WKB energy quantization and first-order perturbation theory

The WKB quantization condition can be used to estimate the quantized bound state energies, E_n, in a potential V(x). If a small perturbation of the form λ V′ (x) is added to this potential, we show that the resulting shift in the quantized energy eigenvalues is given by a form which is quite similar to standard first-order perturbation theory. Several examples are presented in which the large n form of the lowest-order energy shift is more easily evaluated using WKB-perturbation techniques than by using standard first-order perturbation methods.