WKB approach to calculating the lifetime of quasistationary states: Harmonic oscillator in a polynomial perturbation

Abstract

A simple and straightforward WKB approach to calculating the lifetime of quasistationary states in spherically symmetric potential wells is suggested. Using this approach, a general formula for the imaginary part of the energy for potentials of the form $V(x)=P(x)\ensuremath{-}\ensuremath{\mu}Q(x)\mathrm{\char22{}}\mathrm{where}$ $P(x)$ is the radial part of the potential for a spherically symmetric harmonic oscillator and $Q(x)$ is an even polynomial\char22{}is derived. Using this formula, the usual tedious procedure of the explicit asymptotic matching of the WKB and perturbative wave functions is avoided, and calculations are substantially simplified. The leading term and a few corrections of the series for the imaginary part of the energy and the related lifetime are analytically calculated.