Foundations of Physics

, Volume 45, Issue 3, pp 333–353 | Cite as

Discrete Excitation Spectrum of a Classical Harmonic Oscillator in Zero-Point Radiation

  • Wayne Cheng-Wei Huang
  • Herman Batelaan


We report that upon excitation by a single pulse, a classical harmonic oscillator immersed in the classical electromagnetic zero-point radiation exhibits a discrete harmonic spectrum in agreement with that of its quantum counterpart. This result is interesting in view of the fact that the vacuum field is needed in the classical calculation to obtain the agreement.


Stochastic electrodynamics Quantum mechanics Classical dynamics Vacuum field Harmonic oscillator Excitation 



We gratefully acknowledge comments from Prof. Peter W. Milonni. The funding support comes from NSF Grant No. 0969506. This work was completed utilizing the Holland Computing Center of the University of Nebraska and the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF Grant No. OCI-1053575.

Supplementary material


  1. 1.
    Boyer, T.H.: Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zero-point radiation. Phys. Rev. D 11, 790 (1975)CrossRefADSGoogle Scholar
  2. 2.
    Boyer, T.H.: General connection between random electrodynamics and quantum electrodynamics for free electromagnetic fields and for dipole oscillator systems. Phys. Rev. D 11, 809 (1975)CrossRefADSGoogle Scholar
  3. 3.
    Milonni, P.W.: The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Academic Press, Boston (1994)Google Scholar
  4. 4.
    Cetto, A.M., de la Peña, L., Valdés-Hernández, A.: Quantization as an emergent phenomenon due to matter-zeropoint field interaction. J. Phys. 361, 012013 (2012)Google Scholar
  5. 5.
    Cetto, A.M., de la Peña, L.: The Quantum Dice, an Introduction to Stochastic Electrodynamics. Kluwer, Dordrecht (1996)Google Scholar
  6. 6.
    Boyer, T.H.: Asymptotic retarded van der Waals forces derived from classical electrodynamics with classical electromagnetic zero-point radiation. Phys. Rev. A 5, 1799 (1972)CrossRefADSGoogle Scholar
  7. 7.
    Dressel, J., Bliokh, K.Y., Nori, F.: Classical field approach to quantum weak measurements. Phys. Rev. Lett. 112, 110407 (2014)CrossRefADSGoogle Scholar
  8. 8.
    Milonni, P.W.: The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Academic Press, Boston (1994)Google Scholar
  9. 9.
    P. W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics, pp. 51–54, 123–128, 487–488. Academic Press, Boston (1994)Google Scholar
  10. 10.
    Griffiths, D.J.: Introduction to Electrodynamics, 3rd edn. Prentice-Hall, Upper Saddle River (1999)Google Scholar
  11. 11.
    Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields, 4th edn, p. 207. Pergamon Press, New York (1987). (Eq. 75.10)Google Scholar
  12. 12.
    Jackson, J.D.: Classical Electrodynamics, 3rd edn, p. 749. Wiley, New York (1998). (Eq. 16.10)Google Scholar
  13. 13.
    Huang, W., Batelaan, H.: Dynamics underlying the Gaussian distribution of the classical harmonic oscillator in zero-point radiation. J. Comput. Methods Phys. 2013, 308538 (2013)CrossRefGoogle Scholar
  14. 14.
    Thornton, S.T., Marion, J.B.: Classical Dynamics of Particles and Systems, 5th edn, pp. 117–128. Brooks/Cole, Belmont (2004)Google Scholar
  15. 15.
    Landau, L.D., Lifshitz, E.M.: Mechanics, 3rd edn, p. 80. Butterworth-Heinemann, Oxford (1976)Google Scholar
  16. 16.
    Yariv, A.: Quantum Electronics, 3rd edn, p. 407. Wiley, New York (1988)Google Scholar
  17. 17.
    Heisenberg, W.: Über quantentheoretische Umdeutung kinematischer und mechanischer Besiehungen. Z. Phys. 33, 879 (1925)CrossRefADSzbMATHGoogle Scholar
  18. 18.
    Aitchison, I.J.R., MacManus, D.A., Snyder, T.M.: Understanding Heisenbergs magical paper of July 1925: a new look at the calculational details. Am. J. Phys. 72, 11 (2004)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Averbukh, V., Moiseyev, N.: Classical versus quantum harmonic-generation spectrum of a driven anharmonic oscillator in the high-frequency regime. Phys. Rev. A 57, 1345 (1998)CrossRefADSGoogle Scholar
  20. 20.
    Griffiths, D.J.: Introduction to Quantum Mechanics, 2nd edn, p. 366. Upper Saddle River, Pearson Prentice Hall (2005)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Physics and AstronomyTexas A&M UniversityCollege StationUSA
  2. 2.Department of Physics and AstronomyUniversity of Nebraska-LincolnLincolnUSA

Personalised recommendations