Skip to main content
Log in

Quantum Damped Oscillator in Probability Representation

  • Published:
Journal of Russian Laser Research Aims and scope

Abstract

The probability distribution describing quantum states of the damped oscillator in the framework of the Caldirola-Kanai model is introduced. The probability distributions for coherent states and Fock states of the damped oscillator are found explicitly. The transition probability for the damped oscillator is expressed in terms of distributions describing initial and final states.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. W. Heisenberg, Ztschr. Phys., 43, 172 (1927).

    Article  Google Scholar 

  2. E. Schrödinger, Sitzungsber. Preuss. Acad. Wiss., 24, 296 (1930).

    Google Scholar 

  3. H. P. Robertson, Phys. Rev., 34, 163 (1929).

    Article  ADS  Google Scholar 

  4. J. E. Moyal, Proc. Cambrige Philos. Soc., 45, 99, (1949).

    Article  ADS  Google Scholar 

  5. E. Wigner, Phys. Rev., 40, 749 (1932).

    Article  ADS  Google Scholar 

  6. R. J. Glauber, Phys. Rev. Lett., 10, 84 (1963).

    Article  ADS  MathSciNet  Google Scholar 

  7. E. C. G. Sudarshan, Phys. Rev. Lett., 10, 277 (1963).

    Article  ADS  MathSciNet  Google Scholar 

  8. K. Husimi, Proc. Phys. Math. Soc. Jpn., 23, 264 (1940).

    Google Scholar 

  9. D. Bohm, Phys. Rev., 85, 166; 180 (1952).

    Article  ADS  Google Scholar 

  10. O. V. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 18, 407 (1997).

    Article  Google Scholar 

  11. R. J. Glauber, “Coherence and quantum detection,” in: Rend. Scoula Intern. Fis. E. Fermi XLII (1967), Academic, New York (1969), p. 32.

    Google Scholar 

  12. P. Ullersma, Physica, 32, 27 (1966).

    Article  ADS  MathSciNet  Google Scholar 

  13. I. R. Senitzky, Phys. Rev., 119, 670 (1960).

    Article  ADS  MathSciNet  Google Scholar 

  14. M. D. Kostin, J. Chem. Phys., 57, 3589 (1983).

    Article  ADS  Google Scholar 

  15. H.-D. Doebner and G. A. Goldin, Phys. Lett. A, 162, 397 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  16. V. V. Dodonov and S. S. Mizrahi, Ann. Phys., 237, 226 (1995).

    Article  ADS  Google Scholar 

  17. P. Caldirola, Nuovo Cimento, 18, 393 (1941).

    Article  Google Scholar 

  18. E. Kanai, Progr. Theor. Phys., 3, 440 (1948).

    Article  ADS  Google Scholar 

  19. J. Messer, Acta Phys. Austr., 50, 75 (1979).

    Google Scholar 

  20. H. Dekker, Phys. Rep., 80, 1 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  21. S. Mancini, V. I. Man’ko and P. Tombesi, Phys. Lett. A, 213, 1 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  22. O. V. Man’ko, J. Russ. Laser Res., 17, 439 (1996).

    Article  Google Scholar 

  23. K. Vogel and H. Risken, Phys. Rev. A, 40, 2847 (1989).

    Article  ADS  Google Scholar 

  24. V. I. Man’ko, “Energy levels of quantum system in classical formulation of quantum mechanics,” in: M. Dremin and A. M. Semikhatov (eds.), Proceedings of the Second Intern. A. D. Sakharov Conference on Physics (Moscow, May, 1996), World Scientific, Singapore (1997), p. 486; “Optical symplectic tomography and classical probability instead of wave function in quantum mechanics,” in: H.-D. Doeb-ner, W. Scherer, and C. Schultz (eds.), GROUP21. Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras (Goslar, Germany, June–July 1996), World Scientific, Singapore (1997), Vol. 2, p. 764; “Transition probability between energy levels in the framework of the classical approach,” invited lecture at the Inaguration Conference of APCTP (Seoul, June 1996), (to appear in the Proceedings of the Conference, World Scientific, 1997); “Classical description of quantum states and tomography,” talk at the Fifth Intern. Conference “Squeezed States and Uncertainly Relations” (Balatonfured, Hungary, May 1997) (to be published in NASA Conference Publication, 1997).

    Google Scholar 

  25. K. E. Cahill and R. J. Glauber, Phys. Rev., 177, 1882 (1969).

    Article  ADS  Google Scholar 

  26. V. V. Dodonov and V. I. Man’ko, Phys. Rev. A, 20, 550 (1979).

    Article  ADS  Google Scholar 

  27. R. K. Colegrave and M. S. Abdulla, J. Phys. A, 16, 3805 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  28. R. K. Colegrave, P. Croxson, U. Ramjit, and E. Kheyrabady, Physica A, 161, 118 (1989).

  29. L. F. Landovitz, A. M. Lezine, and W. M. Schreiber, J. Math. Phys., 21, 2159 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  30. P. Croxson, Phys. Rev. A, 49, 588 (1994).

    Article  ADS  Google Scholar 

  31. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambrige University Press, Cambrige (1987).

    MATH  Google Scholar 

  32. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin (1932).

    MATH  Google Scholar 

  33. V. I. Man’ko, J. Russ. Laser Res., 17, 579 (1996).

    Article  Google Scholar 

  34. V. V. Dodonov and V. I. Man’ko, Invariants and Evolution of Nonstationary Quantum System, Proceeding of the Lebedev Physical Institute, Nova Science, New York (1989), Vol. 183.

  35. I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Sums, Rows, and Products, [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  36. P. Ermakov, Univ. Izv. Kiev, 20, No. 9, 1 (1880).

    Google Scholar 

  37. H. R. Lewis and W. B. Riesenfeld, J. Math. Phys., 10, 1458 (1969).

    Article  ADS  Google Scholar 

  38. V. I. Man’ko, “Quantum Mechanics and Classical Probability Theory,” in: B. Gruber and M. Ramek (eds.), Symmetries in Science IX, Plenum Press, New York (1997), p. 215.

    Google Scholar 

  39. I. A. Malkin, V. I. Man’ko, and D. A. Trifonov, Phys. Rev. D, 2, 1371 (1970).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Man’ko, V.I., Safonov, S.S. Quantum Damped Oscillator in Probability Representation. J Russ Laser Res 18, 537–560 (1997). https://doi.org/10.1007/BF03380174

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03380174

Navigation