In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zangh`Δ± [2, 3, 4], we considered 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 + δE. The thermal equilibrium macrostate at energy E corresponds to a subspace Heq of H such that dim Heq/ 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 Heq. We argue...