  We analyze the solutions of the Schr\"odinger equation with the low frequency
initial data and a time-dependent weakly random potential. We prove a
homogenization result for the low frequency component of the wave field. We
also show that the dynamics generates a non-trivial energy in the high
frequencies, which do not homogenize -- the high frequency component of the
wave field remains random and the evolution of its energy is described by a
kinetic equation. The transition from the homogenization of the low frequencies
to the random limit of the high frequencies is illustrated by understanding the
size of the small random fluctuations of the low frequency component.
