Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem

TL;DR

This paper revisits von Neumann's 1929 work on the quantum ergodic theorem, highlighting its insights into the long-time behavior of macroscopic quantum systems and clarifying its significance in quantum statistical mechanics.

Contribution

It provides an English translation and commentary on von Neumann's original article, emphasizing the theorem's importance and correcting common misunderstandings.

Findings

The quantum ergodic theorem demonstrates typicality in the long-time behavior of quantum systems.

Most initial states evolve so that observables approach micro-canonical distributions.

The quantum H-theorem is weaker than Boltzmann's classical H-theorem.

Abstract

The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann's 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the "quantum H-theorem", is actually a much weaker statement than Boltzmann's classical H-theorem, the other theorem, which he calls the "quantum ergodic theorem", is a beautiful and very non-trivial result. It expresses a fact we call "normal typicality" and can be summarized as follows: For a "typical" finite family of commuting macroscopic observables, every initial wave function $\psi_0$ from a micro-canonical energy shell so evolves that for most times $t$ in the long run, the joint probability distribution of these observables obtained from $\psi_t$ is close to their micro-canonical distribution.