Quantum influence of topological defects on a relativistic scalar particle with Cornell-type potential in cosmic string space-time with a spacelike dislocation
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Quantum influence of topological defects on arelativistic scalar particle with Cornell-type potentialin cosmic string space-time with a spacelike dislocation
Faizuddin Ahmed Ajmal College of Arts and Science, Dhubri-783324, Assam, India
Abstract
We study the relativistic quantum of scalar particles in the cos-mic string space-time with a screw dislocation (torsion) subject to auniform magnetic field including the magnetic quantum flux in thepresence of potential. We solve the Klein-Gordon equation with aCornell-type scalar potential in the considered framework and obtainthe energy eigenvalues and eigenfunctions and analyze a relativisticanalogue of the Aharonov-Bohm effect for bound states. keywords:
Relativistic wave equations, electromagnetic field, Aharonov-Bohm effect, topological defects, scalar potential.
PACS Number:
In relativistic quantum mechanics, study of spin-0 scalar particles via theKlein-Gordon equation on curved the background with the cosmic stringhas been of current research interest. Several authors have investigated thephysical properties of a series of background with G¨odel-type geometries,such as, the relativistic quantum dynamics of a scalar particle [1, 2], spin-0massive charged particles in the presence of a uniform magnetic field withthe cosmic string [3], quantum influence of topological defects [4], linear [email protected] ; faiz4U.enter@rediffmail.com e. g. , [1, 19, 20,21, 22]). The Dirac equation in (1 + 2)-dimensional rotational symmetryspace-time was investigated in [23].The cosmic string space-time in the polar coordinates ( t, r, φ, z ) is de-scribed by the following line element [19, 24, 25, 26, 27, 28, 29] : ds = − dt + dr + α r dφ + dz , (1)where α = 1 − µ is the topological parameter with µ being the linear massdensity of the cosmic string. In cosmic string space-time, the parameter µ assumes values in the interval 0 < µ < < r < ∞ ,0 ≤ φ ≤ π and −∞ < z < ∞ . Cosmic string may have been produced byphase transitions in the early universe [32] as it is predicted in the extensionsof the standard model of particle physics [24, 25]. Several authors havestudied the relativistic quantum mechanics in the cosmic string space-time( e. g. , [19, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]).2arious potentials have been used to investigate the bound state solutionsto the relativistic wave-equations. Among them, much attention has givenon the Cornell potential. The Cornell potential, which consists of a linear po-tential plus a Coulomb potential, is a particular case of the quark-antiquarkinteraction, one more harmonic type term [48]. The Coulomb potential is re-sponsible for the interaction at small distances and linear potential leads tothe confinement. Recently, the Cornell potential has studied in the groundstate of three quarks [49]. However, this type of potential is worked onspherical symmetry; in cylindrical symmetry, which is our case, this type ofpotential is known as Cornell-type potential [3]. Investigation of the relativis-tic wave-equations with this type of potential are the Klein-Gordon scalarfield in spinning cosmic string space-time [50], relativistic quantum dynamicsof scalar particle subject to a uniform magnetic field in cosmic string space-time [19], spin-0 scalar particle in (1 + 2)-dimensional rotational symmetryspace-time [51], Aharonov-Bohm effect for bound states [52], quantum effectsof confining potential on the Klein-Gordon oscillator [53], effects of potentialon a position-dependent