Abstract
From classical stochastic equations of motion, we derive the quantum Schrödinger equation. The derivation is carried out by assuming that the real and imaginary parts of the wave function \(\phi\) are proportional to the coordinates and momenta associated with the degrees of freedom of an underlying classical system. The wave function \(\phi\) is assumed to be a complex time-dependent random variable that obeys a stochastic equation of motion that preserves the norm of \(\phi\). The quantum Liouville equation is obtained by considering that the stochastic part of the equation of motion changes the phase of \(\phi\) but not its absolute value. The Schrödinger equation follows from the Liouville equation. The wave function \(\psi\) obeying the Schrödinger equation is related to the stochastic wave function by \(|\psi |^2=\langle |\phi |^2\rangle\).
Similar content being viewed by others
References
D. Landau, E.M. Lifshitz, Quantum mechanics (Pergamon Press, London, 1958)
E. Merzbacher, Quantum mechanics (Wiley, New york, 1961)
A. Messiah, Quantum mechanics, 2 vols (Wiley, New York, 1961)
J.J. Sakurai, Advanced quantum mechanics (Addison-Wesley, Reading, 1967)
J.J. Sakurai, Modern quantum mechanics (Addison-Wesley, Reading, 1994)
D.J. Griffiths, Introduction to quantum mechanics (Prentice Hall, Upper Saddle River, 1995)
A.F.R. de Toledo Piza, Mecânica Quântica (Editora da Universidade de São Paulo, São Paulo, 2002)
C. Lanczos, The variational principles of mechanics (University of Toronto Press, Toronto, 1949)
H. Goldstein, Classical mechanics (Addison-Wesley, Cambridge, 1950)
L.D. Landau, E.M. Lifshitz, Mechanics (Pergamon, Oxford, 1960)
V.I. Arnold, Mathematical methods of classical mechanics (Springer, New York, 1978)
B.O. Koopman, Proc. Natl. Acad. Sci. 17, 315 (1931)
J. von Neumann, Ann. Math. 33(587), 789 (1932)
E. Nelson, Phys. Rev. 150, 1079 (1966)
T.C. Wallstrom, Found. Phys. Lett. 2, 113 (1989)
N.G. van Kampen, Stochastic processes in physics and chemistry (North-Holland, Amsterdam, 1981)
H. Risken, The Fokker-Planck equation, 2nd edn. (Springer, Berlin, 1989)
C. Gardiner, Stochastic methods, 4th edn. (Springer, Berlin, 2009)
T. Tomé, M.J. de Oliveira, Stochastic dynamics and irreversibility (Springer, Cham, 2015)
G. Lindblad, Commun. Math. Phys. 48, 119 (1976)
H.-P. Breuer, F. Petruccione, The theory of open quantum systems (Clarendon Press, Oxford, 2002)
R. Omnès, The interpretation of quantum mechanics (Princeton University Press, Princeton, 1994)
G. Auletta, Foundations and interpretation of quantum mechanics (World Scientific, Singapore, 2001)
O. Freire Jr. (ed.), The history of quantum interpretations (Oxford University Press, Oxford, 2022)
L. de Broglie, Journal de Physique 8, 225 (1927)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The author declares no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
de Oliveira, M.J. Derivation of the Schrödinger Equation from Classical Stochastic Dynamics. Braz J Phys 54, 52 (2024). https://doi.org/10.1007/s13538-024-01416-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13538-024-01416-y