  We investigate the time evolution of a generic and finite isolated quantum
many-body system starting from a pure quantum state. We find the kinematical
general canonical principle proposed by Popescu-Short-Winter for statistical
mechanics can be built in a more solid ground by studying the thermalization,
i.e. comparing the density matrices themselves rather than the measures of
distances. In particular, this allows us to explicitly identify that, from any
instantaneous pure state after thermalization, the state of subsystem is like
from a microcanonical ensemble or a generalized Gibbs ensemble, but neither a
canonical nor a thermal ones due to finite-size effect. Our results are
expected to bring the task of characterizing the state after thermalization to
completion. In addition, thermalization of coupled systems with different
temperatures corresponding to mixed initial states is studied.
