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

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 $O(1\mathrm{/}\sqrt{N})$ convergence rate in some dual Sobolev norm for the Wigner transform of the single-particle marginal of the $N$-particle density operator, uniform in $\mathrm{\hslash }\in (0,1]$ (where $\mathrm{\hslash }$ is the Planck constant) provided that $V$ and $(-\mathrm{\Delta }{)}^{3+d\mathrm{/}2}V$ have integrable Fourier transforms.