  In this note we present an example from undergraduate quantum mechanics
designed to highlight the versatility of the Dirac $\delta$-function. Namely,
we compute the expectation value of the Hamiltonian of a free-particle in a
state described by a triangular wave function $\psi(x)$. Since the first
derivative of $\psi(x)$ is piecewise constant, and because this Hamiltonian is
proportional to the second order spatial derivative, students often end up
finding the expectation value to be zero --an unphysical answer. This problem
provides a pedagogical application of the Dirac $\delta$-function. By arriving
at the same result via alternate pathways, this exercise reinforces students'
confidence in the Dirac $\delta$-function and highlights its efficiency and
elegance.
