  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 analyzing the phase cancellations of multiple
scatterings on the random potential by expanding the propagator into a sum of
Feynman graphs. In this paper we consider the non-recollision graphs and prove
that the amplitude of the {\it non-ladder} diagrams is smaller than their
"naive size" by an extra $\lambda^c$ factor {\em per non-(anti)ladder vertex}
for some $c > 0$. This is the first rigorous result showing that the
improvement over the naive estimates on the Feynman graphs grows as a power of
the small parameter with the exponent depending linearly on the number of
vertices. This estimate allows us to prove the convergence of the perturbation
series.
