  We consider random Schr\"odinger equations on $\bZ^d$ for $d\ge 3$ with
identically distributed 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 a solution of a heat equation in the space variable $x$ for arbitrary $L^2$
initial data. The diffusion coefficient is uniquely determined by the kinetic
energy associated to the momentum $v$.
  This work is an extension to the lattice case of our previous result in the
continuum \cite{ESYI}, \cite{ESYII}. Due to the non-convexity of the level
surfaces of the dispersion relation, the estimates of several Feynman graphs
are more involved.
