid:0) − ξ r + ξ r − ξ (cid:1) ψ ( r ) = 0 , (64)where ξ = − ˜ λ , ξ = − m η c , ξ = ˜ χ . (65)Following the similar technique as done earlier, we get the following energyeigenvalues E n,l : E n,l = ± m vuuut − η c (cid:18) n + q α ( l − q Φ2 π ) + η c + (cid:19) + k m + q m , (66)where n = 0 , , , .. is the quantum number associated with radial modes, l = 0 , ± , ± , .... are the quantum number associated with the angular mo-mentum, k and q are constants. Equation (66) corresponds to the relativisticenergy levels for a free-scalar particle subject to Coulomb-type scalar poten-tial in the background of magnetic cosmic string in a Kaluza-Klein theory.The radial wave-function is given by ψ n,l ( r ) = | N | r ˜ χ e − r L (˜ j ) n ( r )= | N | r , r ( l − q Φ2 π )2 α + η c e − r L ( r ( l − q Φ2 π )2 α + η c ) n ( r ) . (67)Here | N | is the normalization constant and L ( r ( l − q Φ2 π )2 α + η c ) n ( r ) is the general-ized Laguerre polynomial.For α →
1, the energy eigenvalues (66) becomes E n,l = ± m vuuut − η c (cid:18) n + q ( l − q Φ2 π ) + k c + (cid:19) + k m + q m , (68)which is similar to the energy eigenvalue obtained in [30] (see Eq. (12) in[30]). Thus we can see that the cosmic string α modify the relativistic energyeigenvalue (66) in comparison to those results obtained in [30].19bserve that the relativistic energy eigenvalues Eq. (66) depend on thecosmic string parameter α , the magnetic quantum flux Φ, and potential pa-rameter η c . We can see that E n,l (Φ + Φ ) = E n,l ∓ τ (Φ) where, Φ = ± πq τ with τ = 0 , , .. . This dependence of the relativistic energy eigenvalues onthe geometric quantum phase gives rise to a relativistic analogue of theAharonov-Bohm effect for bound states [26, 29, 18, 19, 20, 21]. By following [36, 37, 38], the expression for the total persistent currents isgiven by I = X n,l I n,l , (69)where I n,l = − ∂E n,l ∂ Φ (70)is called the Byers-Yang relation [36].Therefore, the persistent current that arises in this relativistic systemusing Eq. (58) is given by I n,l = − ∂E n,l ∂ Φ= ∓ m ω a ( ∂ χ∂ Φ ) s k + q + m + 2 m ω ab + 2 mωa (cid:18) n + 2 + q ( l − q Φ2 π ) α + m ω b + η c (cid:19) , (71)where ∂ χ∂ Φ = − q ( l − q Φ2 π )2 α π q ( l − q Φ2 π ) α + m ω b + η c . (72)Similarly, for the relativistic system discussed in case A in this section,20his current using Eq. (66) is given by I n,l = ± m q η c ( l − q Φ2 π )2 π α (cid:18) n + + q ( l − q Φ2 π ) α + η c (cid:19) q ( l − q Φ2 π ) α + η c × r − η c (cid:16) n + q α ( l − q Φ2 π ) + η c + (cid:17) + k m + q m . (73)For α →
1, the persistent currents expression given by Eq. (73) reduces to theresult obtained in Ref. [30]. Thus we can see that the presence of the cosmicstring parameter modify the persistent currents Eq. (73) in comparison tothose results in Ref. [30].By introducing a magnetic flux through the line element of the cosmicstring space-time in five dimensions, we see that the relativistic energy eigen-value Eq. (58) depend on the geometric quantum phase [21] which givesrise to a relativistic analogue of the Aharonov-Bohm effect for bound states[26, 29, 18, 19, 20, 21]. Moreover, this dependence of the relativistic energyeigenvalues on the geometric quantum phase has yielded persistent currentsin this relativistic quantum system.