mass system [54] etc.. Other investigations with thistype of interaction are in [55, 56, 57, 58]. The Cornell-type potential is givenby S = η c r + η L r, (2)where η c , η L are the potential parameters.In [19], authors studied the relativistic quantum dynamics of bosoniccharged particles in the presence of an external fields in a cosmic stringspace-time. They solved the Klein-Gordon equation and obtained the rela-tivistic energy eigenvalues and wave-function. In addition, they introduceda Cornell-type scalar potential by modifyong the mass term in the Klein-Gordon equation and obtained the bound states solution of the relativisticquantum system. In [59], authors studied a spin-0 scalar massive chargedparticle in the presence of an external fields including a magnetic quantumflux in the space-time with a spacelike dislocation under the influence of3inear potential. They solved the Klein-Gordon equation and evaluated theenergy eigenvalues and analyze a relativistic analogue of the Aharonov-Bohmeffect for bound states. In addition, they introduced a linear scalar potentialby modifying the mass term in the Klein-Gordon equation and obtained thebound states solution of the relativistic quantum system. In [60], authorsinvestigated a spin-0 scalar charged particles in the presence of an externalfields including a magnetic quantum flux in the space-time with a spacelikedislocation subject to a Coulomb-type potential. They solved the Klein-Gordon equation and evaluated the bound states solution of the relativisticquantum system and analyze a relativistic analogue of the Aharonov-Bohmeffect for bound states.Our main motivation in this work is to investigate a relativistic ana-logue of the Aharonov-Bohm effect [61, 62] for bound states of a relativisticscalar charged particle subject to a homogeneous magnetic field includinga magnetic quantum flux in the presence of a Cornell-type potential in thecosmic string space-time with a spacelike dislocation. We solve the Klein-Gordon equation in the considered framework and obtain the relativisticenergy eigenvalues and eigenfunctions and analyze the effects on the eigen-values. In addition, we check the role play by the torsion parameter in thisrelativistic system and see that the presence of torsion parameter modify theenergy levels and break their degeneracy in comparison to the result obtainedin the cosmic string space-time case. In [63, 64], examples of topological defects in the space-time associated withtorsion are given. We start this section by considering the cosmic stringspace-time with a spacelike dislocation, whose line element is given by ( x = t , x = r , x = φ , x = z ) ds = − dt + dr + α r dφ + ( dz + χ dφ ) , (3)4here α > χ is the dislocation (torsion) parameter.For zero torsion parameter, χ →
0, the metric (3) reduces to the cosmic stringspace-time. Furthermore, for χ → α →
1, the study space-time reducesto Minkowski flat space metric in cylindrical coordinates. Topological defectsassociated with torsion have investigated in solid state [30, 65, 66, 67, 68],quantum scattering [69], bound states solutions [52, 59, 70], and in relativisticquantum mechanics [71, 72].The metric tensor for the space-time (3) to be g µν ( x ) = − α r + χ χ χ (4)with its inverse g µν ( x ) = − α r − χα r − χα r χ α r (5)The metric has signature ( − , + , + , +) and the determinant of the correspond-ing metric tensor g µν is det g = − α r . (6)The relativistic quantum dynamics of spin-0 charged scalar particles ofmass m is described by the Klein-Gordon (KG) equation [19, 73] (cid:20) √− g D µ ( √− g g µν D ν ) − m (cid:21) Ψ = 0 , (7)where the minimal substitution is defined by D µ ≡ ∂ µ − i e A µ , (8)where e is the electric charge, and