Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T00:37:34.834Z Has data issue: false hasContentIssue false

Note on Liouville's Theorem and the Heisenberg Uncertainty Principle

Published online by Cambridge University Press:  14 March 2022

J. H. Van Vleck*
Affiliation:
Harvard University

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
Copyright
Copyright © The Philosophy of Science Association 1941

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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