On the classical limit of a time-dependent self-consistent field system: analysis and computation

TL;DR

This paper analyzes the classical limit of a time-dependent self-consistent field system using Wigner transformations, establishing rigorous results and developing numerical methods that efficiently capture quantum-classical dynamics.

Contribution

It provides a rigorous analysis of the classical limit for coupled Schrödinger systems and introduces second-order numerical methods with stability independent of semi-classical parameters.

Findings

Rigorous derivation of classical limit as coupled Vlasov equations.

Development of second-order numerical schemes for semi-classical Schrödinger equations.

Numerical methods effectively capture physical observables with semi-classical parameter independence.

Abstract

We consider a coupled system of Schr\"odinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a mixed quantum-classical model which is closely connected to the well-known Ehrenfest method in molecular dynamics. In the second case, the classical limit of the full system is rigorously established, resulting in a system of coupled Vlasov-type equations. In the second part of our work, we provide a numerical study of the coupled semi-classically scaled Schr\"odinger equations and of the mixed quantum-classical model obtained via Ehrenfest's method. A second order (in time) method is introduced for each case. We show that the proposed methods allow time steps independent of the semi-classical parameter(s) while still capturing the correct behavior of physical observables. It also becomes clear that the order of accuracy of our methods can be improved in a straightforward way.