  We outline a procedure for using matrix mechanics to compute energy
eigenvalues and eigenstates for two and three interacting particles in a
confining trap, in one dimension. Such calculations can bridge a gap in the
undergraduate physics curriculum between single-particle and many-particle
quantum systems, and can also provide a pathway from standard quantum mechanics
course material to understanding current research on cold-atom systems. In
particular we illustrate the notion of "fermionization" and how it occurs not
only for the ground state in the presence of strong repulsive interactions, but
also for excited states, in both the strongly attractive and strongly repulsive
regimes.
