A quantitative comparison of the results from quantum and classical computations of the dissociation of H-2 on the ab initio Cu(111) surface is presented. In order to initiate a detailed comparison of the classical and quantum methods, we chose to investigate motion on the vibrationally adiabatic ground state. This subtracts from the problem the largest quantum degree of freedom with the consequence that broadly speaking, quantum and classical results for the dissociation as a function of initial rotational state and incidence angle agree well. One feature arising from the calculations is that for the first time we have shown how the quantized nature of motion near the transition state affects experimental observables. We demonstrate that hitherto neglected structure in dissociation probabilities may be traced to quantized hindered rotational and translational motion. An additional consequence of the quantization is a pronounced J-dependent broadening in the dissociation probabilities when plotted as a function of the translational energy which should be experimentally observable.