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 Ω 0)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 〈(δx 2)〉〈(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 hΩ 0 /2 and 〈(δx) 2 〉〈(δp) 2 〉=h 2 /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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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).
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).
P. W. Milonni, inThe Quantum Vacuum: An Introduction to Quantum Electrodynamics (Academic, Boston, 1994).
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.
W. Nernst,Ver. Deutsch. Phys. Ges. 18, 83 (1916).
H. M. FranÇa, T. W. Marshall, and E. Santos,Phys. Rev. A 45, 6436 (1992).
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.
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).
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).
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).
P. Carruthers and M. M. Nieto,Am. J. Phys. 33, 537 (1965).
E. A. Power, inNew Frontiers in Quantum Electrodynamics and Quantum Optics, A. O. Barut, ed. (Plenum, New York, 1990), p. 555.
H. M. FranÇa and T. W. Marshall,Phys. Rev. A 38, 3258 (1988).
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).
E. Schrödinger, The continuous transition form micro-to macro-mechanics, inCollected Papers on Wave Mechanics by E. Schrödinger (Blackie, London, 1928), p. 41.
G. H. Goedecke,Found. Phys. 14, 41 (1984), and references therein. This author uses an auxiliary parameterh′ with a different meaning.
H. M. FranÇa and M. T. Thomaz,Phys. Rev. D 31, 1337 (1985);38, 2651 (1988).
P. Schramm and H. Grabert,Phys. Rev. A 34, 4515 (1986), which discusses the effect of dissipation in phase space.
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.
S. Chandrasekhar,Rev. Mod. Phys. 15, 1 (1943).
M. C. Wang and G. E. Uhlenbeck,Rev. Mod. Phys. 17, 323 (1945).
R. W. Davies and K. T. R. Davies,Ann. Phys. 89, 261 (1975).
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).
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).
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.”
D. Delande,Phys. Scri. T 34, 52 (1991). See also D. Kleppner,Phys. Today 44, August (1991), p. 9.
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.
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).
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.
T. W. Marshall,Nuovo Cimento 38, 206 (1965). See also P. W. Milonni and P. L. Knight,Opt. Comm. 9, 119 (1973).
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).
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).
E. A. Hinds,Ad. At. Mol. Opt. Phys. 28, 237 (1991).
W. Jhe, A. Anderson, E. A. Hinds, D. Mesched, L. Moi, and S. Haroche,Phys. Rev. Lett. 58, 666 (1987).
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.
T. H. Boyer,Phys. Rev. A 29, 2389 (1984).
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.
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).
P. L. Knight and L. Allen, inConcepts of Quantum Optics (Pergamon, New York, 1985), Chap. 1.
P. W. Milonni and M. L. Shih,Contemp. Phys. 33, 313 (1993). See also Ref. 3.
D. Cole and H. E. Puthoff,Phys. Rev. E 48, 1562 (1993).
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.
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).
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.
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.
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.
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.
A. V. Barranco and H. M. FranÇa,Found. Phys. Lett. 5, 25 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dechoum, K., FranÇa, H.M. Non-Heisenberg states of the harmonic oscillator. Found Phys 25, 1599–1620 (1995). https://doi.org/10.1007/BF02055510
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02055510