The time-dependent Schrödinger equation with piecewise constant potentials

The linear Schrodinger 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 tunnelling, attention is restricted to the time-independent Schrodinger equation. This paper combines the unified transform method and recent insights for interface problems to present fully explicit solutions for the time-dependent problem.