In Ref. [30], Aharonov-Bohm effects for bound states of a relativistic scalarparticle by solving the Klein-Gordon equation subject to a Coulomb-typepotential in the Minkowski space-time within the Kaluza-Klein theory werestudied. They obtained the relativistic bound states solutions and calcu-lated the persistent currents. In Ref. [16], it is shown that the cosmic stringspace-time and the magnetic cosmic string space-time can have analogue infive dimensions. In Ref. [18], quantum mechanics of a scalar particle inthe background of a chiral cosmic string using the Kaluza-Klein theory wasstudied. They shown that the wave functions, the phase shifts, and scatter-ing amplitudes associated with the particle depend on the global features of21hose space-times. These dependence represent the gravitational analoguesof the well-known Aharonov-Bohm effect. In addition, they discussed theLandau levels in the presence of a cosmic string within the framework ofKaluza-Klein theory. In Ref. [31], the Klein-Gordon oscillator on the curvedbackground within the Kaluza-Klein theory were studied. The problem of theinteraction between particles coupled harmonically with topological defectsin the Kaluza-Klein theory were studied. They considered a series of topo-logical defects and then treated the Klein-Gordon oscillator coupled to thisbackground, and obtained the energy eigenvalue and corresponding eigen-functions in this cases. They have shown that the energy eigenvalue dependon the global parameters characterizing these space-times. In Ref. [32], ascalar particle with position-dependent mass subject to a uniform magneticfield and a quantum magnetic flux, both coming from the background whichis governed by the Kaluza-Klein theory were investigated. They inserted aCornell-type scalar potential into this relativistic systems and determinedthe relativistic energy eigenvalue of the system in this background of extradimension. They analyzed particular cases of this system and a quantumeffect were observed: the dependence of the magnetic field on the quantumnumbers of the solutions. In Ref. [34], the relativistic quantum dynam-ics of a scalar particle subject to linear potential on the curved backgroundwithin the Kaluza-Klein theory was studied. We have solved the general-ized Klein-Gordon oscillator in the cosmic string and magnetic cosmic stringspace-time with a linear potential within the Kaluza-Klein theory. We haveshown that the energy eigenvalues obtained there depend on the global pa-rameters characterizing these space-times and the gravitational analogue tothe Aharonov-Bohm effect for bound states [26, 29, 18, 19, 20, 21] of a scalarparticle was analyzed.In this work, we have investigated the relativistic quantum dynamics ofa scalar particle interacting with gravitational fields produced by topologicaldefects via the Klein-Gordon oscillator of the Klein-Gordon equation in the22resence of cosmic string and magnetic cosmic string within the Kaluza-Kleintheory with scalar potential. We have determined the manner in which thenon-trivial topology due to the topological defects and a quantum magneticflux modifies the energy spectrum and wave-functions of a scalar particle. Wethen have studied the quantum dynamics of a scalar particle interacting withfields by introducing a magnetic flux through the line element of a cosmicstring space-time using the five-dimensional version of the General Relativity.The quantum dynamics in the usual as well as magnetic cosmic string casesallow us to obtain the energy eigenvalues and corresponding wave-functionsthat depend on the external parameters characterize the background space-time, a result known by gravitational analogue of the well studied Aharonov-Bohm effect.In section 2 , we have chosen a Cornell-type function f ( r ) = a r + br andCornell-type potential S ( r ) = η L r + η c r into the relativistic systems. Wehave solved the generalized Klein-Gordon oscillator in the cosmic string back-ground within the Kaluza-Klein theory and obtained the energy eigenvaluesEq. (23). We have plotted graphs of the energy eigenvalues Eq. (23) w.r. t. different parameters by figs. 1–5. By imposing the additional recur-rence condition c n +1 = 0 on the relativistic eigenvalue problem, for example n = 1, we have obtained the ground state energy levels and wave-functionsby Eqs. (26)–(27). We have discussed a special case corresponds to η L → section 3 , we have studied the relativistic quantum dynamics of ascalar particle in the background of magnetic cosmic string in the Kaluza-Klein theory with a scalar potential. By choosing the same function f ( r ) =23 r + br and a Coulomb-type scalar potential S ( r ) = η c r , we have solvedthe radial wave-equation in the considered system and obtained the boundstates energy eigenvalues Eq. (58). We have plotted graphs of the energyeigenvalues Eq. (58) w. r. t. different parameters by figs. 6–9. Subsequently,the ground state energy levels Eq. (59) and corresponding wave-functionsEq. (60) for the radial mode n = 1 by imposing the additional condition c n +1 = 0 on the eigenvalue problem is obtained. Furthermore, a specialcase corresponds to a → b → α →
1, we have seen that theenergy eigenvalues Eq. (66) reduces to the result obtained in Ref. [30].As there is an effective angular momentum quantum number, l → l eff = α ( l − q Φ2 π ), thus the relativistic energy eigenvalues Eqs. (58) and (66) dependon the geometric quantum phase [21]. Hence, we have that E n,l (Φ + Φ ) = E n,l ∓ τ (Φ) where, Φ = ± πq τ with τ = 0 , , , . . This dependence of therelativistic energy eigenvalues on the geometric quantum phase gives rise toa relativistic analogue of the Aharonov-Bohm effect for bound states [29,19, 20, 21]. Finally, we have obtained the persistent currents by Eqs. (71)–(73) for this relativistic quantum system because of the dependence of therelativistic energy eigenvalues on the geometric quantum phase.So in this paper, we have shown some results which are in addition tothose results obtained in Refs. [18, 29, 30, 31, 32, 33, 34] presents manyinteresting effects. Data Availability
No data has been used to prepare this paper.24 onflict of Interest
Author declares that there is no conflict of interest regarding publication thispaper.
