  We study a special kind of semiclassical limit of quantum dynamics on a
circle and in a box (infinite potential well with hard walls) as the Planck
constant tends to zero and time tends to infinity. The results give detailed
information about all stages of evolution of quantum wave packets:
semiclassical motion, collapses, revivals, as well as intermediate stages. In
particular, we rigorously justify the fact that the spatial distribution of a
wave packet is most of the time close to uniform. This fact was previously
known only from numerical calculations.
  We apply the obtained results to a problem of classical mechanics: deciding
whether recently suggested functional classical mechanics is preferable to
traditional Newtonian one from the quantum-mechanical point of view. To do
this, we study the semiclassical limit of the Husimi functions of quantum
states. We show that functional mechanics remains valid at larger time scales
than Newtonian one and, therefore, is preferable.
  Finally, we analyse the quantum dynamics in a box in case when the size of
the box is known with a random error. We show that, in this case, the
probability distribution of the position of a quantum particle is not almost
periodic, but tends to a limit distribution as time indefinitely increases.
