Skip to main content

Advertisement

SpringerLink
Log in

Relativistic Quantum Dynamics of a Neutral Dirac Fermion in the Presence of an Electromagnetic Field

  • Condensed Matter
  • Published:
Brazilian Journal of Physics Aims and scope Submit manuscript

Abstract

In this work, we study the (2 + 1)-dimensional Dirac equation for a neutral fermion with magnetic dipole moment in the presence of an electromagnetic field. Next, we explicitly determine the eigenfunctions and the relativistic energy spectrum of the fermion. As result, we verified that these eigenfunctions are written in terms of the generalized Laguerre polynomials and the energy spectrum depends on the quantum numbers, n = 0,1,2,… and mj = 0,± 1,± 2,…, homogeneous magnetic field B, and of the cyclotron frequency ωAC generated by the electric field. Besides that, this energy spectrum may have finite or infinite degeneracy depending on the values of mj. In particular, we also verified that in the absence of the electric field (ωAC = 0), the energy spectrum reduces to a physical quantity (energy) that depends on the rest mass of the fermion and antifermion and of the magnetic field, already in the absence of the magnetic field (B = 0), the energy spectrum still remains quantized in terms of the quantum numbers n and mj; on the other hand, in the absence of the electromagnetic field (ωAC = B = 0), we get the rest energy of the fermion and antifermion. Finally, we compare our results with the literature, where we observe a similarity in some results of the Dirac oscillator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Koc, M. Koca, Spectrum of the relativistic particles in various potentials. Mod. Phys. Lett. A. 20, 911–921 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. N. Ferkous, A. Bounames, Energy spectrum of 2D Dirac oscillator inthe presence of the Aharonov-Bohm effect. Phys. Lett. A. 325, 21–29 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. A.M. Schakel, Relativistic quantum Hall effect. Phys. Rev. D. 43, 1428 (1991)

    Article  ADS  Google Scholar 

  4. F.D.M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  5. J.A. Neto, M.J. Bueno, C. Furtado, Two-dimensional quantum ring in a graphene layer in the presence of a Aharonov- Bohm flux. Ann. Phys. 373, 273–285 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. A. Neagu, A.M. Schakel, Induced quantum numbers in the (2 + 1)-dimensional electron gás. Phys. Rev. D. 48, 1785 (1993)

    Article  ADS  Google Scholar 

  7. A. Jellal, A.D. Alhaidari, H. Bahlouli, Dynamical mass generation via space compactification in graphene. Phys. Rev. A. 80, 012109 (2009)

    Article  ADS  Google Scholar 

  8. K.S. Novoselov, A.K. Geim, S. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Twodimensional gas of massless Dirac fermions in graphene. Nature. 438, 197 (2005)

    Article  ADS  Google Scholar 

  9. M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Chiral tunneling and the Klein paradox in graphene. Nat. Phys. 2, 620 (2006)

    Article  Google Scholar 

  10. A.C. Neto, F. Guinea, N.M. Peres, K.S. Novoselov, A.K. Geim, The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009)

    Article  ADS  Google Scholar 

  11. A.K. Geim, Graphene: Status and prospects. Science. 324, 1530–1534 (2009)

    Article  ADS  Google Scholar 

  12. A.K. Geim, K.S. Novoselov, in . Nanoscience and Technology: A Collection of Reviews from Nature Journals, pp. 11–19, (2010)

  13. Y. Aharonov, A. Casher, Topological quantum effects for neutral particles. Phys. Rev. Lett. 53, 319 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  14. C.R. Hagen, Exact equivalence of spin-1/2 Aharonov-Bohm and Aharonov-Casher effects. Phys. Rev. Lett. 64, 2347 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. E.O. Silva, F.M. Andrade, C. Filgueiras, H. Belich, Bound states of massive fermions in Aharonov-Bohm-like fields. Eur. Phys. J. C. 73, 2402 (2013)

    Article  ADS  Google Scholar 

  16. S. Bruce, L. Roa, C. Saavedra, A.B. Klimov, A.B. Klimov, Unbroken supersymmetry in the Aharonov-Casher effect. Phys. Rev. A. 60, R1 (1999)

    Article  ADS  Google Scholar 

  17. B. Mirza, M. Zarei, Non-commutative quantum mechanics and the Aharonov-Casher effect. Eur. Phys. J. C. 32, 583–586 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. K. Li, J. Wang, . Eur. Phys. J. C. 50, 1007–1011 (2007)

    Article  ADS  Google Scholar 

  19. K. Li, J. Wang, The topological AC effect on non-commutative phase space. Eur. Phys. J. C. 50, 1007–1011 (2007)

    Article  ADS  MATH  Google Scholar 

  20. K. Bakke, H. Belich, E.O. Silva, Relativistic Landau-Aharonov-Casher quantization based on the Lorentz symmetry violation background. J. Math. Phys. 52, 063505 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Q.G. Lin, Aharonov-Bohm effect on Aharonov-Casher scattering. Phys. Rev. A. 72, 042103 (2005). 81, 012710 (2010)

    Article  ADS  Google Scholar 

  22. F.S. Azevedo, E.O. Silva, L.B. Castro, C. Filgueiras, D. Cogollo, Relativistic quantum dynamics of neutral particle in external electric fields: An approach on effects of spin. Ann. Phys. 362, 196–207 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. K. Bakke, Relativistic bounds states for a neutral particle confined to a parabolic potential induced by noninertial effects. Phys. Lett. A. 374, 4642–4646 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. K. Bakke, On noninertial effects inducing a confinement of a neutral particle to a hard-wall confining potential. Open Phys. 11, 1589–1597 (2013)

