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. © 2010 The Royal Society.
Normal Typicality and von Neumann's Quantum Ergodic Theorem
ZANGHI', PIERANTONIO
2010-01-01
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 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. © 2010 The Royal Society.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.