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Foundations of Physics

, Volume 25, Issue 11, pp 1599–1620 | Cite as

Non-Heisenberg states of the harmonic oscillator

  • K. Dechoum
  • H. M. FranÇa
Article

Abstract

The effects of the vacuum electromagnetic fluctuations and the radiation reaction fields on the time development of a simple microscopic system are identified using a new mathematical method. This is done by studying a charged mechanical oscillator (frequency Ω0)within the realm of stochastic electrodynamics, where the vacuum plays the role of an energy reservoir. According to our approach, which may be regarded as a simple mathematical exercise, we show how the oscillator Liouville equation is transformed into a Schrödinger-like stochastic equation with a free parameter h′ with dimensions of action. The role of the physical Planck's constant h is introduced only through the zero-point vacuum electromagnetic fields. The perturbative and the exact solutions of the stochastic Schrödinger-like equation are presented for h′>0. The exact solutions for which h′<h are called sub-Heisenberg states. These nonperturbative solutions appear in the form of Gaussian, non-Heisenberg states for which the initial classical uncertainty relation takes the form 〈(δx2)〉〈(δp)2〉=(h′/2)2,which includes the limit of zero indeterminacy (h → 0). We show how the radiation reaction and the vacuum fields govern the evolution of these non-Heisenberg states in phase space, guaranteeing their decay to the stationary state with average energy hΩ0/2 and 〈(δx)2〉〈(δp)2〉=h2/4 at zero temperature. Environmental and thermal effects-are briefly discussed and the connection with similar works within the realm of quantum electrodynamics is also presented. We suggest some other applications of the classical non-Heisenberg states introduced in this paper and we also indicate experiments which might give concrete evidence of these states.

