Bohr–Sommerfeld, WKB, and modified semiclassical quantization rules

Abstract

The WKB approximation for energy eigenvalues can be advantageously introduced to students when the Bohr–Sommerfeld quantization rule is discussed. A simple heuristic derivation of these quantization rules is given. Exceptional cases occur when conditions necessary for the validity of semiclassical approximations are violated, but appropriate modifications can still provide useful semiclassical approximations even in such cases. Various situations are illustrated by examples.