Acknowledgement
Author sincerely acknowledge the anonymous kind referee(s) for their valu-able comments and suggestions and thanks the editor.
Appendix A : Brief review of the Nikiforov-Uvarov (NU) method
The Nikiforov-Uvarov method is helpful in order to find eigenvalues andeigenfunctions of the Schr¨odinger like equation, as well as other second-orderdifferential equations of physical interest. According to this method, theeigenfunctions of a second-order differential equation [91] d ψ ( s ) ds + ( α − α s ) s (1 − α s ) dψ ( s ) ds + ( − ξ s + ξ s − ξ ) s (1 − α s ) ψ ( s ) = 0 . (A.1)are given by ψ ( s ) = s α (1 − α s ) − α − α α P ( α − , α α − α − n (1 − α s ) . (A.2)And that the energy eigenvalues equation α n − (2 n + 1) α + (2 n + 1) ( √ α + α √ α ) + n ( n − α + α +2 α α + 2 √ α α = 0 . (A.3)25he parameters α , . . . , α are obtained from the six parameters α , . . . , α and ξ , . . . , ξ as follows: α = 12 (1 − α ) , α = 12 ( α − α ) ,α = α + ξ , α = 2 α α − ξ ,α = α + ξ , α = α + α α + α α ,α = α + 2 α + 2 √ α , α = α − α + 2 ( √ α + α √ α ) ,α = α + √ α , α = α − ( √ α + α √ α ) . (A.4)A special case where α = 0, as in our case, we findlim α → P ( α − , α α − α − n (1 − α s ) = L α − n ( α s ) , (A.5)and lim α → (1 − α s ) − α − α α = e α s . (A.6)Therefore the wave-function from (A.2) becomes ψ ( s ) = s α e α s L α − n ( α s ) , (A.7)where L ( α ) n ( x ) denotes the generalized Laguerre polynomial.The energy eigenvalues equation reduces to n α − (2 n + 1) α + (2 n + 1) √ α + α + 2 √ α α = 0 . (A.8)Noted that the simple Laguerre polynomial is the special case α = 0 of thegeneralized Laguerre polynomial: L (0) n ( x ) = L n ( x ) . (A.9) References [1] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. (Math. Phys.) Klasse ,966 (1921). 262] O. Klein, Z. Phys. (12), 895 (1926); Nature , 516 (1926).[3] M. B. Green, J. H. Schwarz and E. Witten, Superstring theory vol. 1-2 ,Cambridge University Press, Cambridge, UK (1987).[4] G. German, Class. Quantum Grav. , 455 (1985).[5] Y.-S. Wu and A. Zee, J. Math. Phys. , 2696 (1984).[6] P. Ellicott and D. J. Toms, Class. Quantum Grav. , 1033 (1989).[7] R. Delbourgo, S. Twisk and R. B. Zhang, Mod. Phys. Lett. A 3 , 1073(1988).[8] R. Delbourgo and R. B. Zhang, Phys. Rev.
D 38 , 2490 (1988).[9] I. M. Benn and R. W. Tucker, J. Phys. A: Math. Gen. , L123 (1983).[10] D. Bailin and A. Love, Rep. Prog. Phys. , 1087 (1987).[11] A. Macias and H. Dehnen, Class. Quantum Grav. , 203 (1991).[12] S. Ichinose, Phys. Rev. D 66 , 104015 (2002).[13] S. M. Carroll and H. Tam, Phys. Rev.
D 78 , 044047 (2008).[14] M. Gomes, J. R. Nascimento, A. Y. Petrov and A. J. da Silva, Phys.Rev.
D 81 , 045018 (2010).[15] A. P. Baeta Scarpelli, T. Mariz, J. R. Nascimento, and A. Y. Petrov,EPJC (2013) : 2526.[16] M. A.-Ainouy and G. Clement, Class. Quantum Grav. , 2635 (1996).[17] M. E. X. Guimaraes, Phys. Lett. B 398 , 281 (1997).[18] C. Furtado, F. Moraes and V. B. Bezerra, Phys. Rev.
