  The main result in this paper is a new inequality bearing on solutions of the
$N$-body linear Schr\"{o}dinger 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.
