  In this paper, we consider the numerical solution of the one-dimensional
Schr\"odinger equation with a periodic lattice potential and a random external
potential. This is an important model in solid state physics where the
randomness is involved to describe some complicated phenomena that are not
exactly known. Here we generalize the Bloch decomposition-based time-splitting
pseudospectral method to the stochastic setting using the generalize polynomial
chaos with a Galerkin procedure so that the main effects of dispersion and
periodic potential are still computed together. We prove that our method is
unconditionally stable and numerical examples show that it has other nice
properties and is more efficient than the traditional method. Finally, we give
some numerical evidence for the well-known phenomenon of Anderson localization.
