TY - JOUR

T1 - Long-time behavior of macroscopic quantum systems

T2 - Commentary accompanying the English translation of John von Neumann's 1929 article on the quantum ergodic theorem

AU - Goldstein, S.

AU - Lebowitz, J. L.

AU - Tumulka, R.

AU - Zanghì, N.

PY - 2010

Y1 - 2010

N2 - The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann's 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the "quantum H-theorem", is actually a much weaker statement than Boltzmann's classical H-theorem, the other theorem, which he calls the "quantum ergodic theorem", is a beautiful and very nontrivial result. It expresses a fact we call "normal typicality" and can be summarized as follows: for a "typical" finite family of commuting macroscopic observables, every initial wave function ?0 from a microcanonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ?t is close to their micro-canonical distribution.

AB - The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann's 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the "quantum H-theorem", is actually a much weaker statement than Boltzmann's classical H-theorem, the other theorem, which he calls the "quantum ergodic theorem", is a beautiful and very nontrivial result. It expresses a fact we call "normal typicality" and can be summarized as follows: for a "typical" finite family of commuting macroscopic observables, every initial wave function ?0 from a microcanonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ?t is close to their micro-canonical distribution.

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U2 - 10.1140/epjh/e2010-00007-7

DO - 10.1140/epjh/e2010-00007-7

M3 - Article

AN - SCOPUS:83755210269

VL - 35

SP - 173

EP - 200

JO - European Physical Journal H

JF - European Physical Journal H

SN - 2102-6459

IS - 2

ER -