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$ J_L({\bf r})$ , $ N_L({\bf r})$ , and $ H_L^\pm ({\bf r}$ )

For convenience, we define the following:

$\displaystyle J_L({\bf r})$ $\displaystyle =$ $\displaystyle j_\ell(kr) Y_L(\hat{r})$ (13.35)
$\displaystyle N_L({\bf r})$ $\displaystyle =$ $\displaystyle n_\ell(kr) Y_L(\hat{r})$ (13.36)
$\displaystyle H^\pm_L({\bf r})$ $\displaystyle =$ $\displaystyle h^\pm_\ell(kr) Y_L(\hat{r})$ (13.37)

These are the basic solutions to the HHE that are also eigenfunctions of $ L^2$ and $ L_z$ . Clearly there is an implicit label of $ k$ (or $ k^2$ ) for these solutions. A general solution (on a suitable domain) can be constructed out of a linear combination of any two of them.

Robert G. Brown 2017-07-11