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Building Confidence in the Dirac $δ$-function

In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac $δ$-function. Namely, we compute the expectation value of the Hamiltonian of a free-particle in a state described by a triangular wave function $ψ(x)$. Since the first derivative of $ψ(x)$ 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 $δ$-function. By arriving at the same result via alternate pathways, this exercise reinforces students' confidence in the Dirac $δ$-function and highlights its efficiency and elegance.