# Improved Semiclassical Quantization of Bound States

## TL;DR

This paper improves a classical physics method for estimating energy levels in quantum systems, making it more accurate for certain potentials.

## Contribution

A new energy shift based on second-order vibrational perturbation theory is introduced to improve semiclassical quantization.

## Key findings

- The modified formula matches second-order perturbation theory when expanded.
- It provides better energy estimates for the symmetric Rosen-Morse and cubic potentials.
- The results outperform standard BWK and second-order perturbation theory for resonance energies.

## Abstract

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.