• DOI:10.1098/rspa.2009.0635
• Corpus ID: 816619

Normal typicality and von Neumann’s quantum ergodic theorem

@article{Goldstein2010NormalTA,
  title={Normal typicality and von Neumann’s quantum ergodic theorem},
  author={Sheldon Goldstein and Joel L Lebowitz and Christian Mastrodonato and Roderich Tumulka and Nino Zangh{\'i}},
  journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  year={2010},
  volume={466},
  pages={3203 - 3224}
}

• S. Goldstein, J. Lebowitz, +2 authors N. Zanghí
• Published 1 July 2009
• Mathematics
• Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

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.