Dirac's variational approach to semiclassical Kramers problem in Smoluchowski limit

TL;DR

This paper develops a semiclassical approach using Dirac's variational method to analyze quantum effects on the Kramers escape rate in the Smoluchowski limit, providing a simpler alternative to path-integral techniques.

Contribution

It introduces a variational method-based derivation of the semiclassical potential and escape rate, incorporating quantum corrections in a more straightforward manner than path-integral approaches.

Findings

Derived a quantum-corrected semiclassical potential.

Obtained a consistent escape rate expression with existing quantum Smoluchowski results.

Presented a simpler, classical-picture-compatible method for quantum escape problems.

Abstract

Kramers escape from a metastable state in the presence of both thermal and quantum fluctuations under strong damping is treated as a thermally activated process in a quantum modified semiclassical potential. Dirac's time-dependent variational method together with the Jackiw-Kerman function is employed to derive the semiclassical potential. Quantum correction is incorporated in the drift potential, and is determined by quasi-stationary conditions and minimal uncertainty relation. The semiclassical rate obtained here is consistent in form with those from the quantum Smoluchowski equations deduced heuristically by modifying the diffusion coefficient using the path-integral method. Unlike approaches using the path-integral, which involves continuation into imaginary time, the approach here is simpler and more easily understood in terms of classical picture.