  We consider an isolated, macroscopic quantum system. Let H be a
micro-canonical "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 macro-state 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 in 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.

Quantum Physics;Statistical Mechanics