Aharonov-Bohm Effect for Bound States on the Confinement of a Relativistic Scalar Particle to a Coulomb-Type Potential in Kaluza-Klein Theory

Leite, E. V. B.; Belich, H.; Bakke, K.

doi:https://doi.org/10.1155/2015/925846

Advances in High Energy Physics

On this page

Abstract Introduction Conclusions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 925846 | https://doi.org/10.1155/2015/925846

Aharonov-Bohm Effect for Bound States on the Confinement of a Relativistic Scalar Particle to a Coulomb-Type Potential in Kaluza-Klein Theory

E. V. B. Leite,¹H. Belich,²and K. Bakke¹

Academic Editor: Juan José Sanz-Cillero

Received23 Sept 2015

Accepted08 Dec 2015

Published24 Dec 2015

Abstract

Based on the Kaluza-Klein theory, we study the Aharonov-Bohm effect for bound states for a relativistic scalar particle subject to a Coulomb-type potential. We introduce this scalar potential as a modification of the mass term of the Klein-Gordon equation, and a magnetic flux through the line element of the Minkowski spacetime in five dimensions. Then, we obtain the relativistic bound states solutions and calculate the persistent currents.

1. Introduction

The impact of Einstein’s work on the new geometric vision has changed the way of describing the possible symmetries that should lead us to the physical laws. As a consequence of this new approach, gravity was explained by a geometrical perspective. The success of this description of the gravitational interaction inspired Kaluza [1] in 1921, and thereafter Klein [2] in 1926, to create the first proposal of unifying two well-known interactions: gravitation and electromagnetism. This new proposal establishes that the electromagnetism can be introduced through an extra (compactified) dimension in the spacetime, where the spatial dimension becomes five-dimensional. In the original version of the theory there appeared a certain inconsistency that was later removed by Leibowitz and Rosen [3]. The idea behind introducing additional spacetime dimensions has found wide applications in quantum field theory, for instance, in string theory, which is consistent in a space with extra dimensions [4]. The Kaluza-Klein theory has also been investigated in the presence of torsion [5, 6], with fermions [7–9], and in studies of the violation of the Lorentz symmetry [10–12]. In recent decades, generalizations of topological defect spacetimes have been found in the context of the Kaluza-Klein theory, for example, the magnetic cosmic string [13] and magnetic chiral cosmic string [14], both in five dimensions. Based on these generalizations of topological defect spacetimes in the Kaluza-Klein theory, the Aharonov-Bohm effect for bound states [15] has been investigated in [16] and geometric quantum phases have been discussed in graphene [17].

The aim of this work is to investigate the Aharonov-Bohm effect for bound states [15, 16] for a relativistic scalar particle subject to a Coulomb-type potential in the Kaluza-Klein theory [1, 2, 4, 7]. By using the Kaluza-Klein theory [1, 2, 4, 7], we introduce a magnetic flux through the line element of the Minkowski spacetime and thus write the Klein-Gordon equation in the five-dimensional spacetime. Besides, we introduce a Coulomb-type potential as a modification in the mass term of the Klein-Gordon equation; then, we show that the relativistic bound states solutions can be achieved, where the relativistic energy levels depend on the Aharonov-Bohm geometric quantum phase [18]. Due to this dependence of the relativistic energy levels on the geometric quantum phase, we calculate the persistent currents [19, 20] that arise in the relativistic system.

The structure of this paper is as follows: in Section 2, we introduce the scalar potential by modifying the mass term of the Klein-Gordon equation and a magnetic flux through the Kaluza-Klein theory; thus, we analyse an Aharonov-Bohm effect for bound states and the arising of persistent currents in this relativistic quantum system; in Section 4, we present our conclusions.