A µ is the electromagnetic four-vectorpotential by A µ = (0 , ~A ) , ~A = (0 , A φ , . (9)5e choose the angular component of electromagnetic four-vector potential[52, 55, 57, 58, 59, 60, 71, 74] A φ = − α B r + Φ B π , ~B = ~ ∇ × ~A = − B ˆ k. (10)Here Φ B = const. is the internal quantum magnetic flux [75, 76] throughthe core of the topological defects [77]. It is noteworthy that the Aharonov-Bohm effect [61, 62] has been investigated in several branches of physics,such as in, graphene [78], Newtonian theory [79], bound states of massivefermions [80], scattering of dislocated wave fronts [81], torsion effects ona relativistic position-dependent mass system [52, 59], Kaluza-Klein theory[54, 82, 83, 84, 11, 85, 86], and non-minimal Lorentz-violating coupling [87].If one introduces a scalar potential by modifying the mass term in theform m → m + S ( r ) [88] into the above equation, then we have (cid:20) √− g D µ ( √− g g µν D ν ) − ( m + S ) (cid:21) Ψ = 0 , (11)Several authors have studied the relativistic wave-equations with various kindof potentials such as linear, Coulomb-type, Cornell-type etc. ( e. g. ,[3, 5, 7,19, 51, 52, 53, 56, 57, 59, 60]).Using the equation (3), Eq. (11) becomes[ − ∂ ∂t + 1 r ∂∂r ( r ∂∂r ) + 1 α r ( ∂∂φ − i e A φ − χ ∂∂z ) + ∂ ∂z − ( m + S ) ] Ψ = 0 . (12)Since the line-element (3) is independent of t, φ, z , it is appropriate tochoose the following ansatz for the function ΨΨ( t, r, φ, z ) = e i ( − E t + l φ + k z ) ψ ( r ) , (13)where E is the energy of charged particle, l = 0 , ± , ± .... ∈ Z is theeigenvalues of z -component of the angular momentum operator, and k is aconstant. 6ubstituting Eq. (13) into the Eq. (12), we obtain the following radialwave-equation for ψ ( r ): (cid:20) d dr + 1 r ddr + E − α r ( l − e A φ − k χ ) − k − ( m + S ) (cid:21) ψ ( r ) = 0 . (14)Substituting the Eq. (10) and scalar potential (2) into the Eq. (14), weobtain (cid:20) d dr + 1 r ddr + λ − ω r − j r − ar − b r (cid:21) ψ ( r ) = 0 , (15)where λ = E − m − k − η c η L − m ω c α ( l − k χ − Φ) ,ω = q m ω c + η L ,j = r ( l − k χ − Φ) α + η c ,a = 2 m η c ,b = 2 m η L ,ω c = e B m (16)is called the cyclotron frequency of the particle moving in the magnetic field.Transforming x = √ ω r into the Eq. (15), we obtain the following equa-tion: ψ ′′ ( x ) + 1 x ψ ′ ( x ) + (cid:20) ζ − x − j x − ηx − θ x (cid:21) ψ ( x ) = 0 , (17)where we have defined ζ = λω , η = a √ ω , θ = bω . (18)We now use appropriate boundary conditions to investigate the boundstates solutions in this problem. It is require that the wave-functions mustbe regular both at x → x → ∞ . Suppose the possible solution to theEq. (17) is ψ ( x ) = x j e − ( θ + x ) x H ( x ) , (19)7here H ( x ) is an unknown function. Substituting the solution (19) into theEq. (17), we obtain H ′′ ( x ) + h γx − θ − x i H ′ ( x ) + (cid:20) − βx + Θ (cid:21) H ( x ) = 0 , (20)where γ = 1 + 2 j, Θ = ζ + θ − j ) ,β = η + θ j ) . (21)Equation (20) is the biconfluent Heun’s differential equation [3, 5, 7, 12, 19,50, 51, 52, 53, 54, 57, 59, 71, 74, 85, 86, 89, 90] with H ( x ) is the Heunpolynomial function.The above equation (20) can be solved by the Frobenius method. Writingthe solution as a power series expansion around the origin [91]: H ( x ) = ∞ X i =0 c i x i . (22)Substituting the power series solution (22) into the Eq. (20), we get 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 ] . (23)And the various co-efficients are c = (cid:18) η j + θ (cid:19) c ,c = 14 (1 + j ) [( β + θ ) c − Θ c ] . (24)As the function H ( x ) has a power series expansion around the origin inEq. (22), then, the relativistic bound states solution can be achieved by8mposing that the power series expansion becomes a