Empirical Measures and Quantum Mechanics: Applications 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. Applications to the mean-field limit of the N-particle Schrödinger equation include an O(1/N) convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the N-particle density operator, uniform in ħ∈ (0 , 1] provided that V and (- Δ) ^3/2+d/4V have integrable Fourier transforms.