Localization or tunneling in asymmetric double-well potentials

TL;DR

This paper analyzes quantum states in asymmetric double-well potentials, deriving conditions for energy levels and localization, extending previous results to arbitrary asymmetry using WKB and Dekker's method.

Contribution

It generalizes earlier results on double-well potentials to arbitrary asymmetry, providing a quantization condition and localization analysis using WKB and Dekker's method.

Findings

Derives quantization condition matching WKB and exact wave functions.

Shows low-lying states are localized in one well with negligible amplitude in the other.

Extends analysis to arbitrary asymmetry in double-well potentials.

Abstract

An asymmetric double-well potential is considered, assuming that the wells are parabolic around the minima. The WKB wave function of a given energy is constructed inside the barrier between the wells. By matching the WKB function to the exact wave functions of the parabolic wells on both sides of the barrier, for two almost degenerate states, we find a quantization condition for the energy levels which reproduces the known energy splitting formula between the two states. For the other low-lying non-degenerate states, we show that the eigenfunction should be primarily localized in one of the wells with negligible magnitude in the other. Using Dekker's method [Physica 146A (1987) 375], the present analysis generalizes earlier results for weakly biased double-well potentials to systems with arbitrary asymmetry.