Plane Wave Expansion

Plane waves and free spherical waves both form an (on-shell) complete orthnormal set on $\rm I \hspace{-.180em} R^3$ (with or without the origin). That means that one must be able to expand one in terms of the other. Plane waves can be expanded in terms of free spherical waves by:

$$e^{i {\bf k} \cdot {\bf r}} = e^{ikr \cos (\Theta)}$$

$$= \sum_L 4 \pi i^\ell Y_L(\hat{k}) j_\ell(kr) Y_L(\hat{r})^\ast.$$ (11.101)

This is due to Lord Rayleigh and is sometimes called the Rayleigh expansion. Recall that $\Theta$ is the angle betwixt the $\vec{r}$ and the $\vec{k}$ and that $\cos(\Theta) = \cos(-\Theta)$.

There is similarly an (integral) expression for $j_\ell(kr)$ in terms of an integral over the $e^{i {\bf k} \cdot {\bf r}}$ but we will not use it here. It follows from the completeness relation on page 214 in Wyld, the Rayleigh expansion, and the completeness relation on page 212. Derive it for homework (or find it somewhere and copy it, but you shouldn't have to). Check your result by finding it somewhere. I think it might be somewhere in Wyld, but I know it is elsewhere. This will be handed in.