We construct the theory of carriers confined in spherical Si quantum dots with finite energy barriers for electrons and holes in the framework of Luttinger Hamiltonian for holes and taking into account the strong anisotropy of the conduction electron effective mass in Si. As a boundary condition for the electron and hole wave functions we use continuity of the wave functions and the current at the boundary of the nanocrystals. We apply this theory for the case of the SiO₂ matrix surrounding Si quantum dots. We show that for experimentally relevant quantum dots energy spacings between neighbouring electron and hole levels are of the order of hundreds of meV. Therefore the relaxation of excited electrons and holes is damped. Theoretical calculations of probabilities of various radiative transitions are presented.