Semiclassical approximation of the radial equation with two-dimensional potentials

The radial equation for scattering from a cylindrically symmetrical potential is examined, because two-dimensional scattering arises in high-energy electron diffraction from crystals. Particular attention is paid to the case of s waves, where there is a centripetal attractive potential for free particles. After showing that the Langer transformation, which leads to correct semiclassical wavefunctions for all other cases in two and three dimensions, fails for s waves, the method of comparison equations is applied, which enables the phase shifts and bound state conditions to be expressed in a simple form valid for all angular momenta. The theory is tested for s waves by comparison with exactly-calculated energy levels.