  We use separation of variables as a tool to identify and to analyze exactly
soluble time-dependent quantum mechanical potentials. By considering the most
general possible time-dependent re-definition of the spatial coordinate, as
well as general transformations on the wavefunctions, we show that separation
of variables applies and exact solubility occurs only in a very restricted
class of time-dependent models. We consider the formal structure underlying our
findings, and the relationship between our results and other work on
time-dependent potentials. As an application of our methods, we apply our
results to the calculations of propagators.
