  The semiclassical Schr\"{o}dinger equation with multiscale and random
potentials often appears when studying electron dynamics in heterogeneous
quantum systems. As time evolves, the wavefunction develops high-frequency
oscillations in both the physical space and the random space, which poses
severe challenges for numerical methods. In this paper, we propose a multiscale
reduced basis method, where we construct multiscale reduced basis functions
using an optimization method and the proper orthogonal decomposition method in
the physical space and employ the quasi-Monte Carlo method in the random space.
Our method is verified to be efficient: the spatial gridsize is only
proportional to the semiclassical parameter and the number of samples in the
random space is inversely proportional to the same parameter. Several
theoretical aspects of the proposed method, including how to determine the
number of samples in the construction of multiscale reduced basis and
convergence analysis, are studied with numerical justification. In addition, we
investigate the Anderson localization phenomena for Schr\"{o}dinger equation
with correlated random potentials in both 1D and 2D.
