Tunnelings as Catastrophes

TL;DR

This paper derives a semiclassical approximation for the partition function of a quantum system, highlighting how tunneling paths emerge through catastrophes as temperature decreases.

Contribution

It introduces a path-integral-based formula that captures the transition from single to multiple path regimes, including tunneling effects, in a unified framework.

Findings

The formula depends only on integrals involving the potential.

Tunneling paths appear at specific catastrophes as temperature lowers.

The approach accurately accounts for tunneling phenomena in quantum systems.

Abstract

We use path-integrals to derive a general expression for the semiclassical approximation to the partition function of a one-dimensional quantum-mechanical system. Our expression depends solely on ordinary integrals which involve the potential. For high temperatures, the semiclassical expression is dominated by single closed paths. As we lower the temperature, new closed paths appear, including tunneling paths. The transition from single to multiple-path regime corresponds to well-defined catastrophes. Tunneling sets in whenever they occur. (Our formula fully accounts for this feature.)