Eur. Phys. J. H 35, 173–200 (2010)
https://doi.org/10.1140/epjh/e2010-00007-7

Regular Article

Long-time behavior of macroscopic quantum systems

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

¹ Departments of Mathematics and Physics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA
² Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA
³ Dipartimento di Fisica dell’Università di Genova and INFN sezione di Genova, via Dodecaneso 33 16146 Genova, Italy

^a e-mail: oldstein@math.rutgers.edu
^b e-mail: lebowitz@math.rutgers.edu
^c e-mail: tumulka@math.rutgers.edu
^d e-mail: zanghi@ge.infn.it

Received: 2 March 2010
Published online: 7 September 2010

Abstract

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 non-trivial 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 ψ₀ from a micro-canonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ψ_t is close to their micro-canonical distribution.