A Simple Derivation of Mean Field Limits for Quantum Systems

Abstract

We shall present a new strategy for handling mean field limits of quantum mechanical systems. The new method is simple and effective. It is simple, because it translates the idea behind the mean-field description of a many particle quantum system directly into a mathematical algorithm. It is effective because, with less effort, the strategy yields better results than previously achieved. As an instructional example we treat a simple model for the time-dependent Hartree equation which we derive under more general conditions than what has been considered so far. Other mean-field scalings leading, e.g. to the Gross-Pitaevskii equation can also be treated (Pickl in Derivation of the time dependent Gross Pitaevskii equation with external fields, preprint; Pickl in Derivation of the time dependent Gross Pitaevskii equation without positivity condition on the interaction, preprint).