  Boltzmann's ergodic hypothesis furnishes a possible explanation for the
emergence of statistical mechanics in the framework of classical physics. In
quantum mechanics, the Eigenstate Thermalization Hypothesis (ETH) is instead
generally considered as a possible route to thermalization. This is because the
notion of ergodicity itself is vague in the quantum world and it is often
simply taken as a synonym for thermalization. Here we show, in an elementary
way, that when quantum ergodicity is properly defined, it is, in fact,
equivalent to ETH. In turn, ergodicity is equivalent to thermalization, thus
implying the equivalence of thermalization and ETH. This result previously
appeared in [De Palma et al., Phys. Rev. Lett. 115, 220401 (2015)], but becomes
particularly clear in the present context. We also show that it is possible to
define a classical analogue of ETH which is implicitly assumed to be satisfied
when constructing classical statistical mechanics. Classical and quantum
statistical mechanics are built according to the familiar standard
prescription. This prescription, however, is ontologically justified only in
the quantum world.