polynomial of degree n .Through the recurrence relation Eq. (23), we can see that the power seriesexpansion H ( x ) becomes a polynomial of degree n by imposing the followingtwo conditions [3, 5, 7, 12, 19, 50, 51, 52, 53, 54, 57, 59, 71, 74, 85, 86, 92, 93]Θ = 2 n, ( n = 1 , , .... ) c n +1 = 0 . (25)By analyzing the first condition Θ = 2 n , we get second degree equationof the energy eigenvalues E n,l : E n,l = 2 m ω c α ( l − k χ − Φ) + 2 ω ( n + 1 + r ( l − k χ − Φ) α + η c ) − m η L ω + m + k + 2 η c η L . (26)The wave-functions is given by ψ n,l ( x ) = x r ( l − Φ − k χ )2 α + η c e − (cid:18) m ηLω + x (cid:19) x H ( x ) . (27)Note that the Eq. (26) does not represent the general expression foreigenvalues problem. One can obtain the individual energy eigenvalues oneby one, that is, E , E , E by imposing the additional recurrence condition c n +1 = 0 on the eigenvalue. The solution with Heun’s Equation makes itpossible to obtain the eigenvalues one by one as done in [3, 5, 7, 12, 19, 50,51, 52, 53, 54, 57, 59, 71, 74, 85, 86, 92, 93] but not explicitly in the generalform by all eigenvalues n . With the aim of obtaining the energy levels of thestationary states, let us discuss a particular case of n = 1. This means thatwe want to construct a polynomial of first degree to H ( x ). With n = 1, wehave Θ = 2 and c = 0 which implies from Eq. (24) c = 2( β + θ ) c ⇒ η j + θ β + θ ) ⇒ ω ,l − a j ) ω ,l − a b ( 1 + j j ) ω ,l − b j ) = 0 . (28)9 constraint on the physical parameter ω ,l . The relation given in Eq. (28)gives the possible values of the parameter ω ,l that permit us to constructfirst degree polynomial to H(x) for n = 1. Note that its values changesfor each quantum number n and l , so we have labeled ω → ω n,l . Besides,since this parameter is determined by the frequency or the magnetic field B , hence, the magnetic field B ,l is so adjusted that the Eq. (28) can besatisfied and the first degree polynomial to H ( x ) can be achieved, where wehave simplified our notation by labeling: ω c ,l = 1 m q ω ,l − η L ↔ B ,l = 2 e q ω ,l − η L . (29)It is noteworthy that a third-degree algebraic equation (28) has at least onereal solution and it is exactly this solution that gives us the allowed values ofthe magnetic field for the lowest state of the system, which we do not writebecause its expression is very long. We can note, from Eq. (29) that thepossible values of the magnetic field depend on the quantum numbers andthe potential parameter. In addition, for each relativistic energy levels, wehave different relation of the magnetic field associated to the Cornell-typepotential and quantum numbers of the system { l, n } . For this reason, wehave labeled the parameters ω , ω c and B in Eqs. (28) and (29).Therefore, the ground state energy levels for n = 1 is given by E ,l = ±{ m ω c ,l α ( l − k χ − Φ) + 2 m ω ,l (2 + r ( l − k χ − Φ) α + η c ) − m η L ω ,l + m + k + 2 η c η L } , (30)Then, by substituting the real solution ω ,l from Eq. (28) into the Eq. (30)it is possible to obtain the allowed values of the relativistic energy levels forthe radial mode n = 1 of a position dependent mass system. We can seethat the lowest energy state is defined by the real solution of the algebraicequation Eq. (28) plus the expression given in Eq. (30) for the radial mode n = 1, instead of n = 0. This effect arises due to the presence of Cornell-type10otential in the system. Note that, it is necessary physically that the lowestenergy state is n = 1 and not n = 0, otherwise the opposite would implythat c = 0, which requires that the rest mass of the scalar particle be zerothat is contrary to the proposal of this investigation.For Φ B = 0 and χ = 0, we can observe in