Transport in SiO2 under the influence of electric fields of the order of several MV/cm is considered. We perform a Monte Carlo computer simulation in which the effect of higher conduction bands is included. In agreement with recent experiments, we find that the electronic distribution is stable at average energies of a few eV. It is found that for the high energies and collision rates in this energy and field regime, the inclusion of quantum mechanical effects is important. In particular, we model the collisional broadening of the final energy and the intra-collisional field effect, i.e. the acceleration during a finite collision duration. We find that the net effect of these quantum corrections is an increase in average energy for fields up to 3 MV/cm and a decrease in average energy for still higher fields.