  The issue of the thermalization of an isolated quantum system is addressed by
considering a perfect gas confined by gravity and initially trapped above a
certain height. As we are interested in the behavior of truly isolated systems,
we assume the gas is in a pure state of macroscopically well-defined energy. We
show that, in general, for single-particle distributions, such a state is
strictly equivalent to the microcanonical mixed state at the same energy. We
derive an expression for the time-dependent gas density which depends on the
initial gas state only via a few thermodynamic parameters. Though we consider
non-interacting particles, the density relaxes into an asymptotic profile, but
which is not the thermal equilibrium one determined by the gas energy and
particle number.
