Normal typicality and von Neumann's quantum ergodic theorem
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 microcanonical 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.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 November 2010
 DOI:
 10.1098/rspa.2009.0635
 arXiv:
 arXiv:0907.0108
 Bibcode:
 2010RSPSA.466.3203G
 Keywords:

 Quantum Physics;
 Condensed Matter  Statistical Mechanics
 EPrint:
 18 pages LaTeX, no figures