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Falling of a quantum particle in an inverse square attractive potential

  • Vasyl M. Vasyuta
  • Volodymyr M. Tkachuk
Regular Article

Abstract

Evolution of a quantum particle in an inverse square potential is studied by analysis of the equation of motion for 〈r2〉. In such a way we identify the conditions of falling of a particle into the center. We demonstrate the existence of a purely quantum limit of falling, namely, a particle does not fall, when the coupling constant is smaller than a certain critical value. Also the time of falling of a particle into the center is estimated. Although there are no stationary energy levels for this potential, we show that there are quasi-stationary states which evolve with 〈r2〉 being constant in time. Our results are compared with measurements of neutral atoms falling in the electric field of a charged wire. Modifications of the experiment, which may help in observing quantum limit of falling, are proposed.

Graphical abstract

Keywords

Atomic Physics 

References

  1. 1.
    G.H. Shortley, Phys. Rev. 38, 120 (1931)ADSCrossRefGoogle Scholar
  2. 2.
    K.M. Case, Phys. Rev. 80, 797 (1950)ADSCrossRefGoogle Scholar
  3. 3.
    E.A. Guggenheim, Proc. Phys. Soc. 89, 491 (1966)ADSCrossRefGoogle Scholar
  4. 4.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory (Fizmatlit, Moscow, 2004)Google Scholar
  5. 5.
    V.N. Efimov, Sov. J. Nucl. Phys. 12, 589 (1971)Google Scholar
  6. 6.
    L.V. Hau, M.M. Burns, J.A. Golovchenko, Phys. Rev. A 45, 6468 (1992)ADSCrossRefGoogle Scholar
  7. 7.
    J. Schmiedmayer, Appl. Phys. B 60, 169 (1995)ADSCrossRefGoogle Scholar
  8. 8.
    J. Denschlag, J. Schmiedmayer, Europhys. Lett. 38, 405 (1997)ADSCrossRefGoogle Scholar
  9. 9.
    J. Denschlag, G. Umshaus, J. Schmiedmayer, Phys. Rev. Lett. 81, 737 (1998)ADSCrossRefGoogle Scholar
  10. 10.
    V.M. Tkachuk, Phys. Rev. A 60, 4715 (1999)ADSCrossRefGoogle Scholar
  11. 11.
    T.R. Govindarajan, V. Suneeta, S. Vaidya, Nucl. Phys. B 583, 291 (2000)ADSCrossRefGoogle Scholar
  12. 12.
    D. Birmingham, K.S. Gupta, S. Sen, Phys. Lett. B 505, 191 (2001)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    K.S. Gupta, S. Sen, Phys. Lett. B 526, 121 (2002)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    S.K. Chakrabarti, K.S. Gupta, S. Sen, Int. J. Mod. Phys. A 23, 2547 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    H.E. Camblong, C.R. Ordóñez, Class. Quantum Grav. 30, 175007 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    M. Bawin, Phys. Rev. A 70, 022505 (2004)ADSCrossRefGoogle Scholar
  17. 17.
    M. Bawin, S.A. Coon, B.R. Holstein, Int. J. Mod. Phys. A 22, 4901 (2007)ADSCrossRefGoogle Scholar
  18. 18.
    A.D. Alhaidari, J. Phys. A: Math. Theor. 40, 14843 (2007)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    P.R. Giri, K.S. Gupta, S. Meljanac, A. Samsarov, Phys. Lett. A 372, 2967 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    H. Narnhofer, Acta Phys. Austriaca 40, 306 (1974)MathSciNetGoogle Scholar
  21. 21.
    M. Bawin, S.A. Coon, Phys. Rev. A 67, 042712 (2003)ADSCrossRefGoogle Scholar
  22. 22.
    D. Bouazis, M. Bawin, Phys. Rev. A 89, 022113 (2014), and references thereinADSCrossRefGoogle Scholar
  23. 23.
    K.S. Gupta, S.G. Rajeev, Phys. Rev. D 48, 5940 (1993)ADSCrossRefGoogle Scholar
  24. 24.
    H.E. Camblong, L.N. Epele, H. Fanchiotti, C.A. García Canal, Phys. Rev. Lett. 85, 1590 (2000)ADSCrossRefGoogle Scholar
  25. 25.
    S.R. Beane, P.F. Bedaque, L. Childress, A. Kryjevski, J. McGuire, U. van Kolck, Phys. Rev. A 64, 042103 (2001)ADSCrossRefGoogle Scholar
  26. 26.
    S.A. Coon, B.R. Holstein, Am. J. Phys. 70, 513 (2002)ADSCrossRefGoogle Scholar
  27. 27.
    H.-W. Hammer, B.G. Swingle, Ann. Phys. 321, 306 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    A.D. Alhaidari, Found. Phys. 44, 1049 (2014), and references thereinADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    D. Bouaziz, M. Bawin, Phys. Rev. A 76, 032112 (2007)ADSCrossRefGoogle Scholar
  30. 30.
    D. Bouaziz, M. Bawin, Phys. Rev. A 78, 032110 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    P.R. Giri, Int. J. Mod. Phys. A, 24, 2655 (2009)ADSCrossRefGoogle Scholar
  32. 32.
    R. Cotes, Harmonia Mensurarum (Cambridge, 1722)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department for Theoretical Physics, Ivan Franko National University of LvivLvivUkraine

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