  We consider a quantum particle interacting with $N$ obstacles, whose
positions are independently chosen according to a given probability density,
through a two-body potential of the form $N^2 V(Nx)$ (Gross-Pitaevskii
potential). We show convergence of the $N$ dependent one-particle Hamiltonian
to a limiting Hamiltonian where the quantum particle experiences an effective
potential depending only on the scattering length of the unscaled potential and
the density of the obstacles. In this sense our Lorentz gas model exhibits a
universal behavior for $N$ large. Moreover we explicitely characterize the
fluctuations around the limit operator. Our model can be considered as a
simplified model for scattering of slow neutrons from condensed matter.
