  We propose a new approach for the study of the time evolution of a factorized
$N$-particle bosonic wave function with respect to a mean-field dynamics with a
bounded interaction potential. The new technique, which is based on the control
of the growth of the correlations among the particles, leads to quantitative
bounds on the difference between the many-particle Schr\"odinger dynamics and
the one-particle nonlinear Hartree dynamics. In particular the one-particle
density matrix associated with the solution to the $N$-particle Schr\"odinger
equation is shown to converge to the projection onto the one-dimensional
subspace spanned by the solution to the Hartree equation with a speed of
convergence of order 1/N for all fixed times.
