The time-dependent Schr\"odinger equation with piecewise constant potentials

TL;DR

This paper presents explicit solutions for the time-dependent Schr"odinger equation with piecewise constant potentials using the Unified Transform Method, advancing analytical approaches for quantum interface problems.

Contribution

It introduces a novel application of the Unified Transform Method to solve the time-dependent Schr"odinger equation with piecewise constant potentials explicitly.

Findings

Explicit solutions for the time-dependent Schr"odinger equation are derived.

The method effectively handles interface conditions in quantum models.

The approach enhances analytical understanding of quantum tunneling and particle confinement.

Abstract

The linear Schr\"odinger equation with piecewise constant potential in one spatial dimension is a well-studied textbook problem. It is one of only a few solvable models in quantum mechanics and shares many qualitative features with physically important models. In examples such as "particle in a box" and tunneling, attention is restricted to the time-independent Schr\"odinger equation. This paper combines the Unified Transform Method and recent insights for interface problems to present fully explicit solutions for the time-dependent problem.