Aharonov-Bohm effect on spin- 0 scalar massive charged particle with a uniform magnetic field in Som-Raychaudhuri space-time with a cosmic string
aa r X i v : . [ phy s i c s . g e n - ph ] O c t Aharonov-Bohm effect on spin- scalar massivecharged particle with a uniform magnetic field inSom-Raychaudhuri space-time with a cosmic string Faizuddin Ahmed Maryam Ajmal Women’s College of Science & Technology,Hojai-782435, Assam, India
Abstract
We study the relativistic quantum dynamics of spin-0 massivecharged particle in a G¨odel-type space-time with electromagnetic in-teractions. We solve the Klein-Gordon equation subject to a uniformmagnetic field in the Som-Raychaudhuri space-time with a cosmicstring. In addition, we include a magnetic quantum flux into therelativistic quantum system and obtain the energy eigenvalues andanalyze an analogue of the Aharonov-Bohm (AB) effect.
Keywords : Som-Raychaudhuri metric, relativistic wave-equations, elec-tromagnetic field, energy spectrum, wave-functions, Aharonov-Bohm effect,special function.
PACS Number(s):
The relativistic quantum dynamics of spin-0 and spin- particles have beeninvestigated by several reseracheres. Spin-0 particles such as bosons, mesonsare described by Klein-Gordon equation and spin- particle such as fermionsby Dirac equation. The exact solution of the wave-equations are very impor-tant since they contain all the necessary information regarding the quantum [email protected] ; faiz4U.enter@rediffmail.com p µ → p µ − e A µ , where e is thecharge and A µ is the four-vector potential of the electromagnetic field.The relativistic quantum dynamics of spin-0 massive charged particles ofmass M is described by the KG-equation [8][ 1 √− g D µ ( √− g g µν D ν ) − ξ R − M ] Ψ = 0 , (1)where D µ = ∂ µ − i e A µ is the minimal substitution with e is the electriccharge, A µ is the potential of electromagnetic field, R is the scalar curvatureand ξ is the non-minimal coupling cosntant.In recent years, several researchers have investigated the relativistic quan-tum dynamics of scalar particles in the background of G¨odel-type geometries.For examples, the relativistic quantum dynamics of scalar particles [9], Klein-Gordon oscillator with an external fields [10], scalar particles with a cosmicstring [11], linear confinement of a scalar particle [12] (see also, [13]), Groundstate of a bosonic massive charged particle in the presence of external fieldsin [14] (see also, [15]). Furthermore, the relativistic quantum dynamics of ascalar particle in the Som-Raychaudhuri metric was investigated in [16, 17]and observed the similarity of the energy eigenvalues with the Landau lev-els in flat space [18, 19]. The behavior of scalar particles with Yukawa-like2onfining potential in the Som-Raychaudhuri space-time in the presence oftopological defects was investigated in [20]. Other works are the scalar fieldsubject to a Cornell potential [21], survey on the Klein-Gordon equation[22], bound states solution of spin-0 massive in a G¨odel-type space-time withCoulomb potential [23]. In addition, spin-half particles have been studied inG¨odel-type space-time [9], in the Som-Raychaudhuri space-time with torsionand cosmic string [24], with topological defect [25], Fermi field and Diracoscillator in the Som-Raychaudhuri space-time [26], Dirac Fermi field withscalar and vector potentials in the Som-Raychaudhuri space-time [27].Our main motivation is to study the relativistic quantum dynamics ofspin-0 scalar charged particles in the presence of an external fields includ-ing magnetic quantum flux in the Som-Raychaudhuri space-time with thecosmic string which wasn’t studied in [11, 16]. We solve the Klein-Gordonequation in the considered framework and evaluate the energy eigenvaluesand eigenfunctions and analyze the relativistic analogue of Aharonov-Bohmeffect for bound states. We compare our results with [8, 11, 16] and see thatthe energy eigenvalues obtain here get modify due to the presence of variousphysical parameters. scalar massive charged particles :The KG-equation Consider the following Som-Raychaudhuri (SR) space-time with a cosmic-string given by [11, 23, 20, 26, 27] ds = − ( dt + α Ω r dφ ) + α r dφ + dr + dz , (2)where α and Ω characterize the cosmic string and the vorticity parameterof the space-time, respectively. The scalar curvature R of the space-time isgiven by R = 2 Ω . (3)3e choose the four-vector potential of electromagnetic fields A µ = (0 , ~A )with ~A = (0 , A φ , . (4)For the geometry (2), KG-equation (1) becomes[ − ∂ ∂t + 1 r ∂∂r (cid:18) r ∂∂r (cid:19) + (cid:26) α r (cid:18) ∂∂φ − i e A φ (cid:19) − Ω r ∂∂t (cid:27) + ∂ ∂z − ( M + 2 ξ Ω )] Ψ( t, r, φ, z ) = 0 . (5)Since the line-element is independent of time and symmetrical by translationsalong the z -axis, as well by rotations. It is reasonable to write the solutionto Eq. (5) as Ψ( t, r, φ, z ) = e i ( − E t + l φ + k z ) ψ ( r ) , (6)where E is the energy of charged particle, l = 0 , ± , ± , .... are the eigenval-ues of the z -component of the angular momentum operator, and k are theeigenvalues of z -component of the linear momentum operator.Substituting the solution (6) into the Eq. (5), we obtain the followingequation for the radial wave-function ψ ( r ): (cid:20) d dr + 1 r ddr + E − M − k − ξ Ω − ( l − e A φ ) α r − Ω E r − Eα ( l − e A φ ) (cid:21) ψ ( r ) = 0 . (7) Let us consider the electromagnetic four-vector potential associated with auniform external magnetic field given by [8] A φ = − α B r (8)such that the magnetic field is along the z -axis ~B = ~ ∇ × ~A = − B ˆ k .Substituting the potential (8) into the Eq. (7), we obtain the followingradial wave equation: ψ ′′ ( r ) + 1 r ψ ′ ( r ) + (cid:20) λ − ω r − l α r (cid:21) ψ ( r ) = 0 , (9)4here we define λ = E − M − k − E + M ω c ) lα − ξ Ω ,ω = p Ω E + 2 M ω c Ω E + M ω c = (Ω E + M ω c ) , and ω c = e B M (10)is called the cyclotron frequency of the charged particle moving in the mag-netic field.Transforming x = ω r into the above Eq. (9), we obtain the followingdifferential equation ψ ′′ ( x ) + 1 x ψ ′ ( x ) + 1 x ( − ξ x + ξ x − ξ ) ψ ( x ) = 0 , (11)where ξ = 14 , ξ = λ ω , ξ = l α . (12)Compairing the equation (11) with (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 . (13)Therefore, the energy eigenvalues expression using Eqs. (12)-(13) intothe Eq. (A.8) in appendix A is E n,l − (cid:18) n + 1 + | l | α + lα (cid:19) E n,l − M − k − ξ Ω − M ω c (cid:18) n + 1 + | l | α + lα (cid:19) = 0 (14)with the energy eigenvalues associated with n th radial modes is E n,l = Ω (cid:18) n + 1 + lα + | l | α (cid:19) ± { Ω (cid:18) n + 1 + lα + | l | α (cid:19) + M + k +2 M ω c (cid:18) n + 1 + | l | α + lα (cid:19) + 2 ξ Ω } . (15)5here n = 0 , , , .... and k is a constant.The corresponding eigenfunctions is ψ n,l ( x ) = | N | n,l x | l | α e − x L ( | l | α ) n ( x ) , (16)where | N | n,l = (cid:18) n ! ( n + | l | α ) ! (cid:19) is the normalization constant and L ( | l | α ) n ( x ) is thegeneralized Laguerre polynomials and are orthogonal over [0 , ∞ ) with respectto the measure with weighting function x | l | α e − x as Z ∞ x | l | α e − x L ( | l | α ) n L ( | l | α ) m dx = (cid:16) n + | l | α (cid:17) ! n ! δ nm . (17)In [16], Klein-Gordon equation in the Som-Raychaudhuri space-time with-out topological defects was studied. The energy eigenvalues is given by E n,l = Ω (2 n + 1 + l + | l | ) ± p Ω (2 n + 1 + l + | l | ) + M + k . (18)Thus by comparing the result obtained in [16], we can see that the energyeigenvalues Eq. (15) get modify (increases) due to the presence of a uniformmagnetic field B , the topological defect parameter α , and the non-minimalcoupling constant ξ with the background curvature in the relativistic system.In [11], Klein-Gordon equation in the Som-Raychaudhuri space-time witha cosmic string was studied. The energy eigenvalues is given by E n,l = Ω (cid:18) n + 1 + lα + | l | α (cid:19) ± s Ω (cid:18) n + 1 + lα + | l | α (cid:19) + M + k . (19)By comparing the result without external field as obtained in [11], we can seethat the energy eigenvalues Eq. (15) get modify (increases) due to the pres-ence of a uniform magnetic field B and the non-minimal coupling constant ξ in the relativistic system.In [8], the relativistic quantum dynamics of a charged scalar particles inthe presence of an external fields in the cosmic string space-time was studied.6he energy eigenvalues is given