On the Mean Field and Classical Limits of Quantum Mechanics

Abstract

The main result in this paper is a new inequality bearing on solutions of the $N$-body linear Schrödinger 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.

38 pages. Final version, to appear in Communications in Mathematical Physics

References in corpus (2)

• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the rate of convergence for the mean field approximation of many-body quantum dynamics

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