Torres-Vega distribution function in the extended phase space

  • F. Taati
  • T. Jahani
  • D. JahaniEmail author
Regular Article


Classical physics gives us our everyday perception of the world. However, the break down of this classical intuition could be explained by quantum mechanics. Hence, quantum corrections to the classical statistical mechanics have always remained an important issue, especially, in optics and laser physics for which the correspondence between the classical limit of a large number of photons and the quantum regime is of most importance. We could, therefore, employ distribution functions to find this relation between the quantum and classical solutions. One of these distribution functions which is also of interest in quantum optics and ultrafast laser physics is Wigner distribution function. Interestingly, it could be transferred to the other probability distributions such as the positive Torres-Vega distribution function. In this paper, for the first time, we obtain the Torres-Vega distribution function directly from the extended phase space. We also by means of a set of unitary transformations in the phase space extract the extended Hamiltonian for the Torres-Vega distribution.


  1. 1.
    C.K., Zachos, D.B. Fairlie, T.L. Curtright (Editors), Quantum Mechanics in Phase Space: An Overview with Selected Papers (World Scientific, 2005)Google Scholar
  2. 2.
    S. Nasiri, Y. Sobouti, F. Taati, J. Math. Phys. 47, 092106 (2006)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Błaszak, Z. Domański, Ann. Phys. 327, 167 (2012)CrossRefGoogle Scholar
  4. 4.
    C. Lopez, An extended phase space for Quantum Mechanics, arXiv:1509.07025 (2015)Google Scholar
  5. 5.
    M.J. Bastiaans, Opt. Commun. 25, 26 (1978)ADSCrossRefGoogle Scholar
  6. 6.
    M.J. Bastiaans, J. Opt. Soc. 69, 1710 (1979)ADSCrossRefGoogle Scholar
  7. 7.
    M.O.S.M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Phys. Rep. 106, 121 (1984)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Wollenhaupt, A. Assion, T. Baumert, Femtosecond laser pulses: linear properties, manipulation, generation and measurement, in Springer Handbook of Lasers and Optics (Springer, New York, NY, 2007) pp. 937--983Google Scholar
  9. 9.
    O. Bohigas, S. Tomsovic, D. Ullmo, Phys. Rep. 223, 43 (1993)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Y.S. Kim, Phase Space Picture of Quantum Mechanics: Group Theoretical Approach, Vol. 40 (World Scientific, 1991)Google Scholar
  11. 11.
    G. Torres Vega, J.H. Frederick, J. Chem. Phys. 98, 3103 (1993)ADSCrossRefGoogle Scholar
  12. 12.
    G. Torres Vega, J.H. Frederick, J. Chem. Phys. 98, 3103 (1993)ADSCrossRefGoogle Scholar
  13. 13.
    E.P. Wigner, On the quantum correction for thermodynamic equilibrium, in The Collected Works of Eugene Paul Winger (Part A The Scientific Papers), edited by A.S. Wightman, Vol. A/4, Part I: Physical Chemistry. Part II: Solid State Physics (Springer, Berlin, Heidelberg, 1997)
  14. 14.
    Q.S. Li, G.M. Wei, L.Q. Lü, Phys. Rev. A 70, 022105 (2004)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Mielke, Arch. Ration. Mech. Anal. 181, 401 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    L.Y. Hu, H.Y. Fan, Int. J. Theor. Phys. 48, 1539 (2009)CrossRefGoogle Scholar
  17. 17.
    G. García-Calderón, M. Moshinsky, J. Phys. A 13, L185 (1980)ADSCrossRefGoogle Scholar
  18. 18.
    Y. Sobouti, S. Nasiri, Int. J. Mod. Phys. B 7, 3255 (1993)ADSCrossRefGoogle Scholar
  19. 19.
    Y. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature 438, 201 (2005)ADSCrossRefGoogle Scholar
  20. 20.
    D. Jahani, A. Alidoust Ghatar, L. Abaspour, T. Jahani, J. Appl. Phys. 124, 043104 (2018)ADSCrossRefGoogle Scholar
  21. 21.
    A. Zúñiga-Segundo, H.M. Moya-Cessa, F. Soto-Eguibar, AIP Adv. 6, 015202 (2016)ADSCrossRefGoogle Scholar
  22. 22.
    S. Mancini, V.I. Man’Ko, P. Tombesi, Phys. Lett. A 213, 1 (1996)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Asorey, A. Ibort, G. Marmo, F. Ventriglia, Phys. Scr. 90, 074031 (2015)ADSCrossRefGoogle Scholar
  24. 24.
    D. Chruściński, K. Młodawski, Phys. Rev. A 71, 052104 (2005)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    H. Wu, Z.B. Yang, S.B. Zheng, Phys. Rev. A 88, 043816 (2013)ADSCrossRefGoogle Scholar
  26. 26.
    V.V. Dodonov, V.I. Man’ko, Phys. Lett. A 229, 335 (1997)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    E. Colomés, Z. Zhan, X. Oriols, J. Comput. Electron. 14, 894 (2015)CrossRefGoogle Scholar
  28. 28.
    E.P. Gross, Ann. Phys. 69, 42 (1972)ADSCrossRefGoogle Scholar
  29. 29.
    J. Ehlers, P. Geren, R.K. Sachs, J. Math. Phys. 9, 1344 (1968)ADSCrossRefGoogle Scholar
  30. 30.
    I. Horenko, C. Salzmann, B. Schmidt, C. Schütte, J. Chem. Phys. 117, 11075 (2002)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Materials and Energy Research CenterTehranIran

Personalised recommendations