Building confidence in the Dirac δ -function

@article{Gangopadhyaya2018BuildingCI,
  title={Building confidence in the Dirac
 $\delta$
 -function},
  author={Asim Gangopadhyaya and Constantin Rasinariu},
  journal={European Journal of Physics},
  year={2018}
}
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… 
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