The quantum nature of hydrogen chemisorbed on a metal surface is explored in detail. Starting from the effective-medium construction of the adiabatic potential energy surfaces, we demonstrate that due to zero-point motion chemisorbed hydrogen can be to a large extent delocalized in both ground and excited state configurations. A proper description can then only be given in terms of hydrogen energy bands. The quantitative calculations for Ni surfaces are used to discuss a number of experimental observations, in particular the vibrational excitations of hydrogen as measured by inelastic electron scattering. The further implications of this new description are also commented on.