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 / 4 V have integrable Fourier transforms.