It is possible to apply the WKB approximation to the radial equation using a method by R. E. Langer (1937).
Recall that
and that
satisfies the effective one-dimensional Schrödinger equation,
We now perform the following transformations:
Note that, for
The radial equation becomes
In this case, the Bohr-Sommerfeld quantization rule is as in the purely one-dimensional case, but with an effective potential,
Problem
Use the WKB approximation to estimate the energy spectrum for a Hydrogen atom.
Hint: Use the relation,
where
and
and
are the classical turning points of the (effective) potential appearing in the WKB approximation for this problem, and the integral,
Solution