Normal Typicality and von Neumann's Quantum Ergodic Theorem

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences (Impact Factor: 2). 07/2009; 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

  • [Show abstract] [Hide abstract]
    ABSTRACT: In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro-canonical ``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+δE. The thermal equilibrium macro-state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ɛ H is in thermal equilibrium if ψ is ``close'' to Heq. We argue that for ``typical'' Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann's quantum ergodic theorem of 1929.
  • [Show abstract] [Hide abstract]
    ABSTRACT: The dissipative effects of a quantum harmonic oscillator, initially set in a coherent state and linearly coupled to a continuous distribution of frequency modes, are analyzed over long time scales in relation to the behavior of the spectral density near an arbitrary band gap, arbitrarily shaped at the higher frequencies. The reservoir is initially set either in the vacuum state or in continuous distributions of coherent states. These distributions are arbitrarily shaped at high frequencies and structured in sub- or super-ohmic configurations near an arbitrary band gap frequency. Similarly to certain decoherence processes of a qubit, critical conditions emerge, such that arbitrarily slow inverse power law relaxations of the expectation values of the observables, are obtained by approaching the boundary between the sub- and the super-ohmic regimes. Also, in such critical conditions, a trapping of the number of excitations appears in the super-ohmic regime. The technique of critical frequency control, emerging in the scenario of the environment-induced decoherence of a qubit via the reservoir engineering approach, is extended to the harmonic quantum Brownian motion.
    Journal of Physics A Mathematical and Theoretical 12/2012; 46(1):015304. · 1.77 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: The irreversible relaxation to equilibrium is explained for macroscopic quantum systems with an emphasis on the behavior of expectation values and extremely high dimensionality of the Hilbert space. We consider a large but isolated system that is initially out of equilibrium and eventually relaxes to equilibrium. The relaxation is described by the deviation of the expectation value of the quantity of interest from the long-time average. After relaxation, the amount of deviation from equilibrium is discussed based on probabilistic arguments, which are available for nonintegrable systems. We also evaluate how long the system stays near equilibrium.
    Physica Scripta Volume T. 11/2012;

Full-text (2 Sources)

Available from
May 15, 2014