Double Well Potential: Perturbation Theory, Tunneling, WKB (beyond instantons)

TL;DR

This paper introduces a highly accurate approximate solution for the quantum double-well oscillator, combining perturbation theory, semi-classical methods, and tunneling descriptions to achieve precise energy and wavefunction calculations.

Contribution

It presents a novel integrated approach that surpasses previous methods in accuracy for the double-well potential in quantum mechanics.

Findings

Achieves 9-10 digit precision in energy calculations.

Wavefunction deviations are less than 0.1% in real space.

Provides a comprehensive method applicable to arbitrary coupling g.

Abstract

A simple approximate solution for the quantum-mechanical quartic oscillator $V= m^2 x^2+g x^4$ in the double-well regime $m^2<0$ at arbitrary $g \geq 0$ is presented. It is based on a combining of perturbation theory near true minima of the potential, semi-classical approximation at large distances and a description of tunneling under the barrier. It provides 9-10 significant digits in energies and gives for wavefunctions the relative deviation in real $x$-space less than $\lesssim 10^{-3}$.