Eq. (30) there exists an effectiveangular momentum l → l ′ = α ( l − Φ − k χ ). Thus the relativistic energylevels depend on the geometric phase [75, 76] as well as torsion parameter.This dependence of the energy levels on the geometric quantum phase givesrise to the well-known effect called as the Aharonov-Bohm effect for boundstates [52, 54, 59, 61, 62, 82, 83, 84, 11, 85, 86]. Besides, we have that E n,l (Φ B + Φ ) = E n,l ∓ τ (Φ B ) where, Φ = ± πe τ with τ = 0 , , ... , whichmeans the relativistic energy eigenvalues (30) is a periodic function of theAharonov-Bohm geometric quantum phase.The ground state wave function for n = 1 is given by ψ ,l = x r ( l − Φ − k χ )2 α + η c e − m ηLω ,l + x x × ( c + c x ) , (31)where c = 1 √ ω ,l (cid:20) m η c j + m η L ω ,l (cid:21) c . (32) We discuss a case corresponds to η c →
0, that is, only a linear scalar potentialin the considered relativistic quantum systems. Therefore, the radial wave-equation (15) becomes we obtain ψ ′′ ( r ) + 1 r ψ ′ ( r ) + (cid:20) ˜ λ − ω r − j r − b r (cid:21) ψ ( r ) = 0 , (33)where ˜ λ = E − m − k − m ω c α ( l − k χ − Φ).11y changing the variable x = √ ω r , Eq. (33) becomes ψ ′′ ( x ) + 1 x ψ ′ ( x ) + " ˜ λω − x − j x − θ r ψ ( x ) = 0 , (34)Substituting the solution Eq. (19) into the Eq. (34), we obtain H ′′ ( x ) + h γx − θ − x i H ′ ( x ) + " − ˜ βx + ˜Θ H ( x ) = 0 , (35)where γ = 1 + 2 j, ˜Θ = ˜ λω + θ − j ) , ˜ β = θ j ) . (36)Equation (35) is the biconfluent Heun’s differential equation [3, 5, 7, 12, 19,50, 51, 52, 53, 54, 57, 59, 71, 74, 85, 86, 89, 90] with H ( x ) is the Heunpolynomials function.Substituting the power series solution (22) into the Eq. (35), we get thefollowing recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 j ) h { ˜ β + θ ( n + 1) } c n +1 − ( ˜Θ − n ) c n i . (37)And the various co-efficients are c = θ c ,c = 14 (1 + j ) h ( ˜ β + θ ) c − ˜Θ c i . (38)The power series expansion H ( x ) becomes a polynomial of degree n byimposing the following two conditions [3, 5, 7, 12, 19, 50, 51, 52, 53, 54, 57,59, 71, 74, 85, 86, 92, 93] ˜Θ = 2 n, ( n = 1 , , .... ) c n +1 = 0 . (39)12y analyzing the first condition, we obtain the following energy eigenval-ues expression E n,l : E n,l = m + k + 2 m ω c α ( l − k χ − Φ) + 2 ω ( n + 1 + | l − k χ − Φ | α ) − m η L ω , (40)where n = 1 , , ... .The ground state energy levels associated with the radial mode n = 1 isgiven by E ,l = m + k + 2 m ω c ,l α ( l − k χ − Φ) + 2 ω ,l ( n + 1 + | l − k χ − Φ | α ) − m η L ω ,l , (41)where by using Eq. (39) for n = 1, we obtain the following constraint ω ,l = (cid:2) m η L (3 + 2 j ) (cid:3) ,ω c ,l = 1 m q [ m η L (3 + 2 j )] − η L ,j = r ( l − k χ − Φ) α + η c . (42)Equation (41) is the ground state energy levels associated with the radialmode n = 1 of a relativistic scalar charged particle in the presence of anexternal uniform magnetic field including a magnetic quantum flux in thecosmic string space-time with a spacelike dislocation. For α →
1, the energyeigenvalues Eq. (40) reduce to the result obtained in [59]. We can see that thepresence of cosmic string parameter ( α ) shifts the energy levels in comparisonto those in [59]. Thus, by comparing the energy eigenvalues Eq. (30) withEq. (41), we have the presence of an extra linear potential modifies therelativistic spectrum of energy. 13 .2 Interactions with Coulomb-type potential We discuss another case corresponds to η L →
