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Journal of Mathematical Physics

Mean Ergodic Theorems in Quantum Mechanics

J. E. Moyal

Published Mar 1, 1969• 9 citations
Publisher

Abstract

An application of the abstract mean ergodic theorems to quantum systems is described, which is rather closely analogous to the application of these same theorems in classical statistical mechanics.
DOI
10.1063/1.1664868

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An ergodic theorem is a statement of the equality of the time averages of the physical properties of a system and the averages of these quantities obtained from the consideration of a Gibbsian ensemble of identical systems. It is shown that, in quantum statistical mechanics, a necessary and sufficient condition for the existence of an ergodic theorem can be derived for a system in weak energetic interaction with its surroundings. This necessary and sufficient condition for the equality of time and (microcanonical) ensemble averages is that all operators invariant in time shall reduce to constant multiples of the unit operator when applied to the system itself. It is shown that the conditions for ergodicity are formally analogous to those derived by von Neumann for classical statistical mechanics. No assumption of "molecular chaos" is made.