Building Confidence in the Dirac $\delta$-function

Asim Gangopadhyaya; Constantin Rasinariu

doi:10.1088/1361-6404/aaddcd

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arXiv · 2007.13478

Building Confidence in the Dirac $\delta$-function

Asim Gangopadhyaya·Constantin Rasinariu

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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.

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Asim Gangopadhyaya, Constantin Rasinariu. 2020-07-01. Building Confidence in the Dirac $\delta$-function. https://doi.org/10.1088/1361-6404/aaddcd

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