Foundations of Physics

, 39:256 | Cite as

Complete Hamiltonian Description of Wave-Like Features in Classical and Quantum Physics

Article

Abstract

The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system describing in terms of ray trajectories, for a stationary refractive medium, a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the geometrical optics (“eikonal”) approximation, which is contained as a simple limiting case. Due to the fact, moreover, that the time independent Schrödinger equation is itself a Helmholtz-like equation, the same mathematics holding for a classical optical beam turns out to apply to a quantum particle beam moving in a stationary force field, and leads to a system of Hamiltonian equations providing exact and deterministic particle trajectories and dynamical laws, and containing the laws of Classical Mechanics in the eikonal limit.

Keywords

Geometrical optics Hamilton equations Quantum foundations Indeterminism 

References

  1. 1.
    de Broglie, L.: Nature 118, 441 (1926) CrossRefADSGoogle Scholar
  2. 2.
    de Broglie, L.: J. Phys. Radium 8, 225 (1927) CrossRefGoogle Scholar
  3. 3.
    de Broglie, L.: Une Tentative d’Interprétation Causale et Non-Linéaire de la Mécanique Ondulatoire. Gauthier-Villars (1956) Google Scholar
  4. 4.
    Bohm, D.J.: Phys. Rev. 85, 166 (1952) CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Bohm, D.J.: Phys. Rev. 89, 458 (1953) MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Goldstein, H.: Classical Mechanics. Addison Wesley (1965) Google Scholar
  7. 7.
    Airoldi, A., Orefice, A., Ramponi, G.: Il Nuovo Cimento D 6, 527 (1985) CrossRefADSGoogle Scholar
  8. 8.
    Airoldi, A., Orefice, A., Ramponi, G.: Phys. Fluids B 11, 2143 (1989) CrossRefADSGoogle Scholar
  9. 9.
    Messiah, A.: Mécanique Quantique. Dunod (1959) Google Scholar
  10. 10.
    Fowles, G.R.: Introduction to Modern Optics. Dover (1975) Google Scholar
  11. 11.
    Friberg, A.T., Jaakkola, T., Tuovinen, J.: IEEE Trans. Antennas Propag. 40, 984 (1992) CrossRefADSGoogle Scholar
  12. 12.
    Philippidis, C., Dewdney, D., Hiley, B.J.: Il Nuovo Cimento B 52, 15 (1979) CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Bohm, D.J., Hiley, B.J.: Found. Phys. 12, 1001 (1982) CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1993) Google Scholar
  15. 15.
    Holland, P.R.: Ann. Phys. 315, 505 (2005) MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Sanz, A.S., Borondo, F., Miret-Artès, S.: Phys. Rev. B 61, 7743 (2000) CrossRefADSGoogle Scholar
  17. 17.
    Sanz, A.S., Borondo, F., Miret-Artès, S.: J. Chem. Phys. 120, 8794 (2004) CrossRefADSGoogle Scholar
  18. 18.
    Sanz, A.S., Miret-Artès, S.: Chem. Phys. Lett. 445, 350 (2007) CrossRefADSGoogle Scholar
  19. 19.
    Micha, D.A.: J. Chem. Phys. 78, 7138 (1983) CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Cohen, J.M., Micha, D.A.: J. Chem. Phys. 98, 2023 (1993) CrossRefADSGoogle Scholar
  21. 21.
    Nielsen, S., Kapral, R., Ciccotti, G.: J. Stat. Phys. 101, 225 (2000) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Markovic, N., Bäck, A.: J. Phys. Chem. A 108, 8765 (2004) CrossRefGoogle Scholar
  23. 23.
    Nowak, S., Orefice, A.: Phys. Fluids B 5, 1945 (1993) CrossRefADSGoogle Scholar
  24. 24.
    Nowak, S., Orefice, A.: Phys. Plasmas 1, 1242 (1994) CrossRefADSGoogle Scholar
  25. 25.
    Orefice, A., Nowak, S.: Phys. Essays 10, 364 (1997) CrossRefGoogle Scholar
  26. 26.
    Bohm, D.J.: Wholeness and the Implicate Order. Routledge & Kegan Paul (1980) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Di.Pro.Ve.Università di MilanoMilanItaly

Personalised recommendations