Application of a Higher-Order WKB Approximation to Radial Problems

The radial generalization of Dunham's one-dimensional WKB quantization condition, including second- and third-order corrections is derived using the Langer transformation. It is found that, although the first-order integral can be obtained from Dunham's results by substituting ${(l=\frac{1}{2})}^2$ for $l(l+1\mathrm{})\mathrm{}$ in the effective potential, there is no choice of effective potential that leads to the correct second- and third-order integrals. It is suggested that all previous eigenvalue calculations using higher-order WKB approximations for the radial case should be reinvestigated. It is shown that the second- and third-order integrals identically vanish for the hydrogen atom and the three-dimensional harmonic oscillator, as expected.