Various variational approximations of quantum dynamics

TL;DR

This paper compares different variational principles for approximating quantum dynamics, analyzing their properties, conservation laws, and error estimates, with applications to Hartree and Gaussian wave packet methods.

Contribution

It introduces a unified framework for variational principles in quantum dynamics, including new insights into their geometric and conservation properties.

Findings

Both principles are characterized by metric and symplectic orthogonality conditions.

The paper derives an a-posteriori error estimate for the approximations.

Applications to Hartree and Gaussian wave packets demonstrate the methods' effectiveness.

Abstract

We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan (1964) minimizes the residuum of the time-dependent Schr\"odinger equation, while the second one, originating from the lecture notes of Kramer--Saraceno (1981), imposes the stationarity of an action functional. We characterize both principles in terms of metric and a symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a-posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.