Normal typicality and von Neumann’s quantum ergodic theorem

By Sheldon Goldstein, Joel Lebowitz, C. Mastrodonato, Roderich Tumulka, Nino Zanghí. Proceedings of the Royal Society A, 2009

139 citations recorded by the source corpus.

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 ψ0 from an energy shell is ‘normal’: it evolves in such a way that |ψt⟩⟨ψt| 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.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/bd437017a981e7855e3e077bda568cba87398c12