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.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.

Jan 1, 200979 citations

Normal typicality and von Neumann’s quantum

Sheldon Goldstein, Joel L. Lebowitz, Christian Mastrodonato

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 0 from an energy shell is “normal”: it evolves in such a way that j tih tjis, 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.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 0 from an energy shell is “normal”: it evolves in such a way that j tih tjis, 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.

Jan 1, 20100 citations

On the approach to thermal equilibrium of macroscopic quantum systems

Sheldon Goldstein, Roderich Tumulka, Pedro L. Garrido +2 more

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 ε 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.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 ε 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.

AIP conference proceedingsJan 1, 201115 citations

On the approach to thermal equilibrium of macroscopic quantum systems

Sheldon Goldstein, Roderich Tumulka

AIP conference proceedingsJan 1, 20117 citations

Ergodic Theorem for Interacting Systems

H. Ekstein

von Neumann has proved a quantum-mechanical ergodic theorem which deals with a closed macroscopic system and macroscopic observables. Certain unsatisfactory features of this theorem are probably due to the overly-strong idealization of a completely closed system.Since the claims of statistical mechanics go beyond macroscopic systems, there is a need for a more general theorem. The present paper deals with a system of any number of degrees of freedom and with any observable, during interaction with a temperature bath. In order to avoid an assumption of randomness, the bath must be described explicitly and precisely by a time-independent Hamiltonian. The model of the bath is obtained by taking the Gibbs ensemble seriously, i.e., as a set of $N$ identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials $V$ .von Neumann has proved a quantum-mechanical ergodic theorem which deals with a closed macroscopic system and macroscopic observables. Certain unsatisfactory features of this theorem are probably due to the overly-strong idealization of a completely closed system.Since the claims of statistical mechanics go beyond macroscopic systems, there is a need for a more general theorem. The present paper deals with a system of any number of degrees of freedom and with any observable, during interaction with a temperature bath. In order to avoid an assumption of randomness, the bath must be described explicitly and precisely by a time-independent Hamiltonian. The model of the bath is obtained by taking the Gibbs ensemble seriously, i.e., as a set of $N$ identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials $V$.

Physical ReviewJul 15, 195711 citations

The Ergodic Theorem in Quantum Statistical Mechanics

Martin J. Klein

An ergodic theorem is a statement of the equality of the time averages of the physical properties of a system and the averages of these quantities obtained from the consideration of a Gibbsian ensemble of identical systems. It is shown that, in quantum statistical mechanics, a necessary and sufficient condition for the existence of an ergodic theorem can be derived for a system in weak energetic interaction with its surroundings. This necessary and sufficient condition for the equality of time and (microcanonical) ensemble averages is that all operators invariant in time shall reduce to constant multiples of the unit operator when applied to the system itself. It is shown that the conditions for ergodicity are formally analogous to those derived by von Neumann for classical statistical mechanics. No assumption of "molecular chaos" is made.An ergodic theorem is a statement of the equality of the time averages of the physical properties of a system and the averages of these quantities obtained from the consideration of a Gibbsian ensemble of identical systems. It is shown that, in quantum statistical mechanics, a necessary and sufficient condition for the existence of an ergodic theorem can be derived for a system in weak energetic interaction with its surroundings. This necessary and sufficient condition for the equality of time and (microcanonical) ensemble averages is that all operators invariant in time shall reduce to constant multiples of the unit operator when applied to the system itself. It is shown that the conditions for ergodicity are formally analogous to those derived by von Neumann for classical statistical mechanics. No assumption of "molecular chaos" is made.

Physical ReviewJul 1, 195225 citations

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identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials

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