The periodically driven electron in a quantum well with two characteristic curvatures -- redux

TL;DR

This paper analytically solves the time-dependent Schrödinger equation for an electron in a composite quadratic well under periodic forcing, illustrating key techniques and exploring the limits of analytical methods before resorting to numerical solutions.

Contribution

It provides an analytical approach to a driven quantum well problem, demonstrating techniques for solving the time-dependent Schrödinger equation in one dimension.

Findings

Analytical solutions for a periodically driven quadratic well.

Insights into the applicability of analytical methods in quantum dynamics.

Guidance for researchers on solving time-dependent Schrödinger equations.

Abstract

I develop the solution to the problem of an electron confined in a composite quadratic well subject to a simple, external periodic force. The method of solution illustrates several of the basic techniques useful in formally solving the one-dimensional, time-dependent Schrodinger equation. One of the aims of this exercise is to see how far is it possible to push analytics, before plowing into numerical methods. I hope this presentation of the problem may result useful to others seeking to gain additional experience in the details of solving the time-dependent Schrodinger equation in one space dimension.