Normal Typicality and von Neumann's Quantum Ergodic Theorem

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences (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, Sep 25, 2015
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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) "
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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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    • "The QET asserts that the time average of (20) is no greater than (16) (independently of ψ 0 ), and, moreover, that the average of (16) over all D with dim H ν = d ν is small when d ν is sufficiently large. More detailed discussions of the QET have been provided by Pauli and Fierz in 1937 [50] and by Jancel in 1963 [30]; see also [22] "
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    ABSTRACT: Long-time behavior of macroscopic quantum systems Commentary accompanying the English translation of John von Neumann’s 1929 article on the quantum ergodic theorem
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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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    ABSTRACT: We consider an isolated macroscopic quantum system. Let H be a microcanonical "energy shell," i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+deltaE . The thermal equilibrium macrostate at energy E corresponds to a subspace H(eq) of H such that dim H(eq)/dim H is close to 1. We say that a system with state vector psi is the element of H is in thermal equilibrium if psi is "close" to H(eq). We show that for "typical" Hamiltonians with given eigenvalues, all initial state vectors psi(0) evolve in such a way that psi(t) is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.
    Physical Review E 01/2010; 81(1 Pt 1):011109. DOI:10.1103/PhysRevE.81.011109 · 2.29 Impact Factor
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