2. Aharonov-Bohm Effect for Bound States

In this section, we analyse the Aharonov-Bohm effect for bound states [15, 16] for a relativistic scalar particle subject to a Coulomb-type potential in the Kaluza-Klein theory [1, 2, 4, 7]. It has been reported in the literature that Coulomb-type potentials have interest in several areas of physics [21–23]; for instance, in the context of condensed matter physics, studies have worked with 1-dimensional systems [24–28], molecules [29–31], pseudo-harmonic interactions [23, 32], position-dependent mass systems [33–35], the Kratzer potential [36–38], and topological defects in solids [39–43]. Other studies have dealt with Coulomb-type potential in the propagation of gravitational waves [44], quark models [45], atoms with magnetic quadrupole moment [46], neutral particle with permanent magnetic dipole moment [47], and relativistic quantum mechanics [48–51]. From this perspective, let us write the Coulomb-type potential as where is a constant that characterizes the scalar potential and is the radial coordinate. Here, we follow [52, 53] and introduce a Coulomb-type potential by modifying the mass term in the form, , where is the scalar potential. This procedure in modifying the mass term of the Klein-Gordon equation has been explored in studies of the quark-antiquark interaction [54], in the analysis of the behaviour of a Dirac particle in the presence of a static scalar potential and a Coulomb potential [55], in a relativistic scalar particle in the cosmic string spacetime [48], and in the Klein-Gordon oscillator [51].

On the other hand, the idea behind the Kaluza-Klein theory [1, 2, 4, 7] is that the spacetime is five-dimensional with the purpose of unifying gravitation and electromagnetism. In this way, we can work with general relativity in five dimensions, that is, in -dimensions. The information about the electromagnetism is given by introducing a gauge field in the line element of the spacetime aswhere is the Kaluza constant [14], the Greek indices are the spacetime indices in -dimensions, is the usual spacetime metric tensor in -dimensions, and corresponds to the extra dimension. Here, we wish to investigate the Aharonov-Bohm effect for bound states [15, 16]; then, by following [14, 16], we can introduce a magnetic flux through the line element of the Minkowski spacetime in the form (by working with the units ) where is the magnetic flux and the vector potential is given by the component , which gives rise to a magnetic field [14]. In this five-dimensional spacetime, the Klein-Gordon equation is written in the form where . From (1) and (3), the Klein-Gordon equation (4) becomes

Observe that the quantum operators , , and commute with the Hamiltonian operator given in right-hand side of (5). Thereby, a particular solution to the Klein-Gordon equation (5) can be written in terms of the eigenfunctions of these operators as where and are constants, and is a function of the radial coordinate. By substituting (6) into (5) we have where we have defined the parameters

Let us now discuss the asymptotic behaviour of the possible solutions to (7). For , we can obtain scattering states where the radial wave function behaves as . On the other hand, bound states can be obtained if the radial wave function behaves as [56]. Since our focus is on the bound state solutions, then, we consider in (7). By replacing with in (7), we also perform a change of variables given by ; thus, we have Moreover, a particular solution to (9) which is regular at the origin can be written as , where is an unknown function. Thereby, by substituting this solution into (9), we obtain a second-order differential equation given by which corresponds to the Kummer equation or the confluent hypergeometric equation [57, 58]. Hence, the function is the confluent hypergeometric function:

Besides, by analyzing the asymptotic behavior for , we have that the confluent hypergeometric function diverges for large values of [57, 58]. In order that the wave function can vanish at , we must impose the condition that the confluent hypergeometric series becomes a polynomial of degree (). This occurs when . In this way, by using (8), we have where is the quantum number associated with the radial modes, is the quantum number associated with the angular momentum, and and are constants. Equation (12) corresponds to the relativistic energy levels for a scalar particle subject to a Coulomb-type potential in the Kaluza-Klein theory. Observe that, due to the presence of the parameter in (12), we have that the energy levels depend on the Aharonov-Bohm geometric quantum phase [18], where it has a periodicity ; thus, we have that . For , we have that . This dependence of the relativistic energy levels on the geometric quantum phase corresponds to an Aharonov-Bohm effect for bound states [15, 16].

Furthermore, it is well-known in condensed matter systems [19, 20, 59–61] and in relativistic quantum systems [62, 63] that when there exists dependence of the energy levels on the geometric quantum phase, then, persistent current can arise in the system. Studies of persistent currents have explored systems that deal with the Berry phase [64, 65], the Aharonov-Anandan quantum phase [66, 67], and the Aharonov-Casher geometric quantum phase [68–71]. By following [19, 20, 59], the expression for the total persistent currents is given by , where is called the Byers-Yang relation [19]. Therefore, the persistent current that arises in this relativistic system is given by

Hence, we have that relativistic bound states solutions to the Klein-Gordon equation can be obtained for a scalar particle confined to a Coulomb-type potential in the Kaluza-Klein theory. By introducing a magnetic flux through the line element of the Minkowski spacetime in five dimensions, we have seen that the relativistic energy levels (12) depend on the geometric quantum phase which gives rise to an Aharonov-Bohm effect for bound states in Kaluza-Klein theory [15, 16]. Moreover, this dependence of the relativistic energy levels on the geometric quantum phase has yielded persistent currents in the relativistic quantum system.

