A detailed proof of the von Neumann's Quantum Ergodic Theorem

TL;DR

This paper provides a simplified proof of von Neumann's Quantum Ergodic Theorem, demonstrating that quantum systems evolve in a manner consistent with ergodic behavior over large times in finite-dimensional Hilbert spaces.

Contribution

The paper offers a clearer, more accessible proof of von Neumann's theorem, enhancing understanding of quantum ergodicity in finite-dimensional systems.

Findings

Quantum states tend to explore the available Hilbert space over time.

The proof confirms the ergodic nature of quantum evolution under the Schrödinger equation.

The approach simplifies previous complex proofs, making the theorem more approachable.

Abstract

We present a simplified proof of the von Neumann's Quantum Ergodic Theorem. This important result was initially published in german by J. von Neumann in 1929. We are interested here in the time evolution $\psi_t$, $t\geq 0$, (for large times) under the Schrodinger equation associated to a given fixed Hamiltonian $H : \mathcal{H} \to \mathcal{H}$ and a general initial condition $\psi_0$. The dimension of the Hilbert space $\mathcal{H}$ is finite.