Bound-state solutions of the Dirac oscillator in an Aharonov-Bohm-Coulomb system. (English) Zbl 1415.81020
Summary: In this work, we study the (2+1)-dimensional Dirac oscillator in the presence of a homogeneous magnetic field in an Aharonov-Bohm-Coulomb system. To solve our system, we apply the left-handed and right-handed projection operators in the Dirac oscillator to obtain a biconfluent Heun equation. Next, we explicitly determine the energy spectrum for the bound states of the system and their exact dependence on the cyclotron frequency \(\omega_c\) and on the parameters \(Z\) and \(\Phi_{AB}\) that characterize the Aharonov-Bohm-Coulomb system. As a result, we observe that by adjusting the frequency of the Dirac oscillator to resonate with the cyclotron half-frequency the energy spectrum reduces to the rest energy of the particle. Also, we determine the exact eigenfunctions, angular frequencies, and energy levels of the Dirac oscillator for the ground state \((n=1)\) and the first excited state \((n=2)\). In this case, the energy levels do not depend on the homogeneous magnetic field, and the angular frequencies are real and positive quantities, increase quadratically with the energy and linearly with \(\omega_c\).
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81R25 | Spinor and twistor methods applied to problems in quantum theory |
| 81V10 | Electromagnetic interaction; quantum electrodynamics |
Keywords:
Dirac oscillator; Aharonov-Bohm-Coulomb system; biconfluent Heun equation; relativistic bound statesReferences:
| [1] | Itô, D.; Mori, K.; Carriere, E., Nuovo Cimento A, 51, 1119 (1967) |
| [2] | Moshinsky, M.; Szczepaniak, A., J. Phys. A: Math. Gen., 22, L817 (1989) |
| [3] | Bentez, J.; Martínez-y Romero, R. P.; Núnez-Yépez, H. N.; Salas-Brito, A. L., Phys. Rev. Lett.. Phys. Rev. Lett., Phys. Rev. Lett., 65, 2085 (1990), Erratum; Martínez-y-Romero, R. P.; Núnez-Yépez, H. N.; Salas-Brito, A. L., Eur. J. Phys., 16, 135 (1995) |
| [4] | Pacheco, M. H.; Landim, R. R.; Almeida, C. A.S., Phys. Lett. A, 311, 93 (2003); Pacheco, M. H.; Maluf, R. V.; Almeida, C. A.S.; Landim, R. R., Europhys. Lett., 108, 10005 (2014) |
| [5] | Boumali, A.; Hassanabadi, H., Eur. Phys. J. Plus, 128, 124 (2013) |
| [6] | Hassanabadi, H.; Sargolzaeipor, S.; Yazarloo, B. H., Few-Body Syst., 56, 115 (2015) |
| [7] | Sargolzaeipor, S.; Hassanabadi, H.; Chung, W. S., Can. J. Phys., 96, 25 (2017) |
| [8] | Hatami, N.; Setare, M., Phys. Lett. A, 380, 3469 (2016) · Zbl 1360.81203 |
| [9] | Maluf, R. V., Internat. J. Modern Phys. A, 26, 4991 (2011) |
| [10] | Andrade, F. M.; Silva, E. O.; Ferreira, M. M.; Rodrigues, E. C., Phys. Lett. B, 731, 327 (2014); Andrade, F. M.; Silva, E. O., Phys. Lett. B, 738, 44 (2014) |
| [11] | Lange, O. L., J. Phys. A: Math. Gen., 24, 667 (1991) · Zbl 0736.47039 |
| [12] | Szmytkowski, R.; Gruchowski, M., J. Phys. A: Math. Gen., 34, 4991 (2001) · Zbl 0983.81012 |
| [13] | Quesne, C.; Tkachuk, V. M., J. Phys. A, 39, 10909 (2006) · Zbl 1168.81014 |
| [14] | Moshinsky, M.; Loyola, G., Found. Phys., 23, 197 (1993) |
| [15] | de Lange, O. L., J. Math. Phys., 32, 1296 (1991) · Zbl 0748.47052 |
| [16] | Wang, Y. X.; Cao, J.; Xiong, S. J., Eur. Phys. J. B, 85, 237 (2012) |
| [17] | Longhi, S., Opt. Lett., 35, 1302 (2010) |
| [18] | C. Quimbay, P. Strange, Graphene physics via the Dirac oscillator in \(( 2 + 1 )\) arXiv:1311.2021; C. Quimbay, P. Strange, Graphene physics via the Dirac oscillator in \(( 2 + 1 )\) arXiv:1311.2021 |
| [19] | Boumali, A., Phys. Scr., 90, 045702 (2015) |
