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Foundations of Physics

, Volume 45, Issue 6, pp 644–656 | Cite as

The Stationary Dirac Equation as a Generalized Pauli Equation for Two Quasiparticles

  • Nikolay L. ChuprikovEmail author
Article

Abstract

By analyzing the Dirac equation with static electric and magnetic fields it is shown that Dirac’s theory is nothing but a generalized one-particle quantum theory compatible with the special theory of relativity. This equation describes a quantum dynamics of a single relativistic fermion, and its solution is reduced to solution of the generalized Pauli equation for two quasiparticles which move in the Euclidean space with their effective masses holding information about the Lorentzian symmetry of the four-dimensional space-time. We reveal the correspondence between the Dirac bispinor and Pauli spinor (two-component wave function), and show that all four components of the Dirac bispinor correspond to a fermion (or all of them correspond to its antiparticle). Mixing the particle and antiparticle states is prohibited. On this basis we discuss the paradoxical phenomena of Zitterbewegung and the Klein tunneling.

Keywords

Dirac equation Klein tunneling Dirac sea Potential step 

Notes

Acknowledgments

First of all I would like to thank Prof. I. L. Buchbinder for his useful critical remarks on the first version of the paper. I also thank Prof. V. G. Bagrov, Prof. V. A. Bordovitsyn and Prof. G. F. Karavaev for useful discussions on this subject. Finally, I want to express my deep gratitude to Reviewers for their helpful remarks and questions. This work was supported in part by the Programm of supporting the leading scientific schools of RF (Grant No 88.2014.2) for partial support of this work.

References

  1. 1.
    Barut, A.O.: Combining relativity and quantum mechanics: Schrödinger’s interpretation of \(\psi \). Found. Phys. 18, 95–105 (1988)Google Scholar
  2. 2.
    Holland, P., Brown, H.R.: The non-relativistic limits of the Maxwell and Dirac equations: the role of Galilean and gauge invariance. Stud. Hist. Philos. Mod. Phys. 34, 161–187 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Messiah, A.: Quantum Mechanics, vol. 2. North-Holland Publishing Company, Amsterdam (1965)Google Scholar
  4. 4.
    Hansen, A., Ravndal, F.: Klein’s Paradox and its resolution. Phys. Scr. 23, 1036–1042 (1981)CrossRefADSGoogle Scholar
  5. 5.
    Holstein, B.R.: Klein’s paradox. Am. J. Phys. 66, 507–512 (1998). doi: 10.1119/1.18891 CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Calogeracos, A., Dombey, N.: History and physics of the Klein paradox. Contemp. Phys. 40(5), 313–321 (1999)CrossRefADSGoogle Scholar
  7. 7.
    Bosanac, S.D.: Solution of Dirac equation for a step potential and the Klein paradox. J. Phys. A 40, 8991–9001 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kononets, Y.V.: Charge conservation, Klein’s paradox and the concept of paulions in the Dirac electron theory. New results for the dirac equation in external fields. Found. Phys. 40, 545–572 (2010). doi: 10.1007/s10701-010-9414-6 CrossRefADSzbMATHMathSciNetGoogle Scholar
  9. 9.
    Alhaidari, A.D.: Resolution of the Klein paradox. Phys. Scr. 83, 025001 (4pp) (2011)CrossRefADSGoogle Scholar
  10. 10.
    Payandeh. F., Pur, T.M., Fathi, M. and Moghaddam, Z.Gh.: A Krein quantization approach to Klein paradox. arXiv:1305.1927v3 [gr-qc]
  11. 11.
    Gerritsma, R., Kirchmair, G., Zahringer, F., Solano, E., Blatt, R., Roos, C.F.: Quantum simulation of the Dirac equation. Nature 463, 68–71 (2010)CrossRefADSGoogle Scholar
  12. 12.
    O’Connel, R.F.: Zitterbewegung is not an observable. Mod. Phys. Lett. A 26(7), 469–471 (2011)CrossRefADSGoogle Scholar
  13. 13.
    Schroedinger, E.: Über die kräftfreie bewegung in der relativistischen quantenmechanik. Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 24, 418 (1930)Google Scholar
  14. 14.
    Hestenes, D.: Zitterbewegung in quantum mechanics. Found. Phys. 40(1), 1–54 (2010)CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. 15.
    Greiner, W.: Relativistic Quantum Mechanics: Wave Equations. Springer, Berlin (1994)Google Scholar
  16. 16.
    Bjorken, J.D., Drell, S.D.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1964)Google Scholar
  17. 17.
    Beenakker, W.J.: Colloquium: Andreev reflection and Klein tunneling in graphene. Rev. Mod. Phys. 80, 1337–1354 (2008)CrossRefADSGoogle Scholar
  18. 18.
    Burt, M.G.: The justification for applying the effective-mass approximation to microstructures. J. Phys. Condens. Matter 4, 6651–6690 (1992)CrossRefADSGoogle Scholar
  19. 19.
    Karavaev, G.F., Krivorotov, I.N.: A method of enveloping functions for description of electron states in microstructures with smooth variation of the potential at heterointerfaces. Phys. Tech. Semicond. 30(1), 177–187 (1996)Google Scholar
  20. 20.
    Dodonov, V.V.: Strict lower bound for the spatial spreading of a relativistic particle. Phys. Lett. A 171, 394–398 (1993)Google Scholar
  21. 21.
    Unanyan, R.G., Otterbach, J., Fleischhauer, M.: Confinement limit of Dirac particles in scalar one-dimensional potentials. Phys. Rev. A 79, 044101 (2009)CrossRefADSGoogle Scholar
  22. 22.
    Cheng, J.-Y.: A complete proof of the confinement limit of one-dimensional Dirac particles. Found. Phys. 44, 953–959 (2014)CrossRefADSzbMATHMathSciNetGoogle Scholar
  23. 23.
    Zawadzki, W.: One-dimensional semirelativity for electrons in carbon nanotubes. Phys. Rev. B 74, 205439 (1–4) (2006)Google Scholar
  24. 24.
    Cserti, J., Dávid, G.: Unified description of Zitterbewegung for spintronic, graphene, and superconducting systems. Phys. Rev. B 74, 172305(1–4) (2006)Google Scholar
  25. 25.
    Zarenia, M., Chaves, A., Farias, G.A., Peeters, F.M.: Energy levels of triangular and hexagonal graphene quantum dots: a comparative study between the tight-binding and Dirac equation approach. Phys. Rev. B 84, 245403(1–12) (2011)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Tomsk State Pedagogical UniversityTomskRussia

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