  Courses on undergraduate quantum mechanics usually focus on solutions of the
Schr\"odinger equation for several simple one-dimensional examples. When the
notion of a Hilbert space is introduced only academic examples are used, such
as the matrix representation of Dirac's raising and lowering operators or the
angular momentum operators. We introduce some of the same one-dimensional
examples as matrix diagonalization problems, with a basis that consists of the
infinite set of square well eigenfunctions. Undergraduate students are well
equipped to handle such problems in familiar contexts. We pay special attention
to the one-dimensional harmonic oscillator. This paper should equip students to
obtain the low lying bound states of any one-dimensional short range potential.