3. Nonrelativistic Limit

Let us now discuss the nonrelativistic limit of the Klein-Gordon equation (5). The motivation for discussing this nonrelativistic limit comes from the studies of the violation of the Lorentz symmetry through the Kaluza-Klein theory [10–12]. A great deal of work has investigated and established bounds for the parameters associated with the violation of the Lorentz symmetry at low energy scenarios in recent decades [72]. Interesting examples are the hydrogen atom [73], Weyl semimetals [74] and on the Rashba coupling [75], geometric quantum phases [76], and the quantum Hall effect [77]. Hence, investigating Aharonov-Bohm-type effects [15, 18, 72] through Kaluza-Klein theories at low energy scenarios can be important for studies of the violation of the Lorentz symmetry since it could be a way of establishing bounds for the parameters associated with the Lorentz symmetry breaking effects. Thereby, let us analyse the behaviour of the present system at a low energy regime. By following [14, 52], the wave function can be written in the form ; thus, by assuming that and by substituting this ansatz into (5), we obtain

Performing the steps from (6) to (11), we have

Equation (15) corresponds to the nonrelativistic energy levels for a spinless particle confined to the Coulomb-type potential in Kaluza-Klein theory. Note that the nonrelativistic energy levels (15) depend on the Aharonov-Bohm geometric quantum phase, whose periodicity is ; thus, we have that , and the persistent currents are given by

In view of the introduction of the scalar potential by modifying the mass term of the Klein-Gordon equation, then, the mass term can be considered to be an effective position-dependent mass [52]: . Position-dependent mass systems, Coulomb-type potentials, and the Aharonov-Bohm effect for bound states have a great interest in condensed matter physics [15, 29–31, 78–81]; thereby the analysis of the behaviour of the present system at a low energy regime can open new discussions about quantum effects on nonrelativistic systems by analogy, for instance, with position-dependent mass systems.

4. Conclusions

We have investigated relativistic quantum effects on a scalar particle subject to a Coulomb-type potential due to the presence of the Aharonov-Bohm geometric quantum [18] which is introduced in the system through an extra dimension of the spacetime. Thereby, we have shown that relativistic bound state solutions can be achieved, where the relativistic energy levels depend on the Aharonov-Bohm quantum phase. This dependence of the relativistic energy levels on the geometric quantum phase corresponds to an Aharonov-Bohm effect for bound states [15, 16] and gives rise to the appearance of persistent currents in this relativistic quantum system.

Another interesting context in which the Aharonov-Bohm effect for bound states in Kaluza-Klein theory can be explored is based on topological defect spacetime. In [13] it is shown that the cosmic string spacetime and the magnetic cosmic string spacetime can have analogues in five dimensions. Besides, in [14] a five-dimensional chiral cosmic string is shown. From this perspective, interesting discussions about Aharonov-Bohm effects for bound states and persistent currents can be made from the confinement of a relativistic scalar particle to a Coulomb-type potential with the background of the Kaluza-Klein magnetic cosmic string spacetime [13] and the Kaluza-Klein magnetic chiral cosmic string [14]. We hope to present these discussions in the near future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank C. Furtado for interesting discussions and the Brazilian agencies CNPq and CAPES for financial support.

