On the Mean Field and Classical Limits of Quantum Mechanics

TL;DR

This paper establishes a new inequality linking the solutions of the quantum N-body Schrödinger equation and the mean field Hartree equation, showing the mean field limit is uniform in the classical limit with quantitative estimates.

Contribution

It introduces a novel inequality that quantifies the approximation of quantum N-body dynamics by mean field models, applicable to $C^{1,1}$ potentials, without relying on BBGKY hierarchy or second quantization techniques.

Findings

The mean field limit is uniform in the classical limit.

Provides a quantitative estimate of the approximation quality.

Applicable to $C^{1,1}$ interaction potentials.

Abstract

The main result in this paper is a new inequality bearing on solutions of the $N$-body linear Schr\"{o}dinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of $N$ identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of $C^{1,1}$ interaction potentials. The quantity measuring the approximation of the $N$-body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent $2$. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the $N$-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.