Article

# Normal Typicality and von Neumann's Quantum Ergodic Theorem

(Impact Factor: 2.19). 07/2009; 466(2123). DOI: 10.1098/rspa.2009.0635
Source: arXiv

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. Comment: 18 pages LaTeX, no figures

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Available from: Sheldon Goldstein
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• "Several papers with interesting discussions about this work appeared recently (see for instance [1], [2] and other papers which mention these two) "
##### Article: A detailed proof of the von Neumann's Quantum Ergodic Theorem
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ABSTRACT: We present a simplified proof of the von Neumann's Quantum Ergodic Theorem. This important result was initially published in german by J. von Neumann in 1929. We are interested here in the time evolution $\psi_t$, $t\geq 0$, (for large times) under the Schrodinger equation associated to a given fixed Hamiltonian $H : \mathcal{H} \to \mathcal{H}$ and a general initial condition $\psi_0$. The dimension of the Hilbert space $\mathcal{H}$ is finite.
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• "This implies the same behavior for an arbitrary ρ(0). This behavior of isolated, macroscopic quantum systems is an instance of a phenomenon we call normal typicality [5], a version of which is expressed in von Neumann's quantum ergodic theorem [17]. However, our result falls outside the scope of von Neumann's theorem, because of the technical assumptions made in that theorem. "
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Physical Review E 01/2010; 81(1 Pt 1):011109. DOI:10.1103/PhysRevE.81.011109 · 2.29 Impact Factor
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• "Recently there has been significant progress in understanding the fundamental principles of statistical mechanics [1] [2] [3] [4] [5] [6] [7] [8]. Underlying this progress is the realization that quantum mechanics allows individual quantum states to exhibit statistical properties, and that ensemble or time averages are not needed to obtain a mixed equilibrium state for the system under consideration . "
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New Journal of Physics 07/2009; 12(5). DOI:10.1088/1367-2630/12/5/055021 · 3.56 Impact Factor