0, that is, only Coulomb-typescalar potential in the considered relativistic quantum systems. Therefore,the radial wave-equation (15) becomes we obtain ψ ′′ ( r ) + 1 r ψ ′ ( r ) + (cid:20) ˜ λ − m ω c r − j r − ar (cid:21) ψ ( r ) = 0 , (43)where ˜ λ is given earlier.Let us define x = √ m ω c r , then Eq. (43) becomes ψ ′′ ( x ) + 1 x ψ ′ ( x ) + (cid:20) ˜ λ − x − j x − δr (cid:21) ψ ( x ) = 0 , (44)where δ = a √ m ω c . By imposing that ψ ( x ) → x → x → ∞ , wehave ψ ( x ) = x j e − x H ( x ) . (45)By substituting Eq. (45) into the Eq. (44), we obtain the following equationfor H(x): H ′′ ( x ) + (cid:20) jx − x (cid:21) H ′ ( x ) + " ˜ λm ω c − − j − δx H ( x ) = 0 . (46)Equation (46) is the biconfluent Heun’s differential equation [3, 5, 7, 12, 19,50, 51, 52, 53, 54, 57, 59, 71, 74, 85, 86, 89, 90] with H ( x ) is the Heunpolynomials function.Substituting the power series solution (22) into the Eq. (46), we get thefollowing recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 j ) " δ c n +1 − ( ˜ λm ω c − − j − n ) c n . (47)And the various co-efficients are c = δ j c ,c = 14 (1 + j ) " δ c − ( ˜ λm ω c − − j ) c . (48)14he power series expansion H ( x ) becomes a polynomial of degree n byimposing the following two conditions [3, 5, 7, 12, 19, 50, 51, 52, 53, 54, 57,59, 71, 74, 85, 86, 92, 93]˜ λm ω c − − j = 2 n, ( n = 1 , , .... ) c n +1 = 0 . (49)By analyzing the first condition, we obtain the following equation of eigen-values E n,l : E n,l = ± s m + k + 2 m ω c (cid:20) n + 1 + j + 1 α ( l − k χ − Φ) (cid:21) ( n = 1 , , .... ) . (50)For n = 1, the ground state energy levels is given by E ,l = ± s m + k + 2 m ω c ,l (cid:20) j + 1 α ( l − k χ − Φ) (cid:21) , (51)where by using Eq. (49) for n = 1, we obtain the following constraint ω c ,l = 2 m η c j ↔ B ,l = 4 m η c e (1 + 2 j ) . (52)Here the magnetic field B ,l is so adjusted that the Eq. (52) can be satisfiedand a polynomial of first degree to H ( x ) can be achieved.Equation (51) is the energy levels associated with the radial mode n = 1of a relativistic scalar charged particle in the presence of an external uniformmagnetic field including a magnetic quantum flux in the cosmic string space-time with a spacelike dislocation. For α →
1, the energy levels Eq. (51)reduces to the result obtained in [60]. Thus, we can see that the presenceof the cosmic string parameter ( α ) shifts the energy levels in comparisonto those in [60]. Thus, by comparing the energy eigenvalue expression Eq.(30) with Eq. (51), we can see that the presence of an extra Coulomb-typepotential modifies the relativistic energy spectrum of the system.15 .3 Without torsion parameter We discuss here zero torsion parameter, χ →
0, in the considered relativisticquantum systems.Therefore, the radial wave-equation (15) becomes (cid:20) d dr + 1 r ddr + λ − ω r − j r − ar − b r (cid:21) ψ ( r ) = 0 , (53)where λ = E − k − m − m ω c α ( l − Φ) − η c η L ,j = r ( l − Φ) α + η c . (54)Transforming a new variable x = √ ω r into the Eq. (53), we obtain (cid:20) d dx + 1 x ddx + λ ω − x − j x − ηx − θ x (cid:21) ψ ( r ) = 0 , (55)Suppose the possible solution to the Eq. (55) is ψ ( x ) = x j e − ( θ + x ) x H ( x ) . (56)Substituting the solution (19) into the Eq. (17), we obtain H ′′ ( x ) + (cid:20) j x − θ − x (cid:21) H ′ ( x )+ (cid:20) − β x + λ ω + θ − − j (cid:21) H ( x ) = 0 , (57)where β = η + θ (1 + 2 j ).Equation (57) is the biconfluent Heun’s differential equation [3, 5, 7, 12,19, 50, 51, 52, 53, 54, 57, 59, 71, 74, 85, 86, 89, 90] with H ( x ) is the Heunpolynomial function. 