  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.
