We should all fondly remember this basic undergraduate problem: A quantum particle is incident (from the left) upon a potential barrier of height V and width L. Compute the transmission and reflection coefficients.
However, this isn't the full story. To completely solve the potential barrier, we need a complete set of orthogonal and normalized energy eigenstates. The wavefunctions of the leftward traveling particles are just some of the energy eigenstates. The wavefunctions of the rightward traveling particles are some more energy eigenstates. Are there more possible states? What is a complete energy eigenbasis for the Hilbert Space of a single particle in a step potential?
(I'm thinking about a positive potential barrier +V, but one could ask this question for a finite potential well too.)