Semi-classical formula for quantum tunneling in asymmetric double-well potentials

TL;DR

This paper derives a simple semi-classical formula for quantum tunneling in asymmetric double-well potentials using the WKB approach, addressing a gap in the literature and exploring related phenomena like the quantum Zeno effect.

Contribution

It provides a novel, analytical semi-classical formula for tunneling in asymmetric double-well potentials, including degenerate and non-degenerate cases, with applications to time-dependent quantum effects.

Findings

Derived a succinct WKB-based tunneling formula for asymmetric potentials

Analyzed degenerate and non-degenerate localized levels

Explored quantum Zeno effect in the time-dependent regime

Abstract

Despite quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, we derive an succinct analytical formula based on the WKB semi-classical approach. Two different examples of asymmetric potentials are discussed: when the two localized levels are degenerate or not. For the first case, we also discuss a time-dependent problem showing quantum Zeno effect.