| [20] | Belouad, A.; Jellal, A.; Zahidi, Y., Phys. Lett. A, 380, 773 (2016) · Zbl 1349.81103 |
| [21] | Amaro Neto, José; Bueno, M. J.; Furtado, Claudio, Ann. Physics, 373, 273 (2016); Amaro Neto, José; Oliveira, J. R. de S.; Furtado, Claudio; Sergeenkov, Sergei, Eur. Phys. J. Plus, 133, 185 (2018); Bueno, M. J.; de Melo, J. L.; Furtado, C.; Carvalho, Alexandre M. de M., Eur. Phys. J. Plus, 129, 201 (2014) |
| [22] | Franco-Villafañe, J.-A.; Sadurní, E.; Barkhofen, S.; Kuhl, U.; Mortessagne, F.; Seligman, T. H., Phys. Rev. Lett., 111, 170405 (2013) |
| [23] | Hagen, C. R.; Park, D. K., Ann. Phys., 251, 45 (1996); Hagen, C. R., Phys. Rev. A, 77, 036101 (2008) |
| [24] | Park, D. K.; Yoo, S. K., Ann. Physics, 263, 295 (1998) · Zbl 0919.58014 |
| [25] | Lin, De-Hone, J. Phys. A, 31, 4785 (1998); Lin, De-Hone, J. Math. Phys., 40, 1246 (1999) · Zbl 0934.81022 |
| [26] | Bornales, J.; Bernido, C. C.; Carpio-Bernido, M. V., Phys. Lett. A, 260, 447 (1999) · Zbl 0955.81024 |
| [27] | Lin, Q. G., J. Phys. A: Math. Gen., 33, 5049 (2000) |
| [28] | Hagen, C. R., Phys. Rev. D, 48, 5935 (1993) |
| [29] | Park, D. K.; Oh, Jae Geun, Phys. Rev. D, 50, 7715 (1994) |
| [30] | Villalba, V. M., Phys. Lett. A, 193, 218 (1994); Villalba, V. M., J. Math. Phys., 36, 3332 (1995) · Zbl 0842.35096 |
| [31] | Hai, L. X.; Komarov, L. I.; Romanova, T. S., J. Phys. A: Math. Gen., 25, 6461 (1992) · Zbl 0772.35060 |
| [32] | Khalilov, V. R., Theor. Math. Phys., 158, 210-220 (2009); Khalilov, V. R., Eur. Phys. J. C, 73, 2548 (2013); Khalilov, V. R.; Mamsurov, I. V., Mod. Phys. Lett. A, 31, 1650032 (2016) · Zbl 1183.81117 |
| [33] | Jung, E.; Hwang, M. R.; Park, C. S.; Park, D., J. Phys. A: Math. Theor., 45, 055301 (2012) · Zbl 1235.82112 |
| [34] | Nishida, Y., Phys. Rev. B, 94, 085430 (2016) |
| [35] | Ronveaux, A., Heun’s Differential Equations (1995), Oxford University Press: Oxford University Press Oxford · Zbl 0847.34006 |
| [36] | Vitória, R. L.L.; Belich, H.; Bakke, K., Eur. Phys. J. Plus, 132, 25 (2017); Vitória, R. L.L.; Bakke, K., Eur. Phys. J. Plus, 131, 36 (2016); Vitória, R. L.L.; Furtado, C.; Bakke, K., Ann. Phys., 370, 128 (2016) |
| [37] | Medeiros, E. R.F.; Mello, E. R.B., Eur. Phys. J. C, 72, 2051 (2012) |
| [38] | Bakke, K.; Belich, H., Eur. Phys. J. Plus, 127, 102 (2012) |
| [39] | Greiner, W., Relativistic Quantum Mechanics: Wave Equations (2000), Springer: Springer Berlin · Zbl 0998.81503 |
| [40] | Auvil, P. R.; Brown, L. M., Amer. J. Phys., 46, 679 (1978) |
| [41] | D.B. Kaplan, Chiral Symmetry and Lattice Fermions, in Modern perspectives in lattice QCD: Quantum field theory and high performance computing. Proceedings, International School, 93rd Session, Les Houches, France, August 3-28, 2009, pp. 223-272, 2009, arXiv:0912.2560; D.B. Kaplan, Chiral Symmetry and Lattice Fermions, in Modern perspectives in lattice QCD: Quantum field theory and high performance computing. Proceedings, International School, 93rd Session, Les Houches, France, August 3-28, 2009, pp. 223-272, 2009, arXiv:0912.2560 |
| [42] | Villalba, V. M.; Maggiolo, A. R., Eur. Phys. J. B, 22, 31 (2001); Villalba, V. M.; Pino, R., Mod. Phys. Lett. B, 17, 1331 (2003) |
| [43] | Aharonov, Y.; Bohm, D., Phys. Rev., 115, 485 (1959) · Zbl 0099.43102 |
| [44] | Dong, S. H.; Sun, G. H., Phys. Scr., 69, 161 (2004) · Zbl 1133.81331 |
| [45] | Arfken, G. B.; Weber, H. J., Mathematical Methods for Physicists (2005), Elsevier Academic Press: Elsevier Academic Press New York · Zbl 1066.00001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
