The WKBJ Wavefunctions in the Classically Forbbiden Region: the Connection Formulae

As well known, the WKBJ approximation method provides an approximate solution to the Schrödinger equation of a quantum mechanical system; but the method fails at the classical turning points (classically forbidden region) of the system’s potential energy function. Solutions about these “inaccessible ” regions are derived by the transformation of Schrödinger equation to either the Airy or modified Airy differential equation. The asymptotic expansion of these solutions are appropriately connected (connection formulae) with the WKBJ solutions to provide full range solutions and these are used to derive the standard energy level formula (Semiclassical quantization rule), which is applied to obtain the modified semiclassical quantization rule