Journal of Scientific Computing

, Volume 39, Issue 1, pp 1–27 | Cite as

Subharmonic Resonance Behavior for the Classical Hydrogen Atomic System



Previously unexplored resonance conditions are shown to exist for the classical hydrogen atomic system, where the electron is treated as a classical charged point particle following the nonrelativistic Lorentz-Dirac equation of motion about a stationary nucleus of opposite charge. For circularly polarized (CP) light directed normal to the orbit, very pronounced subharmonic resonance behavior is shown to occur with a variety of interesting properties. In particular, only if the amplitude of the CP light exceeds a critical value, will the resonance continue without radius and energy decay. A perturbation analysis is carried out to illustrate the main features of the behavior. The present phenomena adds to a growing list of other nonlinear dynamical behaviors of this simple system, that may well be important for more deeply understanding classical and quantum connections.


Hydrogen Rydberg Stochastic Electrodynamics Simulation Classical Nonlinear 


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  1. 1.
    Cole, D.C., Zou, Y.: Simulation study of aspects of the classical hydrogen atom interacting with electromagnetic radiation: Circular orbits. J. Sci. Comput. 20(1), 43–68 (2004) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cole, D.C., Zou, Y.: Simulation study of aspects of the classical hydrogen atom interacting with electromagnetic radiation: Elliptical orbits. J. Sci. Comput. 20(3), 379–404 (2004). Preprint available at MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cole, D.C., Zou, Y.: Perturbation analysis and simulation study of the effects of phase on the classical hydrogen atom interacting with circularly polarized electromagnetic radiation. J. Sci. Comput. 21(2), 145–172 (2004). Preprint available at MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cole, D.C., Zou, Y.: Analysis of orbital decay time for the classical hydrogen atom interacting with circularly polarized electromagnetic radiation. Phys. Rev. E 69(1), 16601 (2004) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cole, D.C., Zou, Y.: Quantum mechanical ground state of hydrogen obtained from classical electrodynamics. Phys. Lett. A 317(1–2), 14–20 (2003) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    de la Peña, L., Cetto, A.M.: The Quantum Dice—An Introduction to Stochastic Electrodynamics. Kluwer Academic, Dordrecht (1996) Google Scholar
  7. 7.
    Cole, D.C.: Reviewing and extending some recent work on stochastic electrodynamics. In: Lakhtakia, A. (ed.) Essays on the Formal Aspects of Electromagnetic Theory, pp. 501–532. World Scientific, Singapore (1993) Google Scholar
  8. 8.
    Noel, M.W., Griffith, W.M., Gallagher, T.F.: Classical subharmonic resonances in microwave ionization of lithium Rydberg atoms. Phys. Rev. A 62, 063401 (2000) CrossRefGoogle Scholar
  9. 9.
    Koch, P.M., van Leeuwen, K.A.H.: The importance of resonances in microwave “ionization” of excited hydrogen atoms. Phys. Rep. 255, 289–403 (1995) CrossRefGoogle Scholar
  10. 10.
    Maeda, H., Norum, D.V.L., Gallagher, T.F.: Microwave manipulation of an atomic electron in a classical orbit. Science 307, 1757–1760 (2005) CrossRefGoogle Scholar
  11. 11.
    Teitelboim, C., Villarroel, D., van Weert, Ch.G.: Classical electrodynamics of retarded fields and point particles. Riv. Nuovo Cim. 3(9), 1–64 (1980) CrossRefGoogle Scholar
  12. 12.
    Boyer, T.H.: Random electrodynamics: The theory of classical electrodynamics with classical electromagnetic zero–point radiation. Phys. Rev. D 11(4), 790–808 (1975) CrossRefGoogle Scholar
  13. 13.
    Cole, D.C.: Derivation of the classical electromagnetic zero–point radiation spectrum via a classical thermodynamic operation involving van der Waals forces. Phys. Rev. A 42, 1847–1862 (1990) CrossRefGoogle Scholar
  14. 14.
    Cole, D.C.: Reinvestigation of the thermodynamics of blackbody radiation via classical physics. Phys. Rev. A 45, 8471–8489 (1992) CrossRefGoogle Scholar
  15. 15.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979) MATHGoogle Scholar
  16. 16.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007) MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Dept. of Manufacturing EngineeringBoston UniversityBrooklineUSA
  2. 2.One AMD PlaceSunnyvaleUSA

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