Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics

TL;DR

This paper reformulates key statistical mechanics concepts within quantum mechanics, proving the ergodic and H-theorems without relying on disorder assumptions, thus bridging macroscopic and quantum descriptions.

Contribution

It introduces a quantum-mechanical reinterpretation of statistical mechanics notions and provides rigorous proofs of the ergodic and H-theorems.

Findings

Proves the ergodic theorem in quantum mechanics.

Establishes the H-theorem without disorder assumptions.

Clarifies the physical meaning of mathematical conditions for validity.

Abstract

It is shown how to resolve the apparent contradiction between the macroscopic approach of phase space and the validity of the uncertainty relations. The main notions of statistical mechanics are re-interpreted in a quantum-mechanical way, the ergodic theorem and the H-theorem are formulated and proven (without "assumptions of disorder"), followed by a discussion of the physical meaning of the mathematical conditions characterizing their domain of validity.