References

Th. Kaluza, “On the problem of unity in physics,” Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse, vol. 1, pp. 966–972, 1921.
View at: Google Scholar
O. Klein, “Quantentheorie und fünfdimensionale relativitätstheorie,” Zeitschrift für Physik, vol. 37, no. 12, pp. 895–906, 1926.
View at: Publisher Site | Google Scholar
E. Leibowitz and N. Rosen, “Five-dimensional relativity theory,” General Relativity and Gravitation, vol. 4, no. 6, pp. 449–474, 1973.
View at: Publisher Site | Google Scholar
M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, vol. 1-2, Cambridge University Press, Cambridge, UK, 1987.
G. Germán, “On Kaluza-KLEin theory with torsion,” Classical and Quantum Gravity, vol. 2, no. 4, pp. 455–460, 1985.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-S. Wu and A. Zee, “Massless fermions and Kaluza-Klein theory with torsion,” Journal of Mathematical Physics, vol. 25, no. 9, pp. 2696–2703, 1984.
View at: Publisher Site | Google Scholar | MathSciNet
D. Bailin and A. Love, “Kaluza-Klein theories,” Reports on Progress in Physics, vol. 50, no. 9, pp. 1087–1170, 1987.
View at: Google Scholar
A. Macias and H. Dehnen, “Dirac field in the five-dimensional Kaluza-Klein theory,” Classical and Quantum Gravity, vol. 8, no. 1, p. 203, 1991.
View at: Publisher Site | Google Scholar
S. Ichinose, “Fermions in Kaluza-KLEin and Randall-Sundrum theories,” Physical Review D, vol. 66, no. 10, Article ID 104015, 13 pages, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
S. M. Carroll and H. Tam, “Aether compactification,” Physical Review D, vol. 78, no. 4, Article ID 044047, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. Gomes, J. R. Nascimento, A. Y. Petrov, and A. J. da Silva, “Aetherlike Lorentz-breaking actions,” Physical Review D, vol. 81, no. 4, Article ID 045018, 9 pages, 2010.
View at: Publisher Site | Google Scholar
A. P. Baeta Scarpelli, T. Mariz, J. R. Nascimento, and A. Y. Petrov, “Four-dimensional aether-like Lorentz-breaking QED revisited and problem of ambiguities,” The European Physical Journal C, vol. 73, article 2526, 2013.
View at: Publisher Site | Google Scholar
M. Azreg-Ainou and G. Clément, “Kaluza—Klein and Gauss—Bonnet cosmic strings,” Classical and Quantum Gravity, vol. 13, no. 10, article 2635, 1996.
View at: Publisher Site | Google Scholar
C. Furtado, F. Moraes, and V. B. Bezerra, “Global effects due to cosmic defects in Kaluza-Klein theory,” Physical Review D, vol. 59, no. 10, Article ID 107504, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect, vol. 340 of Lecture Notes in Physics, Springer, Berlin, Germany, 1989.
View at: Publisher Site
C. Furtado, V. B. Bezerra, and F. Moraes, “Aharonov-Bohm effect for bound states in Kaluza-Klein theory,” Modern Physics Letters A, vol. 15, no. 4, p. 253, 2000.
View at: Publisher Site | Google Scholar
K. Bakke, A. Y. Petrov, and C. Furtado, “A Kaluza-KLEin description of geometric phases in graphene,” Annals of Physics, vol. 327, no. 12, pp. 2946–2954, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Physical Review, vol. 115, no. 3, article 485, 1959.
View at: Publisher Site | Google Scholar
N. Byers and C. N. Yang, “Theoretical considerations concerning quantized magnetic flux in superconducting cylinders,” Physical Review Letters, vol. 7, no. 2, pp. 46–49, 1961.
View at: Publisher Site | Google Scholar
W.-C. Tan and J. C. Inkson, “Magnetization, persistent currents, and their relation in quantum rings and dots,” Physical Review B—Condensed Matter and Materials Physics, vol. 60, no. 8, pp. 5626–5635, 1999.
View at: Publisher Site | Google Scholar
M. Dineykhan, G. V. Efimov, G. Ganbold, and S. N. Nedelko, Oscillator Representation in Quantum Physics, Springer, Berlin, Germany, 1995.
M. A. Núñez, “Accurate computation of eigenfunctions for Schrödinger operators associated with Coulomb-type potentials,” Physical Review A, vol. 47, no. 5, pp. 3620–3631, 1993.
View at: Publisher Site | Google Scholar
A. De Souza Dutra, “Conditionally exactly soluble class of quantum potentials,” Physical Review A, vol. 47, no. 4, pp. R2435–R2437, 1993.
