Approach to thermal equilibrium of macroscopic quantum systems

We consider an isolated macroscopic quantum system. Let $\mathcal{H}$ be a microcanonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between $E$ and $E+\delta E$. The thermal equilibrium macrostate at energy $E$ corresponds to a subspace ${\mathcal{H}}_{eq}$ of $\mathcal{H}$ such that $\text{dim}{\mathcal{H}}_{eq}/\text{dim}\mathcal{H}$ is close to 1. We say that a system with state vector $\psi ∊\mathcal{H}$ is in thermal equilibrium if $\psi$ is “close” to ${\mathcal{H}}_{eq}$. We show that for “typical” Hamiltonians with given eigenvalues, all initial state vectors ${\psi }_0$ evolve in such a way that ${\psi }_t$ is in thermal equilibrium for most times $t$. This result is closely related to von Neumann’s quantum ergodic theorem of 1929.