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Note on Liouville's Theorem and the Heisenberg Uncertainty Principle

Published online by Cambridge University Press:  14 March 2022

J. H. Van Vleck
J. H. Van Vleck*
Affiliation:
Harvard University
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Extract

It is well known that, in classical theory, Liouville's theorem shows that if an ensemble of systems is distributed over a small element of volume in phase space, the ensemble fills a region of equal volume at all later instants of time. In quantum mechanics, the uncertainty principle is associated with the products of the errors in conjugate coordinates and momenta, and such products can be interpreted in terms of volume elements in phase space. Comparison of these two facts leads to the assertion in the interesting volume on “Statistical Mechanics” recently published by J. E. and M. G. Mayer that “The Liouville theorem is essential for the complete understanding of the uncertainty principle.

Type
Research Article
Information
Philosophy of Science , Volume 8 , Issue 2 , April 1941 , pp. 275 - 279
Copyright
Copyright © The Philosophy of Science Association 1941

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References

1 Mayer, J. E. and Mayer, M. G., “Statistical Mechanics”, p. 61.Google Scholar

2 Cf., for instance, Heisenberg, W., “The Physical Principles of Quantum Theory”, pp. 18 and 68; Kemble, E. C., “The Principles of Quantum Mechanics”, pp. 223-224.Google Scholar

3 We have inserted the factor 2 in the definition (1) of the fluctuations in order that our usage agree with that of Mayer or Heisenberg. This factor is, however, rather arbitrary, and is not included by Kemble, whose minimum uncertainty is consequently h/4π rather than h/2π.Google Scholar

4 For the underlying theory, see, for example, Kemble, E. C., “The Principles of Quantum Mechanics”, sections 9a and 33b.Google Scholar

5 London, F., Zeitschrift f. Physik, 40, 193 (1927); Dirac, P. A. M., Proc. Roy. Soc., 110, 561; 111, 281 (1926).CrossRefGoogle Scholar