True quantum face of the “exponential” decay law

  • Krzysztof UrbanowskiEmail author
Open Access
Regular Article


Results of theoretical studies of the quantum unstable systems caused that there are rather widespread belief that a universal feature of the quantum decay process is the presence of three time regimes of the decay process: the early time (initial) leading to the Quantum Zeno (or Anti Zeno) Effects, “exponential” (or “canonical”) described by the decay law of the exponential form, and late time characterized by the decay law having inverse-power law form. Based on the fundamental principles of the quantum theory we give the proof that there is no time interval in which the survival probability (decay law) could be a decreasing function of time of the purely exponential form but even at the “exponential” regime the decay curve is oscillatory modulated with a smaller or a large amplitude of oscillations depending on parameters of the model considered.

Graphical abstract


Molecular Physics and Chemical Physics 


  1. 1.
    E. Ruthheford, Philos. Mag. XLIX, 1 (1900)CrossRefGoogle Scholar
  2. 2.
    E. Rutherford, F. Soddy, Philos. Mag. IV, 370 (1902)CrossRefGoogle Scholar
  3. 3.
    V.F. Weisskopf, E.T. Wigner, Zeitschrift für Physik 63, 54 (1930)ADSCrossRefGoogle Scholar
  4. 4.
    V.F. Weisskopf, E.T. Wigner, Zeitschrift für Physik 65, 18 (1930)ADSCrossRefGoogle Scholar
  5. 5.
    L.A. Khalfin, Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki (USSR) 33, 1371 (1957) [in Russian], [Soviet Phys. J. Exp. Theor. Phys. 6, 1053 (1958)]Google Scholar
  6. 6.
    L. Fonda, G.C. Ghirardii, A. Rimini, Rep. Prog. Phys. 41, 587 (1978)ADSCrossRefGoogle Scholar
  7. 7.
    E.B. Norman, S.B. Gazes, S.G. Crane, D.A. Bennett, Phys. Rev. Lett. 60, 2246 (1988)ADSCrossRefGoogle Scholar
  8. 8.
    C. Rothe, S.I. Hintschich, A.P. Monkman, Phys. Rev. Lett. 96, 163601 (2006)ADSCrossRefGoogle Scholar
  9. 9.
    B. Misra, E.C.G. Sudarshan, J. Math. Phys. 18, 745 (1977)CrossRefGoogle Scholar
  10. 10.
    M.C. Fischer, B. Gutiérrez-Medina, M.G. Raizen, Phys. Rev. Lett. 87, 040402 (2001)ADSCrossRefGoogle Scholar
  11. 11.
    W.M. Itano, D.J. Heinzen, J.J. Bollinger, D.J. Wineland, Phys. Rev. A 41, 2295 (1990)ADSCrossRefGoogle Scholar
  12. 12.
    Y.S. Patil, S. Chakram, M. Vengalattore, Phys. Rev. Lett. 115, 140402 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    M. Peshkin, A. Volya, V. Zelevinsky, Europhys. Lett. 107, 40001 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    Yu.A. Litvinov, F. Bosch, N. Winckler, D. Boutin, H.G. Essel, T. Faestermann, H. Geissel, S. Hess, P. Kienle, R. Knöbel, C. Kozhuharov, J. Kurcewicz, L. Maier, K. Beckert, P. Beller, C. Brandau, L. Chen, C. Dimopoulou, B. Fabian, A. Fragner, E. Haettner et al., Phys. Lett. B 664, 162 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    P. Kienle, F. Bosch, P. Bühler, T. Faestermann, Yu.A. Litvinov, N. Winckler, M.S. Sanjari, D.B. Shubina, D. Atanasov, H. Geissel, V. Ivanova, X.L. Yan et al., Phys. Lett. B 726, 638 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    N.S. Krylov, V.A. Fock, Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki (USSR) 17, 93 (1947) [in Russian]Google Scholar
  17. 17.
    V.A. Fock, Fundamentals of Quantum Mechanics (Mir Publishers, Moscow, 1978)Google Scholar
  18. 18.
    N.G. Kelkar, M. Nowakowski, J. Phys. A: Math. Theor. 43, 385308 (2010)ADSCrossRefGoogle Scholar
  19. 19.
    F. Giraldi, Eur. Phys. J. D 69, 5 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    F. Giacosa, Found. Phys. 42, 1262 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    M.L. Goldberger, K.M. Watson, Collision theory (Wiley, 1964)Google Scholar
  22. 22.
    R.E.A.C. Paley, Fourier transforms in the complex domain (American Mathematical Society, New York, 1934)Google Scholar
  23. 23.
    K.M. Sluis, E.A. Gislason, Phys. Rev. A 43, 4581 (1991)ADSCrossRefGoogle Scholar
  24. 24.
    K. Urbanowski, Eur. Phys. J. C 58, 151 (2008)ADSCrossRefGoogle Scholar
  25. 25.
    K. Urbanowski, Open Physics 7, 696 (2009) (Formerly: Central European Journal of Physics)ADSCrossRefGoogle Scholar
  26. 26.
    M. Abramowitz, A.I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications Inc., New York, 1964)Google Scholar
  27. 27.
    K. Urbanowski, Phys. Rev. A 50, 2847 (1994)ADSCrossRefGoogle Scholar
  28. 28.
    K. Urbanowski, K. Raczyńska, Phys. Lett. B 731, 236 (2014)ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.University of Zielona Góra, Institute of PhysicsZielona GóraPoland

Personalised recommendations