Semiclassical approach to states near potential barrier top

TL;DR

This paper develops an instanton-based semiclassical method to analytically analyze quantum states near potential barriers, accurately describing tunneling, resonance phenomena, and crossover behaviors in various anharmonic potentials.

Contribution

It introduces a semiclassical instanton approach that accurately models quantum states near barriers, including crossover regions, outperforming traditional methods like WKB.

Findings

Instanton method reproduces quantum results for barrier penetration.

Accurately describes resonance broadening and collapse.

Provides analytical insights into highly excited states in anharmonic potentials.

Abstract

Within the framework of the instanton approach we present analytical results for the following model problems: (i) particle penetration through a parabolic potential barrier, where the instanton solution practically coincides with the exact (quantum) one; (ii) descriptions of highly excited states in two types of anharmonic potentials: double-well $X^4$, and decay $X^3$. For the former case the instanton method reproduces accurately not only single well and double-well quantization but as well a crossover region (in the contrast with the standard WKB approach which fails to describe the crossover behavior), and for the latter case the instanton method allows to study resonance broadening and collapse phenomena. We investigate also resonance tunneling, playing a relevant role in many semiconducting devices. We show that in a broad region of energies the instanton approach gives exact (quantum) results. Applications of the method and of the results may concern the various systems in physics, chemistry and biology exhibiting double level behavior and resonance tunneling.