  We consider random Schr\"odinger equations on $\bR^d$ for $d\ge 3$ with a
homogeneous Anderson-Poisson type random potential. Denote by $\lambda$ the
coupling constant and $\psi_t$ the solution with initial data $\psi_0$. The
space and time variables scale as $x\sim \lambda^{-2 -\kappa/2}, t \sim
\lambda^{-2 -\kappa}$ with $0< \kappa < \kappa_0(d)$. We prove that, in the
limit $\lambda \to 0$, the expectation of the Wigner distribution of $\psi_t$
converges weakly to the solution of a heat equation in the space variable $x$
for arbitrary $L^2$ initial data. The proof is based on a rigorous analysis of
Feynman diagrams. In the companion paper the analysis of the non-repetition
diagrams was presented. In this paper we complete the proof by estimating the
recollision diagrams and showing that the main terms, i.e. the ladder diagrams
with renormalized propagator, converge to the heat equation.
