WKB Exactness and Generalized Langer Correction for All Conventional Potentials

The WKB semiclassical method can be used to find approximate eigenergies. A modified version of WKB, known as SWKB, is also used in the context of supersymmetric quantum mechanics. SWKB is known to yield exact, rather than approximate, results for an entire class of translationally shape-invariant potentials, known as conventional potentials. However, not all conventional potentials are WKB-exact without modification; some can be made WKB-exact using a correction known as the Langer correction. We propose a generalized Langer correction and show that it generates the exact WKB quantization condition for all conventional potentials. We proved that this generalized correction is directly related to the SWKB exactness of these potentials.