View at: Publisher Site | Google Scholar
P. Gribi and E. Sigmund, “Exact solutions for a quasi-one-dimensional Coulomb-type potential,” Physical Review B, vol. 44, no. 8, pp. 3537–3549, 1991.
View at: Publisher Site | Google Scholar
F. Gesztesy and B. Thaller, “Born expansions for Coulomb-type interactions,” Journal of Physics A: Mathematical and General, vol. 14, no. 3, pp. 639–657, 1981.
View at: Publisher Site | Google Scholar
J. A. Reyes and M. del Castillo-Mussot, “1D Schrödinger equations with Coulomb-type potentials,” Journal of Physics A: Mathematical and General, vol. 32, no. 10, article 2017, 1999.
View at: Google Scholar
Y. Ran, L. Xue, S. Hu, and R.-K. Su, “On the Coulomb-type potential of the one-dimensional Schrödinger equation,” Journal of Physics A: Mathematical and General, vol. 33, no. 50, pp. 9265–9272, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Chargui, A. Dhahbi, and A. Trabelsi, “Exact analytical treatment of the asymmetrical spinless Salpeter equation with a Coulomb-type potential,” Physica Scripta, vol. 90, no. 1, Article ID 015201, 2015.
View at: Publisher Site | Google Scholar
S. M. Ikhdair, B. J. Falaye, and M. Hamzavi, “Nonrelativistic molecular models under external magnetic and AB flux fields,” Annals of Physics, vol. 353, pp. 282–298, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
I. I. Guseinov and B. A. Mamedov, “Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals,” The Journal of Chemical Physics, vol. 121, no. 4, pp. 1649–1654, 2004.
View at: Publisher Site | Google Scholar
I. I. Guseinov, “Unified treatment of multicenter integrals of integer and noninteger u Yukawa-type screened Coulomb type potentials and their derivatives over Slater orbitals,” The Journal of Chemical Physics, vol. 120, no. 20, pp. 9454–9457, 2004.
View at: Publisher Site | Google Scholar
S. M. Ikhdair and M. Hamzavi, “Spectral properties of quantum dots influenced by a confining potential model,” Physica B: Condensed Matter, vol. 407, no. 24, pp. 4797–4803, 2012.
View at: Publisher Site | Google Scholar
A. D. Alhaidari, “Solutions of the nonrelativistic wave equation with position-dependent effective mass,” Physical Review A, vol. 66, no. 4, Article ID 042116, 2002.
View at: Publisher Site | Google Scholar
A. D. Alhaidari, “Solution of the Dirac equation with position-dependent mass in the Coulomb field,” Physics Letters A, vol. 322, no. 1-2, pp. 72–77, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
J. Yu and S.-H. Dong, “Exactly solvable potentials for the Schrödinger equation with spatially dependent mass,” Physics Letters A, vol. 325, no. 3-4, pp. 194–198, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
A. Kratzer, “Die ultraroten Rotationsspektren der Halogenwasserstoffe,” Zeitschrift für Physik, vol. 3, no. 5, pp. 289–307, 1920.
View at: Publisher Site | Google Scholar
M. R. Setare and E. Karimi, “Algebraic approach to the Kratzer potential,” Physica Scripta, vol. 75, no. 1, pp. 90–93, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
G. D. A. Marques and V. B. Bezerra, “Non-relativistic quantum systems on topological defects spacetimes,” Classical and Quantum Gravity, vol. 19, no. 5, p. 985, 2002.
View at: Publisher Site | Google Scholar
C. Furtado, B. G. C. da Cunha, F. Moraes, E. R. Bezerra de Mello, and V. B. Bezzerra, “Landau levels in the presence of disclinations,” Physics Letters A, vol. 195, no. 1, pp. 90–94, 1994.
View at: Publisher Site | Google Scholar
S. Mil'shtein, “Application of dislocation-induced electric potentials in Si and Ge,” Le Journal de Physique Colloques, vol. 40, no. 6, pp. C6-207–C6-211, 1979.
View at: Publisher Site | Google Scholar
M. Kittler, X. Yu, T. Mchedlidze et al., “Regular dislocation networks in silicon as a tool for nanostructure devices used in optics, biology, and electronics,” Small, vol. 3, no. 6, pp. 964–973, 2007.
View at: Publisher Site | Google Scholar
M. Reich, M. Kittler, D. Buca et al., “Dislocation-based Si-nanodevices,” Japanese Journal of Applied Physics, vol. 49, no. 4S, Article ID 04DJ02, 2010.
View at: Publisher Site | Google Scholar