16ubstituting the power series solution (22) into the Eq. (57), we get thefollowing recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 j ) [ { η + θ ( n + j + 32 ) } c n +1 − ( λ ω + θ − − j − n ) c n ] . (58)And the various co-efficients are c = (cid:18) η j + θ (cid:19) c ,c = 14 (1 + j ) (cid:20) { η + θ ( j + 32 ) } c − ( λ ω + θ − − j ) c (cid:21) . (59)The power series expansion H ( x ) becomes a polynomial of degree n byimposing the following two conditions [3, 5, 7, 12, 19, 50, 51, 52, 53, 54, 57,59, 71, 74, 85, 86, 92, 93] λ ω + θ − − j = 2 n ( n = 1 , , ... ) c n +1 = 0 . (60)By analyzing the first condition. we obtain the following second degreeenergy eigenvalues expression E n,l : E n,l = m + k + 2 m ω c ( l − Φ) α + 2 m ω n + 1 + r ( l − Φ) α + η c ! +2 η c η L − m η L ω . (61)Equation (61) is the energy eigenvalues of a relativistic scalar particle withan external uniform magnetic field including a magnetic quantum flux in thecosmic string space-time subject to Cornell-type scalar potential. For zeromagnetic quantum flux, Φ B →
0, the energy eigenvalues (61) is consistentwith the result in [19]. Thus, we can see that the energy eigenvalue expressionEq. (26) get modify in comparison to those in [19] due to the presence of the17agnetic quantum flux Φ B as well the torsion parameter χ which break thedegeneracy of the energy spectrum.To obtain the individual energy levels, we impose the additional recur-rence condition c n +1 = 0. For example, n = 1, we have from (59) ω ,l − a j ) ω ,l − a b ( 1 + j j ) ω ,l − b j ) = 0 , (62)a constraint on the physical parameter ω ,l . The relation given in Eq. (62)gives the possible values of the parameter ω ,l that permit us to constructfirst degree polynomial to H ( x ) for n = 1. Note that its values changesfor each quantum number n and l , so we have labeled ω → ω n,l . Besides,since this parameter is determined by the frequency or the magnetic field B , hence, the magnetic field B ,l is so adjusted that the Eq. (62) can besatisfied and the first degree polynomial to H ( x ) can be achieved, where wehave simplified our notation by labeling: ω c ,l = 1 m q ω ,l − η L ↔ B ,l = 2 e q ω ,l − η L . (63)Note that the equation (62) has at least one real solution and it is exactly thissolution that gives us the allowed values of the magnetic field for the loweststate of the system, which we do not write because its expression is verylong. We can note, from Eq. (63) that the possible values of the magneticfield depend on the quantum numbers and the potential parameter.The ground state energy levels for n = 1 is E ,l = ±{ m + k + 2 η c η L + 2 m ω c ,l α ( l − Φ)+2 m ω ,l r ( l − Φ) α + η c ! − m η L ω ,l } , (64)Then, by substituting the real solution ω ,l from Eq. (62) into the Eq. (64)it is possible to obtain the allowed values of the relativistic energy levels forthe radial mode n = 1 of a position dependent mass system. We can see18hat the lowest energy state is defined by the real solution of the algebraicequation Eq. (62) plus the expression given in Eq. (64) for the radial mode n = 1, instead of n = 0. This effect arises due to the presence of Cornell-typepotential in the system.The ground state wave function is ψ ,l = x r ( l − Φ)2 α + η c e − m ηLω ,l + x x ( c + c x ) , (65)where c = 1 ω ,l (cid:20) m η c j + m η L ω ,l (cid:21) c . (66)In sub-section 2.1–2.3 , we can see that the relativistic energy eigenvaluesdepend on the geometric quantum phase [75, 76]. This dependence of theenergy eigenvalues on the geometric quantum phase gives rise to a relativisticanalogue of the Aharonov-Bohm effect for bound states [52, 54, 59, 61, 62,82, 83, 84, 11, 85, 86]. Besides, we have that E n,l (Φ B + Φ ) = E n,l ∓ τ (Φ B )where, Φ = ± πe τ with τ = 0 , , ... . In [19], authors studied the relativistic quantum dynamics of bosonic chargedparticle in the presence of an external fields in the cosmic string space-timesubject to a Cornell-type potential. In [59], authors studied a scalar field inthe presence of an external fields including a magnetic quantum flux in thespace-time with a spacelike dislocation subject to a linear potential. In [60],authors investigated