Transition state theory for wave packet dynamics. I. Thermal decay in metastable Schr\"odinger systems

TL;DR

This paper develops a method to apply transition state theory to wave packet dynamics in metastable quantum systems, enabling calculation of thermal decay rates with improved accuracy by mapping variational parameters to classical phase space.

Contribution

It introduces a novel approach to connect variational wave function dynamics with classical transition state theory for metastable Schrödinger systems.

Findings

Decay rates for narrow wave functions match classical normal form results.

Broader wave functions yield decay rates consistent with quantum normal form.

Method successfully applied to a cubic model potential.

Abstract

We demonstrate the application of transition state theory to wave packet dynamics in metastable Schr\"odinger systems which are approached by means of a variational ansatz for the wave function and whose dynamics is described within the framework of a time-dependent variational principle. The application of classical transition state theory, which requires knowledge of a classical Hamilton function, is made possible by mapping the variational parameters to classical phase space coordinates and constructing an appropriate Hamiltonian in action variables. This mapping, which is performed by a normal form expansion of the equations of motion and an additional adaptation to the energy functional, as well as the requirements to the variational ansatz are discussed in detail. The applicability of the procedure is demonstrated for a cubic model potential for which we calculate thermal decay rates of a frozen Gaussian wave function. The decay rate obtained with a narrow trial wave function agrees perfectly with the results using the classical normal form of the corresponding point particle. The results with a broader trial wave function go even beyond the classical approach, i.e., they agree with those using the quantum normal form. The method presented here will be applied to Bose-Einstein condensates in the following paper [A. Junginger, M. Dorwarth, J. Main, and G. Wunner, submitted to J. Phys. A].