Normal Typicality and von Neumann's Quantum Ergodic Theorem

TL;DR

The paper revisits von Neumann's 1929 quantum ergodic theorem, clarifying its significance in quantum mechanics and correcting historical misconceptions about its implications for typical system behavior.

Contribution

It clarifies the original QET's meaning, corrects past criticisms, and presents a stronger, more precise formulation of normal typicality in quantum systems.

Findings

The QET is a significant mathematical result in quantum mechanics.

Previous criticisms misrepresented the QET as vacuous.

A tighter bound on deviations from the average is derived.

Abstract

We discuss the content and significance of John von Neumann's quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e., the statement that, for typical large systems, every initial wave function $\psi_0$ from an energy shell is "normal": it evolves in such a way that $|\psi_t> <\psi_t|$ is, for most $t$, macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.