Skip to main content
Log in

Imprints of the Quantum World in Classical Mechanics

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerous than is usually believed. We show that the Schrödinger equation for a nonrelativistic spinless particle is a classical equation which is equivalent to Hamilton’s equations. Our discussion is quite general, and incorporates time-dependent systems. This gives us the opportunity of discussing the group of Hamiltonian canonical transformations which is a non-linear variant of the usual symplectic group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, vol. 75. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  2. Banyaga, A.: Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. Comment. Math. Helv. 53, 174–227 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Derbes, D.: Feynman’s derivation of the Schrödinger equation. Am. J. Math. Phys. 64(7), 881–884 (1996)

    MathSciNet  ADS  MATH  Google Scholar 

  4. Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)

    Article  MathSciNet  ADS  Google Scholar 

  5. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. III, pp. 16–22. Addison-Wesley, Reading (1965)

    MATH  Google Scholar 

  6. de Gosson, M.: The Principles of Newtonian and Quantum Mechanics. Imperial College Press, London (2001), with a Foreword by B. Hiley

    Book  MATH  Google Scholar 

  7. de Gosson, M.: Symplectic Geometry and Quantum Mechanics. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  8. de Gosson, M.: The symplectic camel and the uncertainty principle: the tip of an iceberg? Found. Phys. 99, 194–214 (2009)

    Article  Google Scholar 

  9. de Gosson, M., Luef, F.: Symplectic capacities and the geometry of uncertainty: the irruption of symplectic topology in classical and quantum mechanics. Phys. Rep. 484, 131–179 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  10. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    MATH  Google Scholar 

  11. Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Guillemin, V., Sternberg, V.S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984)

    MATH  Google Scholar 

  13. Hall, M.J.W., Reginatto, M.: Schrödinger equation from an exact uncertainty principle. J. Phys. A, Math. Gen. 35, 3289 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Kastler, D.: The C -algebras of a free boson field. Commun. Math. Phys. 1, 114–48 (1965)

    Article  MathSciNet  Google Scholar 

  15. Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1949). Reprint 4th edn. New York: Dover Publications (1986)

    MATH  Google Scholar 

  16. Littlejohn, R.G.: The semiclassical evolution of wave packets. Phys. Rep. 138(4–5), 193–291 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  17. Loupias, G., Miracle-Sole, S.: C -algèbres des systèmes canoniques, I. Commun. Math. Phys. 2, 31–48 (1966)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Loupias, G., Miracle-Sole, S.: C -algèbres des systèmes canoniques, II. Ann. Inst. Henri Poincaré 6(1), 39–58 (1967)

    MathSciNet  MATH  Google Scholar 

  19. Mackey, G.W.: The relationship between classical and quantum mechanics. In: Contemporary Mathematics, vol. 214. Am. Math. Soc., Providence (1998)

    Google Scholar 

  20. Narcowich, F.J., O’Connell, R.F.: Necessary and sufficient conditions for a phase-space function to be a Wigner distribution. Phys. Rev. A 34(1), 1–6 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  21. Narcowich, F.J.: Geometry and uncertainty. J. Math. Phys. 31(2) (1990)

  22. Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. A 150(4), 1079–1085 (1966)

    Article  ADS  Google Scholar 

  23. Polterovich, L.: The Geometry of the Group of Symplectic Diffeomorphisms. Lectures in Mathematics. Birkhäuser, Basel (2001)

    Book  Google Scholar 

  24. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York (1972)

    MATH  Google Scholar 

  25. Schmelzer, I.: Why the Hamiltonian operator alone is not enough. Found. Phys. 39(5), 486–498 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Schmelzer, I.: Pure quantum interpretations are not viable. Found. Phys. (2010)

  27. Schrödinger, E.: Quantisierung als Eigenwertproblem. Ann. Phys. 384, 361–376 (1926)

    Article  Google Scholar 

  28. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987) [original Russian edition in Nauka, Moskva (1978)]

    Book  MATH  Google Scholar 

  29. Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  30. Stone, M.H.: Linear transformations in Hilbert space. III: Operational methods and group theory. Proc. Natl. Acad. Sci. USA 172–175 (1930)

  31. Struckmeier, J.: Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems. J. Phys. A, Math. Gen. 38, 1257–1278 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Synge, J.L.: Encyclopedia of Physics, vol. 3/1. Springer, Berlin (1960). ed. S. Flügge

    Google Scholar 

  33. Wong, M.W.: Weyl Transforms. Springer, Berlin (1998)

    MATH  Google Scholar 

  34. Zeilinger, A.: http://www.metanexus.org/ultimate_reality/zeilinger.pdf

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maurice A. de Gosson.

Additional information

Maurice de Gosson has been financed by the Austrian Research Agency FWF (Projekt “Symplectic Geometry and Applications to TFA and QM”, Projektnummer P20442-N13).

Rights and permissions

Reprints and permissions

About this article

Cite this article

de Gosson, M.A., Hiley, B.J. Imprints of the Quantum World in Classical Mechanics. Found Phys 41, 1415–1436 (2011). https://doi.org/10.1007/s10701-011-9544-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-011-9544-5

Keywords

Navigation