The Quantization process for the Driven Quantum Well - Perturbative Expansion and the Classical Limit

TL;DR

This paper investigates the quantum behavior of a driven particle in a 1D potential well, using perturbative expansion to connect quantum results with classical diffusion, highlighting the role of the momentum operator's properties.

Contribution

It introduces a perturbative approach based on the non-self-adjointness of the momentum operator to analyze quantum dynamics in a driven potential well and explores the classical limit.

Findings

First order contribution to the cross section and energy gain calculated.

Classical limit of energy gain matches classical diffusion coefficient.

Quantum and classical results are validated through numerical simulations.

Abstract

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.