Transmission through a potential barrier in quantum mechanics of multiple degrees of freedom: complex way to the top

TL;DR

This paper develops a semiclassical method to calculate tunneling exponents in multi-degree-of-freedom quantum systems, identifying multiple classical solution branches and introducing a regularization technique for physical relevance.

Contribution

It introduces a novel semiclassical approach with a regularization technique to handle multiple solution branches in tunneling problems involving many degrees of freedom.

Findings

Classical solutions form multiple branches with bifurcations.

The regularization technique selects physically relevant solution branches.

Method agrees with exact solutions in a two-degree-of-freedom system.

Abstract

A semiclassical method for the calculation of tunneling exponent in systems with many degrees of freedom is developed. We find that corresponding classical solution as function of energy form several branches joint by bifurcation points. A regularization technique is proposed, which enables one to choose physically relevant branches of solutions everywhere in the classically forbidden region and also in the allowed region. At relatively high energy the physical branch describes tunneling via creation of a classical state, close to the top of the barrier. The method is checked against exact solutions of the Schrodinger equation in a quantum mechanical system of two degrees of freedom.