D 59 , 107504(1999). 2719] M. Peshkin and A. Tonomura,
Lect. Notes Phys. , Springer-Verlag,Berlin, Germany (1989).[20] V. B. Bezerra, J. Math. Phys. , 2895 (1989).[21] Y. Aharonov and D. Bohm, Phys. Rev. , 485 (1959).[22] R. Jackiw, A. I. Milstein, S.-Y. Pi and I. S. Terekhov, Phys. Rev. B 80 ,033413 (2009).[23] M. A. Anacleto, I. G. Salako, F. A. Brito and E. Passos, Phys. Rev.
D92 , 125010 (2015).[24] V. R. Khalilov, Eur. Phys. J. C (2014) : 2708.[25] C. Coste, F. Lund and M. Umeki, Phys. Rev. E 60 , 4908 (1999).[26] R. L. L. Vit´oria and K. Bakke, Int. J. Mod. Phys. D , 1850005 (2018).[27] F. Ahmed, Adv. High Energy Phys. , 5691025 (2020).[28] H. Belich, E. O. Silva, M. M. Ferreira Jr. and M. T. D. Orlando, Phys.Rev. D 83 , 125025 (2011).[29] C. Furtado, V. B. Bezerra and F. Moraes, Mod. Phys. Lett
A 15 , 253( 2000).[30] E. V. B. Leite, H. Belich and K. Bakke, Adv. HEP , 925846 (2015).[31] J. Carvalho, A. M. de M. Carvalho, E. Cavalcante and C. Furtado, Eur.Phys. J. C (2016) : 365.[32] E. V. B. Leite, H. Belich, and R. L. L. Vit´oria, Adv. HEP , 6740360(2019).[33] E. V. B. Leite, R. L. L. Vit´oria and H. Belich, Mod. Phys. Lett. A 34 ,1950319 (2019). 2834] F. Ahmed, Eur. Phys. J. C (2020) : 211.[35] K. Bakke, A. Y. Petrov and C. Furtado, Ann. Phys. (N. Y.) , 2946(2012).[36] N. Byers and C. N. Yang, Phys. Rev. Lett. , 46 (1961).[37] W. -C. Tan and J. C. Inkson, Phys. Rev. B 60 , 5626 (1999).[38] L. Dantas, C. Furtado and A. L. S. Netto, Phys. Lett.
A 379 , 11 (2015).[39] M. B¨uttiker, Y. Imry and R. Landauer, Phys. Lett.
A 96 , 365 (1983).[40] A. C. B.-Jayich, W. E. Shanks, B. Peaudecerf, E. Ginossar, F. V. Oppen,L. Glazman and J. G. E. Harris, Science (5950), 272 (2009).[41] K. Bakke and C. Furtado, Phys. Lett.
A 376 , 1269 (2012).[42] K. Bakke and C. Furtado, Ann. Phys. (N. Y.) , 489 (2013).[43] D. Loss, P. Goldbart and A. V. Balatsky, Phys. Rev. Lett. , 1655(1990).[44] D. Loss and P. M. Goldbart, Phys. Rev. B 45 , 13544 (1992).[45] X. -C. Gao and T. -Z. Qian, Phys. Rev.
B 47 , 7128 (1993).[46] T. -Z. Qian and Z. -B. Su, Phys. Rev. Lett. , 2311 (1994).[47] A. V. Balatsky and B. L. Altshuler, Phys. Rev. Lett. , 1678 (1993).[48] S. Oh and C.-M. Ryu, Phys. Rev. B 51 , 13441 (1995).[49] H. Mathur and A. D. Stone, Phys. Rev.
