Tunneling and energy splitting in an asymmetric double-well potential

TL;DR

This paper develops a WKB-based method to analyze tunneling and energy splitting in asymmetric double-well potentials, revealing resonance phenomena and generalizing symmetric case results.

Contribution

It introduces a wave function matching approach at the local maximum to derive energy splitting formulas for asymmetric potentials, extending previous symmetric models.

Findings

Derived a general energy splitting formula for asymmetric double wells.

Reproduced known instanton results for symmetric potentials.

Identified conditions for tunneling resonances in asymmetric cases.

Abstract

An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency $\omega$ and the difference of the minima is close to a multiple of $\hbar \omega$. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential.