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

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.