by E n,l = ± s M + k + 2 M ω c (cid:18) n + 12 + | l | α + l α (cid:19) . (20)Again by comparing the energy eigenvalues Eq. (15) with those in [8] orEq. (20) here, we can see that the present energy eigenvalues get modifydue to the presence of the vorticity parameter Ω of the space-time and thenon-minimal coupling constant ξ with the background curvature. Let us consider the system described in Eq. (7) in the presence of an externalfields in the z -direction. We have assumed that the topological defects ( e.g. , cosmic string) has an internal magnetic flux field (with magnetic fluxΦ B ) [28, 29, 30]. The electromagnetic four-vector potential is given by thefollowing angular component [10, 38]: A φ = − α B r + Φ B π . (21)Here Φ B = const. is the internal quantum magnetic flux [28, 29, 30] throughthe core of the topological defects [29]. Three-vector potential in symmetricgauge is defined by ~A = ~A + ~A such that ~ ∇ × ~A = ~ ∇ × ~A + ~ ∇ × ~A = ~B = − B ˆ k . It is worth mentioning that this Aharonov-Bohm effect [31, 32] hasinvestigated in graphene [33], in Newtonian theory [34], in bound states ofmassive fermions [35], in scattering of dislocated wave fronts [36], with torsioneffects on a relativistic position-dependent mass system [37, 38, 39], boundstates of spin-0 massive charged particles [23, 40]. In addition, this effect hasinvestigated in the context of the Kaluza-Klein theory [41, 42, 43, 44, 45, 46],and with a non-minimal Lorentz-violating coupling [47].Substituting the potential (21) into the Eq. (7), we obtain the followingequation ψ ′′ ( r ) + 1 r ψ ′ ( r ) + (cid:20) λ − ω r − j r (cid:21) ψ ( r ) = 0 , (22)7here λ = E − M − k − E + M ω c ) j − ξ Ω ,j = ( l − Φ) α . (23)Following the similar technique as done earlier, we obtain the relativisticeigenvalues associated with n th radial modes E n,l = Ω (cid:18) n + 1 + l − Φ + | l − Φ | α (cid:19) ± { Ω (cid:18) n + 1 + l − Φ + | l − Φ | α (cid:19) + k + M + 2 m ω c (cid:18) n + 1 + l − Φ + | l − Φ | α (cid:19) + 2 ξ Ω } . (24)Equation (24) is the energy spectrum of massive charged particles in thepresence of an external uniform magnetic field including a magnetic quan-tum flux in the Som-Raychaudhuri space-time with a cosmic string. Theenergy eigenvalues depend on the cosmic string parameter α , the externalmagnetic field B including the magnetic quantum flux Φ B , and the non-minimal coupling constant ξ . We can see that the energy eigenvalues Eq.(24) get modify in comparison to the result Eq. (15) due to the presenceof the magnetic quantum flux Φ B which causes shifts the energy levels andgives rise to a relativistic analogue of the Aharonov-Bohm effect.The wave-functions are given by ψ n,l ( x ) = | N | n,l x | l − Φ | α e − x L ( | l − Φ | α ) n ( x ) , (25) | N | n,l = (cid:18) n !( n + | l − Φ | α )! (cid:19) is the normalization constant and L ( | l − Φ | α ) n ( x ) is thegeneralized Laguerre polynomial. Special Case
We discuss a special case corresponds to zero vorticity parameter, Ω → ψ ′′ ( r ) + 1 r ψ ′ ( r ) + (cid:20) E − M − k − M ω c j − M ω c r − j r (cid:21) ψ ( r ) = 0 . (26)Transforming x = M ω c r into the above equation (26), we obtain thefollowing equation ψ ′′ ( x ) + 1 x ψ ′ ( x ) + 1 x ( − ξ x + ξ x − ξ ) ψ ( x ) = 0 , (27)where ξ = 14 , ξ = E − M − k − M ω c j M ω c , ξ = j . (28)We obtained the following energy eigenvalues expression associated with n th radial modes: E n,l = ± (cid:18) M + k + 4 M ω c (cid:18) n + 12 + | l − Φ | α + ( l − Φ)2 α (cid:19)(cid:19) , (29)where n = 0 , , , .... and the corresponding eigenfunction is given by Eq.