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+. The thermal equilibrium macrostate at energy E corresponds to a subspace H₄ₐ of H such that dim H₄ₐ/dim H is close to 1. We say that a system with state vector ∊H is in thermal equilibrium if is ``close'' to H₄ₐ. We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ₀ evolve in such a way that ₓ is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.