We study the time evolution of a system of N spinless fermions in R 3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system. MSC class: 35Q40, 35Q55, 81Q05, 82C10

We consider the semiclassical limit of nonrelativistic quantum many-boson systems with delta potential in one dimensional space. We prove that time evolved coherent states behave semiclassically as squeezed states by a Bogoliubov time-dependent affine transformation. This allows us to obtain properties analogous to those proved by Hepp and Ginibre-Velo ([Hep], [GiVe1, GiVe2]) and also to show propagation of chaos for Schrödinger dynamics in the mean field limit. Thus, we provide a derivation of the cubic NLS equation in one dimension.