Quantization Process for the Driven Well - Where P Fails to Commute with P²

We consider the quantum mechanical behavior of a driven particle in an infinite 1D potential well. We show that the quantum dynamics of the system is induced by the delicate nontrivial properties of the momentum operator in this case, namely, its non-self-adjointness. Using this, we calculate the first order contribution to the cross section and the energy gain, and discuss their classical limit.