Citation: von Neumann, John. "Proof of the ergodic theorem and the H-theorem in quantum mechanics: Translation of: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik." The European Physical Journal H 35.2 (2010): 201-237.
Web: https://arxiv.org/abs/1003.2133
Tags: Foundational, Statistical-mechanics
In this work, von Neumann solves some early foundational problems towards building a theory of quantum statistical mechanics. The issue von Neumann addresses is that the Gibbs formulation of quantum statistical mechanics requires a notion of a phase space which has definite positions and momentums. This is in contrast to quantum mechanics, where we cannot simultaneously measure position and momentum. The solution von Neumann proposes is as follows. Even though position and momentum don't commute, they very nearly commute! The size of their commutator is controlled by the Plank constant. This means that there are exactly commuting matrices which are approximately equal to position and momentum. These operators can be readily simultanesouly measured. In fact, von Neumann argues that this is what happens at a physical level. When we simultaneously measure the position and momentum of an atom, actually we are measuring operators which do commute but are not exactly equal to position and momentum.
After settling this difficulty philosophically, he then moves on to using this resolution to build a theory of quantum statistical mechanics. This means defining an object which behaves like the canonical ensemble in classical mechanics. He then proves an ergodic theorem for quantum statistical mechanics, which turns out to actually be a lot easier than in the classical case. A more modern (and more correct) treatment of this theorem is due to Ogata:
> Ogata, Yoshiko. "Approximating macroscopic observables in quantum spin systems with commuting matrices." Journal of Functional Analysis 264.9 (2013): 2005-2033.