Empirical Measures and Quantum Mechanics: Application to 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 Schrödinger 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 $\hslash \in (0,1]$ (where $\hslash$ is the Planck constant) provided that $V$ and $(-\Delta {)}^{3+d/2}V$ have integrable Fourier transforms.

35 pages, no figure

References in corpus (3)

• Mean field evolution of fermions with Coulomb interaction
• Wave Packets and the Quadratic Monge-Kantorovich Distance in Quantum Mechanics
• On the derivation of the Hartree equation in the mean field limit: Uniformity in the Planck constant

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