(25).Equation (29) is the relativistic energy eigenvalue of a massive chargedparticle in the presence of an external fields including a magnetic quantumflux in static cosmic string space-time. Observe that the energy eigenvalueEq. (29) in comparison to those result [8] get modify due to the presenceof the magnetic quantum flux Φ B which causes shifts the energy levels andgives rise to a relativistic analogue of the Aharonov-Bohm effect.We can see in the above expression of the energy eigenvalues Eqs. (24)and (29) that the z -component of the angular momentum l is shifted, thatis, l eff = 1 α ( l − Φ) , (30)an effective angular momentum due to both the boundary condition, whichstates that the total angle around the string is 2 π α , and the minimal cou-pling with the electromagnetic fields. We can see that the relativistic energy9igenvalues Eqs. (24) and (29) depend on the geometric quantum phase[28, 29]. Thus, we have that E n,l (Φ B + Φ ) = E n,l ± τ (Φ B ), where Φ = ∓ πe τ with τ = 0 , , , .. . This dependence of the relativistic energy eigenvalueson the geometric quantum phase gives rise to a relativistic analogue of theAharonov-Bohm effect.Formula (25) suggests that, when the particle circles the string, the wave-function changes according toΨ → Ψ ′ = e i π l eff Ψ =
Exp { π iα ( l − e Φ B π ) } Ψ . (31)An immediate consequence of Eq. (31) is that the angular momentum oper-ator may be redefined as ˆ l eff = − iα ( ∂ φ − i e Φ B π ) , (32)where the additional term, − e Φ B π α , takes into account the Aharonov-Bohmmagnetic flux Φ B (internal magnetic field). In this paper, we have investigated spin-0 massive charged particles in thepresence of an external fields including a magnetic quantum flux in the Som-Raychaudhuri space-time with a cosmic string. We have introduced the elec-tromagnetic interactions into the Klein-Gordon equation through the mini-mal substitution. In section 2.1 , Klein-Gordon field in the background of theSom-Raychaudhuri space-time with a cosmic string in the presence of exter-nal uniform magnetic field is considered, and derived the final form of theradial wave equation. We then solved it using the Nikiforov-Uvarov methodand obtained the relativistic energy eigenvalues Eq. (15) and correspondingeigenfunctions Eq. (16). We have seen that the relativistic energy eigenvaluesdepend on the cosmic string ( α ), the parameter (Ω) that characterise vortic-ity of the space-time, the external magnetic field ( B ), and the non-minimal10oupling constant ( ξ ). We have seen the energy eigenvalues Eq. (15) getmodify (increases) in comparison to those results obtained in [11, 16] due tothe presence of an external uniform magnetic field as well as the comsic stringwith the non-minimal coupling constant. We have also seen that the energyeigenvalues Eq. (15) in comparison to the result in [8] get modify (increases)due to the presence of vorticity parameter (Ω) of the space-time. In section2.2 , we have considered an external uniform magnetic field including a mag-netic quantum flux and drived the final form of the radial wave-equation. Wehave solved this equation using the same method and obtained the relativis-tic energy eigenvalues Eq. (24) and corresponding eigenfunctions Eq. (25).The expression for the relativistic energy eigenvalues Eqs. (24) reveals thepossibility of establishing a quantum condition between the energy eigenval-ues of a massive charged particle and the parameter that characterize thevorticity of the space-time (Ω). There we have discussed a special case cor-responds to zero vorticity parameter and seen that the energy eigenvaluesEq. (29) get modify (decreases) in comparison to the results in [8] due tothe presence of a magnetic quantum flux. We have seen that the relativisticeigenvalues depend on the geometric quantum phase [28, 29] and we havethat E n,l (Φ B + Φ ) = E n,l ∓ τ (Φ B ), where Φ = ± πe τ with τ = 0 , , , .. .This dependence of the energy eigenvalues on the geometric quantum phasegives rise to an analogue of the Aharonov-Bohm effect.In this paper, we have shown some results which are in addition to theprevious results obtained in [8, 11, 16] present many interesting effects. Thisis the fundamental subject in physics and connection between these theories(gravitation and quantum mechanics) are not well understood.11 ppendix 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 [48] 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)The parameters α , . . . , α are obatined 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)12herefore 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 ( s ) 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 polynomila: L (0) n ( s ) = L n ( s ) . (A.9) References [1] L. D. Landau and E. M. Lifshitz,
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