A Modern Approach to Quantum Mechanics · Problem 14.2

Photons and Atoms — John S. Townsend

Physics — Quantum Mechanics · plane-wave-orthonormality

Problem

Show that

\int d^3r\,\frac{e^{-i\mathbf{k}\cdot\mathbf{r}}}{\sqrt{V}}\frac{e^{i\mathbf{k}'\cdot\mathbf{r}}}{\sqrt{V}} = \delta_{\mathbf{k},\mathbf{k}'}

for plane waves in a box of volume

V

with periodic boundary conditions.