Approach to thermal equilibrium of macroscopic quantum systems

We consider an isolated macroscopic quantum system. Let 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 H-eq of H such that dim H-eq/dim H is close to 1. We say that a system with state vector psi is an element of H is in thermal equilibrium if psi is close to 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.