Y. Ran, Y. Zhang, and A. Vishwanath, “One-dimensional topologically protected modes in topological insulators with lattice dislocations,” Nature Physics, vol. 5, no. 4, pp. 298–303, 2009.
View at: Publisher Site | Google Scholar
H. Asada and T. Futamase, “Propagation of gravitational waves from slow motion sources in a Coulomb-type potential,” Physical Review D, vol. 56, no. 10, pp. R6062–R6066, 1997.
View at: Publisher Site | Google Scholar
C. L. Critchfield, “Scalar potentials in the Dirac equation,” Journal of Mathematical Physics, vol. 17, no. 2, pp. 261–266, 1976.
View at: Publisher Site | Google Scholar
I. C. Fonseca and K. Bakke, “Quantum aspects of a moving magnetic quadrupole moment interacting with an electric field,” Journal of Mathematical Physics, vol. 56, no. 6, Article ID 062107, 2015.
View at: Publisher Site | Google Scholar
P. M. T. Barboza and K. Bakke, “An angular frequency dependence on the Aharonov-Casher geometric phase,” Annals of Physics, vol. 361, pp. 259–265, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
E. R. Figueiredo Medeiros and E. R. Bezerra de Mello, “Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential,” The European Physical Journal C, vol. 72, article 2051, 2012.
View at: Publisher Site | Google Scholar
V. R. Khalilov, “Relativistic Aharonov-Bohm effect in the presence of planar Coulomb potentials,” Physical Review A, vol. 71, no. 1, Article ID 012105, 6 pages, 2005.
View at: Publisher Site | Google Scholar
H. W. Crater, J.-H. Yoon, and C.-Y. Wong, “Singularity structures in Coulomb-type potentials in two-body Dirac equations of constraint dynamics,” Physical Review D, vol. 79, no. 3, Article ID 034011, 2009.
View at: Publisher Site | Google Scholar
K. Bakke and C. Furtado, “On the Klein–Gordon oscillator subject to a Coulomb-type potential,” Annals of Physics, vol. 355, pp. 48–54, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
W. Greiner, Relativistic Quantum Mechanics: Wave Equations, Springer, Berlin, Germany, 3rd edition, 2000.
View at: MathSciNet
H. G. Dosch, J. H. D. Jensen, and V. F. Muller, “Einige bemerkungen zum Kleinschen paradoxon,” Physica Norvegica, vol. 5, p. 151, 1971.
View at: Google Scholar
M. K. Bahar and F. Yasuk, “Exact solutions of the mass-dependent Klein-Gordon equation with the vector quark-antiquark interaction and harmonic oscillator potential,” Advances in High Energy Physics, vol. 2013, Article ID 814985, 6 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
G. Soff, B. Müller, J. Rafelski, and W. Greiner, “Solution of the dirac equation for scalar potentials and its implications in atomic physics,” Zeitschrift für Naturforschung, vol. 28, pp. 1389–1396, 1973.
View at: Google Scholar
E. R. Bezerra de Mello, “Effects of anomalous magnetic moment in the quantum motion of neutral particle in magnetic and electric fields produced by a linear source in a conical spacetime,” Journal of High Energy Physics, vol. 2004, no. 6, article 016, 2004.
View at: Publisher Site | Google Scholar
G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Elsevier Academic Press, New York, NY, USA, 6th edition, 2005.
View at: MathSciNet
M. Abramowitz and I. A. Stegum, Handbook of Mathematical Functions, Dover, New York, NY, USA, 1965.
L. Dantas, C. Furtado, and A. L. Silva Netto, “Quantum ring in a rotating frame in the presence of a topological defect,” Physics Letters A, vol. 379, no. 1-2, pp. 11–15, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
M. Büttiker, Y. Imry, and R. Landauer, “Josephson behavior in small normal one-dimensional rings,” Physics Letters A, vol. 96, no. 7, pp. 365–367, 1983.
View at: Publisher Site | Google Scholar
A. C. Bleszynski-Jayich, W. E. Shanks, B. Peaudecerf et al., “Persistent currents in normal metal rings,” Science, vol. 326, no. 5950, pp. 272–275, 2009.
View at: Publisher Site | Google Scholar
K. Bakke and C. Furtado, “On the confinement of a Dirac particle to a two-dimensional ring,” Physics Letters A, vol. 376, no. 15, pp. 1269–1273, 2012.
View at: Publisher Site | Google Scholar