Keywords

Average Energy Quantum Electrodynamic Liouville Equation Stochastic Equation Mechanical Oscillator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    T. H. Boyer,Phys. Rev. D 11, 790 (1975);11, 809 (1975). See also the remarkable paper by T. W. Marshall,Proc. R. Soc. London Ser. A 273, 475 (1963).Google Scholar
  2. 2.
    L. de la Peña, inStochastic Processes Applied to Physics and Other Related Fields, B. Gomezet al., ed. (World Scientific, Singapore, 1982), p. 428. See also L. de la Peña and A. M. Cetto,Found. Phys. 12, 1017 (1982) and P. W. Milonni,Phys. Rep. 25, 1 (1976).Google Scholar
  3. 3.
    P. W. Milonni, inThe Quantum Vacuum: An Introduction to Quantum Electrodynamics (Academic, Boston, 1994).Google Scholar
  4. 4.
    See S. Bergia, P. Lugli, and N. Zamboni,Ann. Found. Louis de Broglie 5, 39 (1980) for a commented translation of the Einstein and Stern 1913 paper. See also P. W. Milonni and M. L. Shih,Am. J. Phys. 59, 684 (1991) for interesting comments concerning the zero-point energy in early quantum theory.Google Scholar
  5. 5.
    W. Nernst,Ver. Deutsch. Phys. Ges. 18, 83 (1916).Google Scholar
  6. 6.
    H. M. FranÇa, T. W. Marshall, and E. Santos,Phys. Rev. A 45, 6436 (1992).Google Scholar
  7. 7.
    H. M. FranÇa, G. C. Marques, and A. J. Silva,Nuovo Cimenta A 48, 65 (1978). See also H. M. FranÇa and G. C. Santos,Nuovo Cimento B 86, 51 (1985) which discusses the radiation reaction in an extended charge within SED.Google Scholar
  8. 8.
    R. Schiller and H. Tesser,Phys. Rev. A 3, 2035 (1971); P. W. Milonni,Am. J. Phys. 52, 340 (1984); W. Eckhardt,Z. Phys. B 64, 515 (1986); and P. W. Milonni,Phys. Scri. T 21, 102 (1988).Google Scholar
  9. 9.
    E. Fichbach, G. L. Greene, and R. L. Hughes,Phys. Rev. Lett. 66, 256 (1991). See also F. Battaglia,Int. J. Theor. Phys. 32, 1401 (1993).Google Scholar
  10. 10.
    E. Wigner,Phys. Rev. 40, 749 (1932). See also C. W. Gardiner, inQuantum Noise (Springer-Verlag, Berlin, 1991), Chapter 4, and G. Manfredi, S. Mola, and M. R. Feix,Eur. J. Phys. 14, 101 (1993).Google Scholar
  11. 11.
    P. Carruthers and M. M. Nieto,Am. J. Phys. 33, 537 (1965).Google Scholar
  12. 12.
    E. A. Power, inNew Frontiers in Quantum Electrodynamics and Quantum Optics, A. O. Barut, ed. (Plenum, New York, 1990), p. 555.Google Scholar
  13. 13.
    H. M. FranÇa and T. W. Marshall,Phys. Rev. A 38, 3258 (1988).Google Scholar
  14. 14.
    J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji,J. Phys. (Paris) 43, 1617 (1982). See also Claude Cohen-Tannoudji,Phys. Scr. T 12, 19 (1986).Google Scholar
  15. 15.
    E. Schrödinger, The continuous transition form micro-to macro-mechanics, inCollected Papers on Wave Mechanics by E. Schrödinger (Blackie, London, 1928), p. 41.Google Scholar
  16. 16.
    G. H. Goedecke,Found. Phys. 14, 41 (1984), and references therein. This author uses an auxiliary parameterh′ with a different meaning.Google Scholar
  17. 17.
    H. M. FranÇa and M. T. Thomaz,Phys. Rev. D 31, 1337 (1985);38, 2651 (1988).Google Scholar
  18. 18.
    P. Schramm and H. Grabert,Phys. Rev. A 34, 4515 (1986), which discusses the effect of dissipation in phase space.Google Scholar
  19. 19.
    M. M. Nieto, in Proceedings of NATO Advanced Study Institute:Frontiers of Non-equilibrium Statistical Physics, G. T. Moore and M. O. Scully, eds. (Plenum, New York, 1986). This paper does not include dissipation.Google Scholar
  20. 20.
    S. Chandrasekhar,Rev. Mod. Phys. 15, 1 (1943).Google Scholar
  21. 21.
    M. C. Wang and G. E. Uhlenbeck,Rev. Mod. Phys. 17, 323 (1945).Google Scholar
  22. 22.
    R. W. Davies and K. T. R. Davies,Ann. Phys. 89, 261 (1975).Google Scholar
  23. 23.
    See M. Kleber,Phys. Rep. 236, 333 (1994), and M. Suárez Barnes, M. Navenberg, M. Nockleby, and S. Tomsovic,J. Phys. A 27, 3299 (1994).Google Scholar
  24. 24.
    F. H. J. Cornish,J. Phys. A 17, 323 (1984). The reduction of the Kepler problem to that of a harmonic oscillator is also discussed by the same author inJ. Phys. A 17, 2191 (1984).Google Scholar
  25. 25.
    J. Ford and G. Mantica,Am. J. Phys. 60, 1086 (1992). In this paper “an experiment, well within current laboratory capability, is proposed which can expose the inability of quantum mechanics to adequately describe macroscopic chaos.”Google Scholar
  26. 26.
