  We show that a quantum particle in $\mathbb{R}^d$, for $d \geq 1$, subject to
a white-noise potential, moves super-ballistically in the sense that the mean
square displacement $\int \|x\|^2 \langle \rho(x,x,t) \rangle ~dx$ grows like
$t^{3}$ in any dimension. The white noise potential is Gaussian distributed
with an arbitrary spatial correlation function and a delta correlation function
in time. This is a known result in one dimension (see refs. Fischer, Leschke,
M\"uller and Javannar, Kumar}. The energy of the system is also shown to
increase linearly in time. We also prove that for the same white-noise
potential model on the lattice $\mathbb{Z}^d$, for $d \geq 1$, the mean square
displacement is diffusive growing like $t^{1}$. This behavior on the lattice is
consistent with the diffusive behavior observed for similar models in the
lattice $\mathbb{Z}^d$ with a time-dependent Markovian potential (see ref.
Kang, Schenker).
