On the Thermalization Hypothesis of Quantum States

The eigenstate thermalization hypothesis (ETH) is discussed. We note that one common formulation of the ETH does not necessarily imply thermalization of an observable of an isolated many-body quantum system. We show that to get thermalization, one has to postulate the canonical or microcanonical distribution in the ETH ansatz. More generally, any other average can be postulated in the generalized ETH ansatz, which leads to a corresponding equilibration condition.