Empirical Measures and Quantum Mechanics: Application to the Mean-Field Limit

TL;DR

This paper introduces a quantum analogue of empirical measures, derives their evolution equation, and applies it to establish an $O(1/\sqrt{N})$ convergence rate for the mean-field limit of the Schrödinger equation, uniform in Planck's constant.

Contribution

It defines a quantum empirical measure, derives its evolution, and applies it to prove convergence rates in the mean-field limit of quantum many-body systems.

Findings

Established a quantum analogue of empirical measure.

Derived an evolution equation containing the Hartree equation.

Proved an $O(1/\sqrt{N})$ convergence rate in the mean-field limit.

Abstract

In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of $N$-particle systems. We establish an equation governing the evolution of our quantum analogue of the $N$-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Our main application of this new object to the mean-field limit of the $N$-particle Schr\"odinger equation is an $O(1/\sqrt{N})$ convergence rate in some dual Sobolev norm for the Wigner transform of the single-particle marginal of the $N$-particle density operator, uniform in $\hbar\in(0,1]$ (where $\hbar$ is the Planck constant) provided that $V$ and $(-\Delta)^{3+d/2}V$ have integrable Fourier transforms.