B 44 , 10957 (1991).[50] H. Mathur and A. D. Stone, Phys. Rev. Lett. , 2964 (1992).[51] M. J. Bueno, J. L. de Melo, C. Furtado and A. M. de M. Carvalho, EPJPlus (2014) : 201. 2952] S. Bruce and P. Minning, II Nuovo Cimento A 106 , 711 (1993).[53] V. V. Dvoeglazov, II Nuovo Cimento
A 107 , 1413 (1994).[54] M. Moshinsky, J. Phys. A : Math. Theor. , L817 (1989).[55] N. A. Rao and B. A. Kagali, Phys. Scr. , 015003 (2008).[56] A. Boumali and N. Messai, Can. J. Phys. , 1460 (2014).[57] K. Bakke and C. Furtado, Ann. Phys. (N. Y.) , 48 (2015).[58] R. L. L. Vit´oria, C. Furtado and K. Bakke, Ann. Phys. (N. Y.) , 128(2016).[59] L. C. N. Santosa and C. C. Barros Jr., Eur. Phys. J. C (2018) : 13.[60] Z. Wang, Z. Long, C. Long and M. Wu, Euro. Phys. J. Plus (2015) : 36.[61] J. Carvalho, A. M. de M. Carvalho and C. Furtado, Eur Phys. J. C(2014) : 2935.[62] B. Mirza, R. Narimani and S. Zare, Commun. Theor. Phys. , 405(2011).[63] F. Ahmed, Ann. Phys. (N. Y.) , 1 (2019).[64] F. Ahmed, Gen. Relativ. Grav. (2019) : 69.[65] R. L. L. Vit´oria and K. Bakke, Euro. Phys. J. Plus (2016) : 36.[66] L. -F. Deng, C. -Y. Long, Z. -W. Long and T. Xu, Adv. HEP ,2741694 (2018).[67] Z. -L. Zhao, Z. -W. Long and M. -Y. Zhang, Adv. HEP , 3423198(2019). 3068] L. -F. Deng, C. -Y. Long, Z. -W. Long and T. Xu, Eur. J. Phys. Plus(2019) : 355.[69] S. Zare, H. Hassanabadi and Marc de Montigny, Gen. Relativ. Grav.(2020) : 25.[70] E. R. F. Medeiros and E. R. B. de Mello, Eur. Phys. J. C (2012) :2051.[71] W. Greiner, Relativistic quantum mechanics: wave equations , Springer-Verlag, Berlin, Germany (2000).[72] B. Mirza and M. Mohadesi, Commun. Theor. Phys. , 664 (2004).[73] H. Asada and T. Futamase, Phys. Rev. D 56 , R6062 (1997).[74] C. L. Chrichfield, J. Math. Phys. , 261 (1976).[75] S. M. Ikhdair, B.J. Falaye, M. Hamzavi, Ann. Phys. (N. Y.) , 282(2015).[76] A. D. Alhaidari, Phys. Rev. A 66 , 042116 (2002).[77] A. D. Alhaidari, Phys. Lett.
A 322 , 72 (2004).[78] J. Yu, S.-H. Dong, Phys. Lett.
A 325 , 194 (2004).[79] M. K. Bahar and F. Yasuk, Adv. High Energy Phys. , 814985(2013).[80] C. Alexandrou, Ph. de. Forcrand and O Jahn, Nuc. Phys. B (Proc.Supp.) , 667 (2003).[81] R. L. L. Vit´oria and K. Bakke, Euro. Phys. J. Plus (2018) : 490.[82] A. Vilenkin, Phys. Lett.
B 133 , 177 (1983).3183] A. Ronveaux,
Heuns differential equations , Oxford University Press, Ox-ford (1995).[84] K. Bakke and H. Belich, Ann. Phys. (N. Y.) , 596 (2015).[85] K. Bakke and H. Belich, Euro. Phys. J. Plus (2012) : 102.[86] A. B. Oliveira and K. Bakke, Proc. R. Soc.
A 472 , 20150858 (2016).[87] K. Bakke, Ann. Phys. (N. Y.) , 86 (2014).[88] G. B. Arfken and H. J. Weber,
Mathematical Methods For Physicists ,Elsevier Academic Pres, London (2005).[89] A. Vercin, Phys. Lett.
B 260 , 120 (1991).[90] J. Myrhein, E. Halvorsen and A. Vercin, Phys. Lett.
B 278 , 171 (1992).[91] A. F. Nikiforov and V. B. Uvarov,
Special Functions of MathematicalPhysics , Birkh¨auser, Basel (1988).32 Η c E , H Η c L Figure 1: n = l = k = M = q = a = b = η L = 1, α = 0 . ω = 0 . Η L E , H Η L L Figure 2: n = l = k = M = q = a = b = η c = 1, α = 0 . ω = 0 . E , H M L Figure 3: n = l = k = q = a = b = η c = η L = 1, α = 0 . ω = 0 . Ω E , H Ω L Figure 4: n = l = k = q = a = b = η c = η L = M = 1, α = 0 . W E , H W L Figure 5: n = l = k = q = a = b = η c = η L = M = 1, α = 0 . ω = 0 . Η c E , H Η c L Figure 6: n = l = k = q = a = b = M = 1, α = 0 . ω = 0 .
5, Φ = π E , H M L Figure 7: n = l = k = q = a = b = η c = 1, α = 0 . ω = 0 .
5, Φ = π Ω E , H Ω L Figure 8: n = l = k = q = a = b = η c = M = 1, α = 0 .
5, Φ = π F E , H F L Figure 9: n = l = k = q = a = b = η c = M = 1, α = 0 . ω = 0 ..