a spin-0 massive charged particle in the presence of anexternal fields including a magnetic quantum flux in the space-time with aspacelike dislocation subject to a Coulomb-type potential.In this paper, we have investigated quantum effects of torsion and topo-logical defects that stems from a space-time with a spacelike dislocation underthe influence of a Cornell-type potential in the relativistic quantum system.19y solving the Klein-Gordon equation subject to a uniform magnetic fieldincluding a magnetic flux in the presence of a Cornell-type potential, we haveobtained the energy eigenvalues Eq. (26) and corresponding eigenfunctionsEq. (27). By imposing the additional recurrence condition c n +1 = 0, we haveobtained the individual energy levels and corresponding wave-function, as forexample, n = 1 and others are in the same way. The presence of the torsionparameter modify the energy levels and break their degeneracy. We havediscussed three cases ( sub-section 2.1 ) for zero Coulomb potential, η c → sub-section 2.2 ) zero linear potential, η L →
0, and ( sub-section 2.3 ) zerotorsion parameter χ → sub-section 2.1–2.2 , we have seen that for α →
1, the energy eigenvalues Eq.(40) and Eq. (51) are consistent with those results obtained in [59] and [60],respectively. Thus, the presence of the cosmic string parameter shifts theenergy levels in comparison to those obtained in [59, 60]. Furthermore, bycomparing the energy eigenvalues Eq. (26) with those results obtained in [59]and [60], we have seen that the presence of an extra potential term as well thecosmic string parameter modifies the energy eigenvalues. In sub-section 2.3 ,for zero zero magnetic quantum flux, Φ B →
0, we have seen that the energyeigenvalues Eq. (61) is consistent with those result obtained in [19]. Hence,the relativistic energy eigenvalues Eq. (61) is the extended result in com-parison to those in [19] due to the presence of a magnetic quantum flux Φ B .In addition, by comparing the energy eigenvalues Eq. (26) with the resultobtained in [19], we have seen that the presence of the torsion parameter χ aswell as the magnetic quantum flux Φ B modify the energy eigenvalues where,the degeneracy of the energy levels is broken by the torsion parameter.We have seen in each cases that the angular quantum number l is shifted, l → l eff = α ( l − Φ − k χ ), an effective angular quantum number. Thus,the relativistic energy eigenvalues obtained in sub-section 2.1–2.2 dependson the geometric quantum phase [75, 76] as well as the torsion parameterand only the geometric quantum phase in sub-section 2.3 . This dependence20f the relativistic energy eigenvalues on the geometric quantum phase givesrise to a relativistic analogue of the Aharonov-Bohm effect for bound states[52, 54, 59, 61, 62, 82, 83, 84, 11, 85, 86]. Besides, we have that E n,l (Φ B +Φ ) = E n,l ∓ τ (Φ B ) where, Φ = ± πe τ with τ = 0 , , ... .It is well known in non-relativistic quantum mechanics that the Lan-dau quantization is the simplest system that would work with the studies ofquantum Hall effect. Therefore, the relativistic quantum systems analyzedin this work would used for investigating the influence of torsion, the cosmicstring as well as the Cornell-type potential on the thermodynamic proper-ties of quantum systems [94, 95, 96, 97], searching a relativistic analogue ofthe quantum Hall effect [98, 99], and the displaced Fock states [100] in atopological defects space-time with a spacelike dislocation. Data Availability
No data has been used to prepare this paper.
Conflict of Interest
Author declares that there is no conflict of interest regarding publication thispaper.
Acknowledgment
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