  We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG)
methods for a class of highly oscillatory transport equations that arise in
semiclassical modeling of non-adiabatic quantum dynamics. These models contain
uncertainties, particularly in coefficients that correspond to the potentials
of the molecular system. We first focus on a highly oscillatory scalar model
with random uncertainty. Our method is built upon the nonlinear geometrical
optics (NGO) based method, developed in \cite{NGO} for numerical approximations
of deterministic equations, which can obtain accurate pointwise solution even
without numerically resolving spatially and temporally the oscillations. With
the random uncertainty, we show that such a method has oscillatory higher order
derivatives in the random space, thus requires a frequency dependent
discretization in the random space. We modify this method by introducing a new
"time" variable based on the phase, which is shown to be non-oscillatory in the
random space, based on which we develop a gPC-SG method that can capture
oscillations with the frequency-independent time step, mesh size as well as the
degree of polynomial chaos. A similar approach is then extended to a
semiclassical surface hopping model system with a similar numerical conclusion.
Various numerical examples attest that these methods indeed capture accurately
the solution statistics {\em pointwisely} even though none of the numerical
parameters resolve the high frequencies of the solution.
