  We investigate the relation between the classical ergodicity and the quantum
eigenstate thermalization in the fully connected Ising ferromagnets. In the
case of spin-1/2, an expectation value of an observable in a single energy
eigenstate coincides with the long-time average in the underlying classical
dynamics, which is a consequence of the Wentzel-Kramers-Brillouin
approximation. In the case of spin-1, the underlying classical dynamics is not
necessarily ergodic. In that case, it turns out that, in the thermodynamic
limit, the statistics of the expectation values of an observable in the energy
eigenstates coincides with the statistics of the long-time averages in the
underlying classical dynamics starting from random initial states sampled
uniformly from the classical phase space. This feature seems to be a general
property in semiclassical systems, and the result presented here is crucial in
discussing equilibration, thermalization, and dynamical transitions of such
systems.
