Improved Semiclassical Quantization of Bound States

Almost a century has passed since the publication of the seminal Brillouin, Wentzel, and Kramers (BWK) papers on the semiclassical quantization of vibrations, yet the BWK semiclassical quantization formula does not lead to the correct zero point energy estimate of the energy except for a few special cases. In this Letter, a simple energy shift is introduced into the expression for the action, whose magnitude is determined by second order vibrational perturbation theory, removing this deficiency. The resulting modified semiclassical quantization formula, when appropriately expanded, is shown to be identical to second order vibrational perturbation theory. It improves the resulting energy eigenvalues for the symmetric Rosen-Morse potential and is shown to provide rather accurate energy estimates for resonance energies in a cubic potential, better than those predicted by the standard semiclassical BWK expression or second order perturbation theory.