    D. Delande,Phys. Scri. T 34, 52 (1991). See also D. Kleppner,Phys. Today 44, August (1991), p. 9.Google Scholar
  27. 27.
    M. Berry, Some quantum-to-classical asymptotic, inChaos and Quantum Physics, Les Houches (1989, M. J. Giannoni, A. Voros, and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1991), p. 251.Google Scholar
  28. 28.
    T. Matsumoto, L. O. Chua and S. Tanaka,Phys. Rev. A 30, 1155 (1984). See also L. Kocarev, K. S. Halle, K. Eckert, and L. O. Chua,Int. J. Bifurc. Chaos 3, 1051 (1993).Google Scholar
  29. 29.
    S. Haroche and D. Kleppner,Phys. Today 42(1), 24 (1989). See also the article “Cavity quantum electrodynamics,” by S. Haroche,Sci. Am., April 1993, p. 26.Google Scholar
  30. 30.
    T. W. Marshall,Nuovo Cimento 38, 206 (1965). See also P. W. Milonni and P. L. Knight,Opt. Comm. 9, 119 (1973).Google Scholar
  31. 31.
    A. M. Cetto and L. de la Peña,Phys. Rev. A 37, 1952 (1988);37, 960 (1988). Jonathan P. Dowling,Found. Phys. 23, 895 (1993).Google Scholar
  32. 32.
    I. R. Senitzky,Phys. Rev. 119, 670 (1960). The stationary regime (γt≫1) is clearly discussed by P. W. Milonni,Am. J. Phys. 49, 177 (1981).Google Scholar
  33. 33.
    E. A. Hinds,Ad. At. Mol. Opt. Phys. 28, 237 (1991).Google Scholar
  34. 34.
    W. Jhe, A. Anderson, E. A. Hinds, D. Mesched, L. Moi, and S. Haroche,Phys. Rev. Lett. 58, 666 (1987).Google Scholar
  35. 35.
    I. M. Suarez Barnes, M. Nauenberg, M. Nockleby, and S. Tomsovic,Phys. Rev. Lett. 71, 1961 (1993). See also “The classical limit of an atom” by M. Nauenberg, C. Stroud, and J. Yeazell,Sce. Am. 270, June 1994, p. 24, and M. Courtney, H. Jiao, N. Spellmeyer, and D. Kleppner,Phys. Rev. Lett. 74, 1538 (1995), which report an experimental and theoretical study of the effect of bifurcation of closed classical orbits in continuum Stark spectra.Google Scholar
  36. 36.
    T. H. Boyer,Phys. Rev. A 29, 2389 (1984).Google Scholar
  37. 37.
    A. V. Barranco, S. A. Brunini, and H. M. FranÇa,Phys. Rev. A 39, 5492 (1989). See also H. M. FranÇa, T. W. Marshall, E. Santos, and E. J. Watson,Phys. Rev. A 46, 2265 (1992) for a semiclassical description of the Stern-Gerlach phenomenon.Google Scholar
  38. 38.
    M. O. Scully, B. G. Englert, and H. Walther,Nature 351, 111 (1991). See also P. Storey, S. Tan, M. Collet, and D. Walls,Nature 367, 626 (1994).Google Scholar
  39. 39.
    P. L. Knight and L. Allen, inConcepts of Quantum Optics (Pergamon, New York, 1985), Chap. 1.Google Scholar
  40. 40.
    P. W. Milonni and M. L. Shih,Contemp. Phys. 33, 313 (1993). See also Ref. 3.Google Scholar
  41. 41.
    D. Cole and H. E. Puthoff,Phys. Rev. E 48, 1562 (1993).Google Scholar
  42. 42.
    R. H. Koch, D. J. Harlinger, and John Clarke,Phys. Rev. Lett. 47, 1216 (1981). See also G. Y. Hu and R. F. O'Connell,Phys. Rev. B 46, 14219 (1992) for other experimental observations of Nyquist noise and zero-point fluctuations in electric circuits.Google Scholar
  43. 43.
    See theScientific Programme, Abstracts, and Outlines of the “International Workshop on the Zeropoint Electromagnetic Field” A. M. Cetto and L. de la Peña, eds. Cuernavaca, México (1993).Google Scholar
  44. 44.
    B. Haisch, A. Rueda, and H. E. Puthoff,Phys. Rev. A 49, 678 (1994). See also the comment “Unbearable lightness” by C. S. Powell inSci. Am., May 1994, p. 14.Google Scholar
  45. 45.
    Claudia Eberlein, in “Sonoluminescence as Quantum Vacuum Radiation,” University of Illinois, Urbana, Illinois 61801-3080, USA, preprint (May 1995). See also S. J. Putterman,Sci. Am. 272, February 1995, p. 32.Google Scholar
  46. 46.
    H. M. FranÇa and A. Maia, Jr., in “Maxwell electromagnetic theory, Planck's radiation law, and Bose-Einstein statistics,” preprint IFUSP (May, 1995), submitted toFound. Phys. Google Scholar
  47. 47.
    K. Dechoum, H. M. FranÇa, and A. Maia, Jr., in “Some observable effects of the current fluctuations in a long solenoid: the significance of the vector potential,” preprint IFUSP (September, 1995), submitted toFound. Phys. Google Scholar
  48. 48.
    A. V. Barranco and H. M. FranÇa,Found. Phys. Lett. 5, 25 (1992).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • K. Dechoum
    • 1
  • H. M. FranÇa
    • 1
  1. 1.Instituto de FisicaUniversidade de SÃo PauloSÃo Paulo, SPBrazil

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