Expanding on an earlier Communication [M. H. Alexander, H.-J. Werner, and D. E. Manolopoulos, J. Chem. Phys. 109, 5710 (1998)], we present here the full framework for the quantum treatment of reactions of the fluorine atom with molecular hydrogen. This involves four potential energy surfaces (PESs) and two, coordinate-dependent spin-orbit interaction terms, all of which were fitted to the results of ab initio calculations. Quantum scattering calculations, based on a time-independent method formulated in hyperspherical coordinates, were carried out to determine initial and final state-resolved reactive cross sections, for reaction of F in its ground (P-2(3/2)) and excited (P-2(1/2)) spin-orbit state with H-2 in j=0 and j=2(pH(2)) and j=1(oH(2)). The overall reactivity of the excited state of F, which can occur only through nonadiabatic transitions, is found to be small, at most 25% of the reactivity of the ground spin-orbit state, which is adiabatically allowed. In addition, when compared with results of earlier calculations, based on a single, electronically adiabatic, PES, our calculations show that even fine details of the dynamics of the F+H-2 reaction will be well described by calculations on a single PES. The contribution of the excited spin-orbit state can be seen most clearly in the formation of HF products in the v=3 vibrational manifold, which are nearly thermoneutral (or even slightly endoergic) in the reaction of ground-state F atoms. The cross section for the near resonant electronic-rotational process [F*+H-2(j=0)-->F+H-2(j=2)] is found to be large, in confirmation of earlier work.