    Article  ADS  Google Scholar 

  25. K. Bakke, J.R. Nascimento, C. Furtado, Geometric phase for neutral particle in the presence of a topological defect. Phys. Rev. D. 78, 064012 (2008)

    Article  ADS  Google Scholar 

  26. K. Bakke, C. Furtado, On the interaction of the Dirac oscillaton with the Aharonov-Casher system in topological defect backgrounds. Ann. Phys. 336, 489–504 (2013)

    Article  ADS  MATH  Google Scholar 

  27. K. Bakke, On the effects of curvature on the confinement of a neutral particle to a quantum dot induced by noninertial effects. Int. J. Theor. Phys. 51, 759–771 (2012)

    Article  MATH  Google Scholar 

  28. K. Bakke, A geometric approach to confining a Dirac neutral particle in analogous way to a quantum dot. Eur. Phys. J. B. 85, 354 (2012)

    Article  ADS  Google Scholar 

  29. W. Greiner, Relativistic Quantum Mechanics. Wave Equations (Springer, Berlin, 1997)

    Google Scholar 

  30. M. Moshinsky, A. Szczepaniak, . J. Phys. A, Math. Gen. 22, L817 (1989)

    Article  ADS  Google Scholar 

  31. J. Bentez, R.M. y Romero, H.N. Nú,ez-Yépez, A.L. Salas-Brito, Solution and hidden supersymmetry of a Dirac oscillator. Phys. Rev. Lett. 64, 1643 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  32. R.P. Martinez-y-Romero, H.N. Núnez-Yépez, A.L. Salas-Brito, Relativistic quantum mechanics of a Dirac oscillator. Eur. J. Phys. 16, 135 (1995)

    Article  MathSciNet  Google Scholar 

  33. S. Bruce, M. D. Campos, J. Diaz-Valdes, Low-Energy neutron interaction with a classical electric field. Braz. J. Phys. 44, 356–360 (2014)

    Article  ADS  Google Scholar 

  34. M. Ericsson, E. Sjöqvist, Towards a quantum hall effect for atoms using electric fields. Phys. Rev. A. 65, 013607 (2001)

    Article  ADS  Google Scholar 

  35. L.R. Ribeiro, C. Furtado, J.R. Nascimento, Landau levels analog to electric dipole. Phys. Lett. A. 348, 135–140 (2006)

    Article  ADS  Google Scholar 

  36. C. Furtado, J.R. Nascimento, L.R. Ribeiro, Landau quantization of neutral particles in an external field. Phys. Lett. A. 358, 336–338 (2006)

    Article  ADS  MATH  Google Scholar 

  37. L.R. Ribeiro, E. Passos, C. Furtado, J.R. Nascimento, Landau analog levels for dipoles in non-commutative space and phase space. Eur. Phys. J. C. 56, 597–606 (2008)

    Article  ADS  Google Scholar 

  38. L. Dantas, C. Furtado, Induced electric dipole in a quantum ring. Phys. Lett. A. 377, 2926–2930 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. I.C. Fonseca, K. Bakke, On an atom with a magnetic quadrupole moment subject to harmonic and confining potentials. Proc. R. Soc. A. 471, 20150362 (2015)

    Article  ADS  Google Scholar 

  40. E.O. Silva, On planar quantum dynamics of a magnetic dipole moment in the presence of electric and magnetic fields. Eur. Phys. J. C. 74, 3112 (2014)

    Article  Google Scholar 

  41. V.M. Villalba, A.R. Maggiolo, Energy spectrum of a 2D Dirac electron in the presence of a magnetic field. Eur. Phys. J. B. 22, 31 (2001)

    ADS  Google Scholar 

  42. F.M. Andrade, E.O. Silva, Remarks on the Dirac oscillator in (2 + 1) dimensions. Europhys. Europhys. Lett. 108, 30003 (2014)

    Article  ADS  Google Scholar 

  43. D.J. Griffiths. Introduction to Electrodynamics, 4th edn. (Pearson Education, London, 2012)

    Google Scholar 

  44. J.J. Sakurai. Modern Quantum Mechanics (Addison-Wesley Publishing Company, Boston, 1994)

    Google Scholar 

  45. K. Bakke, C. Furtado, On the confinement of a Dirac particle to a two-dimensional ring. Phys. Lett. A. 376, 1269–1273 (2012)

    Article  ADS  MATH  Google Scholar 

  46. M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions (Dover Publications Inc., New York, 1965)

    MATH  Google Scholar 

  47. P. Strange. Relativistic Quantum Mechanics: with Applications in Condensed Matter and Atomic Physics (Cambridge University Press, Cambridge, 1998)

    Book  Google Scholar 

Download references

Funding

The authors received financial support from the Fundação Cearense de apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP) the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

Author information

Authors and Affiliations

  1. Departamento de Física, Universidade Federal do Ceará (UFC), Campus do Pici, Fortaleza, CE, C.P. 6030, 60455-760, Brazil

    R. R. S. Oliveira

  2. Departamento de Física, Universidade Federal de Campina Grande (UFCG), Campus de Campina Grande, Campina Grande, PB, C.P. 10071, 58429-900, Brazil

    M. F. Sousa

Authors
  1. R. R. S. Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. F. Sousa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to R. R. S. Oliveira or M. F. Sousa.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oliveira, R.R.S., Sousa, M.F. Relativistic Quantum Dynamics of a Neutral Dirac Fermion in the Presence of an Electromagnetic Field. Braz J Phys 49, 315–320 (2019). https://doi.org/10.1007/s13538-019-00660-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13538-019-00660-x

Keywords

Search

Navigation