Building Confidence in the Dirac $\delta$-function

TL;DR

This paper presents a pedagogical example demonstrating the effective use of the Dirac delta function in quantum mechanics, specifically in calculating the expectation value of a free-particle Hamiltonian with a triangular wave function.

Contribution

It offers an educational example that reinforces confidence in the Dirac delta function's application and showcases its versatility in quantum mechanics calculations.

Findings

Expectation value of Hamiltonian is zero for the given state

Multiple calculation pathways reinforce the delta function's utility

Enhances pedagogical understanding of the delta function's role

Abstract

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.