# 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 that

$u(r)=rR(r),\!$

and that $u(r)\!$ satisfies the effective one-dimensional Schrödinger equation,

$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}+V(r)-E\right] u(r)=0.$

We now perform the following transformations:

$\ r = e^{s};$ $\ u(r) = W(s)e^{\frac{1}{2}s}$

Note that, for $0 $-\infty The radial equation becomes

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

In this case, the Bohr-Sommerfeld quantization rule is as in the purely one-dimensional case, but with an effective potential,

$V_{\text{eff}}(r)=V(r)+\frac{\hbar^2(\ell+\frac{1}{2})^2}{2mr^2}.$

## Problem

Use the WKB approximation to estimate the energy spectrum for a Hydrogen atom.

Hint: Use the relation,

$r^{2}-Vr+T=(r_{1}-r)(r_{2}-r),\!$

where

$T=-\frac{\hbar^2(l+\tfrac{1}{2})^2}{2mE},\,V=\frac{e^2}{E},$

and $r_1\!$ and $r_2\!$ are the classical turning points of the (effective) potential appearing in the WKB approximation for this problem, and the integral,

$\int_{r_1}^{r_2}\sqrt{{\frac{(x-r_1)(x-r_2)}{x^{2}}}}\,dx=\frac{\pi }{2}(\sqrt {r_2} -\sqrt {r_1} )^{2}.$