WKB in Spherical Coordinates

It is possible to apply the WKB approximation to the radial equation using a method by R. E. Langer (1937).

Recall: $\ u(r)=rR(r)$, $\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}+V(r)-E\right]u(r)=0$

Now apply the transformations: $\ r=e^{s};$ $\ u(r)=W(s)e^{{\frac {1}{2}}s}$

Note that for $\ r$ varying from 0 to infinity, $\ s$ will vary from minus infinity to plus infinity.

The radial equation then transforms into:

$\ {\frac {d^{2}W}{ds^{2}}}+{\frac {2m}{\hbar ^{2}}}\left[E-V(e^{s})-{\frac {\hbar ^{2}}{2m}}\left(l+{\frac {1}{2}}\right)^{2}e^{-2s}\right]e^{2s}=0$

In this case the Bohr-Sommerfeld quantization rule becomes:

$\ \int _{r_{1}}^{r_{2}}{\sqrt {2m\left(E_{n}-V(r)-{\frac {\hbar ^{2}}{2m}}{\frac {(\ell +{\frac {1}{2}})^{2}}{r^{2}}}\right)}}dr=\left(n+{\frac {1}{2}}\right)\pi \hbar$