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+δE. 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 ψ∊H is in thermal equilibrium if ψ is “close” to H_eq. We show that for “typical” Hamiltonians with given eigenvalues, all initial state vectors ψ₀ evolve in such a way that ψ_t is in thermal equilibrium for most times t. This result is closely related to von Neumann’s quantum ergodic theorem of 1929.