Foundations of Physics

, Volume 41, Issue 9, pp 1415–1436

Imprints of the Quantum World in Classical Mechanics



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.


Quantization Schrödinger’s equation Hamiltonian flows Symplectic covariance of Weyl calculus Stone’s theorem 


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  1. 1.
    Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, vol. 75. Springer, Berlin (1988) MATHCrossRefGoogle Scholar
  2. 2.
    Banyaga, A.: Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. Comment. Math. Helv. 53, 174–227 (1978) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Derbes, D.: Feynman’s derivation of the Schrödinger equation. Am. J. Math. Phys. 64(7), 881–884 (1996) MathSciNetADSMATHGoogle Scholar
  4. 4.
    Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948) MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. III, pp. 16–22. Addison-Wesley, Reading (1965) MATHGoogle Scholar
  6. 6.
    de Gosson, M.: The Principles of Newtonian and Quantum Mechanics. Imperial College Press, London (2001), with a Foreword by B. Hiley MATHCrossRefGoogle Scholar
  7. 7.
    de Gosson, M.: Symplectic Geometry and Quantum Mechanics. Birkhäuser, Basel (2006) MATHGoogle Scholar
  8. 8.
    de Gosson, M.: The symplectic camel and the uncertainty principle: the tip of an iceberg? Found. Phys. 99, 194–214 (2009) CrossRefGoogle Scholar
  9. 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) MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001) MATHGoogle Scholar
  11. 11.
    Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985) MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Guillemin, V., Sternberg, V.S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984) MATHGoogle Scholar
  13. 13.
    Hall, M.J.W., Reginatto, M.: Schrödinger equation from an exact uncertainty principle. J. Phys. A, Math. Gen. 35, 3289 (2002) MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Kastler, D.: The C -algebras of a free boson field. Commun. Math. Phys. 1, 114–48 (1965) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1949). Reprint 4th edn. New York: Dover Publications (1986) MATHGoogle Scholar
  16. 16.
    Littlejohn, R.G.: The semiclassical evolution of wave packets. Phys. Rep. 138(4–5), 193–291 (1986) MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Loupias, G., Miracle-Sole, S.: C -algèbres des systèmes canoniques, I. Commun. Math. Phys. 2, 31–48 (1966) MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Loupias, G., Miracle-Sole, S.: C -algèbres des systèmes canoniques, II. Ann. Inst. Henri Poincaré 6(1), 39–58 (1967) MathSciNetMATHGoogle Scholar
  19. 19.
    Mackey, G.W.: The relationship between classical and quantum mechanics. In: Contemporary Mathematics, vol. 214. Am. Math. Soc., Providence (1998) Google Scholar
  20. 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) MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Narcowich, F.J.: Geometry and uncertainty. J. Math. Phys. 31(2) (1990) Google Scholar
  22. 22.
    Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. A 150(4), 1079–1085 (1966) ADSCrossRefGoogle Scholar
  23. 23.
    Polterovich, L.: The Geometry of the Group of Symplectic Diffeomorphisms. Lectures in Mathematics. Birkhäuser, Basel (2001) CrossRefGoogle Scholar
  24. 24.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York (1972) MATHGoogle Scholar
  25. 25.
    Schmelzer, I.: Why the Hamiltonian operator alone is not enough. Found. Phys. 39(5), 486–498 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Schmelzer, I.: Pure quantum interpretations are not viable. Found. Phys. (2010) Google Scholar
  27. 27.
    Schrödinger, E.: Quantisierung als Eigenwertproblem. Ann. Phys. 384, 361–376 (1926) CrossRefGoogle Scholar
  28. 28.
    Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987) [original Russian edition in Nauka, Moskva (1978)] MATHCrossRefGoogle Scholar
  29. 29.
    Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993) MATHGoogle Scholar
  30. 30.
    Stone, M.H.: Linear transformations in Hilbert space. III: Operational methods and group theory. Proc. Natl. Acad. Sci. USA 172–175 (1930) Google Scholar
  31. 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) MathSciNetADSMATHCrossRefGoogle Scholar
  32. 32.
    Synge, J.L.: Encyclopedia of Physics, vol. 3/1. Springer, Berlin (1960). ed. S. Flügge Google Scholar
  33. 33.
    Wong, M.W.: Weyl Transforms. Springer, Berlin (1998) MATHGoogle Scholar
  34. 34.

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.NuHAG, Fakultät für MathematikUniversität WienWienAustria
  2. 2.TPRU, BirkbeckUniversity of LondonLondonUK

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