  We consider random Schr\"odinger equations on $\bR^d$ or $\bZ^d$ for $d\ge 3$
with uncorrelated, identically distributed random potential. Denote by
$\lambda$ the coupling constant and $\psi_t$ the solution with initial data
$\psi_0$. Suppose that the space and time variables scale as $x\sim \lambda^{-2
-\kappa/2}, t \sim \lambda^{-2 -\kappa}$ with $0< \kappa \leq \kappa_0$, where
$\kappa_0$ is a sufficiently small universal constant. We prove that the
expectation value of the Wigner distribution of $\psi_t$, $\bE W_{\psi_{t}} (x,
v)$, converges weakly to a solution of a heat equation in the space variable
$x$ for arbitrary $L^2$ initial data in the weak coupling limit $\lambda \to
0$. The diffusion coefficient is uniquely determined by the kinetic energy
associated to the momentum $v$.
