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

For convenience, we define the following:

$\displaystyle J_L({\bf r})$ $\textstyle =$ $\displaystyle j_\ell(kr) Y_L(\hat{r})$ (11.102)
$\displaystyle N_L({\bf r})$ $\textstyle =$ $\displaystyle n_\ell(kr) Y_L(\hat{r})$ (11.103)
$\displaystyle H^\pm_L({\bf r})$ $\textstyle =$ $\displaystyle h^\pm_\ell(kr) Y_L(\hat{r})$ (11.104)

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 2007-12-28