  We consider the quantum mechanical behavior of a driven particle in an
infinite 1D potential well. We show that the time dependent perturbation series
is induced by the delicate non-trivial properties of the momentum operator in
this case, namely, its non-self-adjointness. Using this expansion, we calculate
the first order contribution to the cross section and the energy gain, and
discuss their classical limit. In this limit the one-period energy gain
converges to its classical analog - the classical local (momentum space)
diffusion coefficient. Both the classical and quantum mechanical results are
compared with numerical simulations.