K. Bakke and C. Furtado, “On the interaction of the Dirac oscillator with the Aharonov–Casher system in topological defect backgrounds,” Annals of Physics, vol. 336, pp. 489–504, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
D. Loss, P. Goldbart, and A. V. Balatsky, “Berry's phase and persistent charge and spin currents in textured mesoscopic rings,” Physical Review Letters, vol. 65, no. 13, pp. 1655–1658, 1990.
View at: Publisher Site | Google Scholar
D. Loss and P. M. Goldbart, “Persistent currents from Berrys phase in mesoscopic systems,” Physical Review B, vol. 45, no. 23, article 13544, 1992.
View at: Publisher Site | Google Scholar
X.-C. Gao and T.-Z. Qian, “Aharonov-Anandan phase and persistent currents in a mesoscopic ring,” Physical Review B, vol. 47, no. 12, pp. 7128–7131, 1993.
View at: Publisher Site | Google Scholar
T.-Z. Qian and Z.-B. Su, “Spin-orbit interaction and Aharonov-Anandan phase in mesoscopic rings,” Physical Review Letters, vol. 72, no. 15, pp. 2311–2315, 1994.
View at: Publisher Site | Google Scholar
A. V. Balatsky and B. L. Altshuler, “Persistent spin and mass currents and aharonov-casher effect,” Physical Review Letters, vol. 70, no. 11, pp. 1678–1681, 1993.
View at: Publisher Site | Google Scholar
S. Oh and C.-M. Ryu, “Persistent spin currents induced by the Aharonov-Casher effect in mesoscopic rings,” Physical Review B, vol. 51, no. 19, pp. 13441–13448, 1995.
View at: Publisher Site | Google Scholar
H. Mathur and A. D. Stone, “Persistent-current paramagnetism and spin-orbit interaction in mesoscopic rings,” Physical Review B, vol. 44, no. 19, pp. 10957–10960, 1991.
View at: Publisher Site | Google Scholar
H. Mathur and A. D. Stone, “Quantum transport and the electronic Aharonov-Casher effect,” Physical Review Letters, vol. 68, no. 19, pp. 2964–2967, 1992.
View at: Publisher Site | Google Scholar
K. Bakke and H. Belich, Spontaneous Lorentz Symmetry Violation and Low Energy Scenarios, LAMBERT Academic Publishing, Saarbrücken, Germany, 2015.
M. M. Ferreira Jr. and F. M. O. Moucherek, “Influence of Lorentz violation on the hydrogen spectrum,” Nuclear Physics A, vol. 790, no. 1–4, pp. 635c–638c, 2007.
View at: Publisher Site | Google Scholar
A. G. Grushin, “Consequences of a condensed matter realization of Lorentz-violating QED in Weyl semi-metals,” Physical Review D, vol. 86, no. 4, Article ID 045001, 2012.
View at: Publisher Site | Google Scholar
M. A. Ajaib, “Understanding lorentz violation with rashba interaction,” International Journal of Modern Physics A, vol. 27, no. 24, Article ID 1250139, 2012.
View at: Publisher Site | Google Scholar
L. R. Ribeiro, E. Passos, C. Furtado, and J. R. Nascimento, “Geometric phases modified by a Lorentz-symmetry violation background,” International Journal of Modern Physics A, vol. 30, no. 14, Article ID 1550072, 2015.
View at: Publisher Site | Google Scholar
L. R. Ribeiro, E. Passos, and C. Furtado, “An analogy of the quantum hall conductivity in a Lorentz-symmetry violation setup,” Journal of Physics G: Nuclear and Particle Physics, vol. 39, no. 10, Article ID 105004, 2012.
View at: Publisher Site | Google Scholar
L. Dekar, L. Chetouani, and T. F. Hammann, “Wave function for smooth potential and mass step,” Physical Review A—Atomic, Molecular, and Optical Physics, vol. 59, no. 1, pp. 107–112, 1999.
View at: Publisher Site | Google Scholar
O. von Roos, “Position-dependent effective masses in semiconductor theory,” Physical Review B, vol. 27, no. 12, pp. 7547–7552, 1983.
View at: Publisher Site | Google Scholar
O. von Roos and H. Mavromatis, “Position-dependent effective masses in semiconductor theory. II,” Physical Review B, vol. 31, no. 4, pp. 2294–2298, 1985.
View at: Publisher Site | Google Scholar
S.-H. Dong and M. Lozada-Cassou, “Exact solutions of the Schrödinger equation with the position-dependent mass for a hard-core potential,” Physics Letters A, vol. 337, no. 4–6, pp. 313–320, 2005.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2015 E. V. B. Leite et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

PDF Download Citation

Download other formats

Order printed copies

Views

1543

Downloads

1107

Citations