According to the Heisenberg uncertainty principle of quantum mechanics, particles which are localized in space by a bounding potential must have a finite distribution of momenta. This leads, even in the lowest possible energy state, to vibrations, and thus, to the so-called zero-point energy. For chemically bound hydrogen the zero-point energy can be quite substantial. For example, for a free H-2 molecule it is 0.26 eV, a significant value in the realm of chemistry, where often an energy of the order of 0.1 eV/atom (or 2.3 kcal/mol) decides whether or not a chemical reaction takes place with an appreciable rate. Yet, in many theoretical studies the dynamics of chemical reactions involving hydrogen has been treated classically or quasiclassically, assuming that the quantum mechanical nature of H nuclei, i.e., the zero-point effects, will not strongly affect the relevant physical or chemical properties. In this article we show that this assumption is not justified. We will demonstrate that for very basic and fundamental catalytic reaction steps, namely the dissociative adsorption of molecular hydrogen at transition metal surfaces and its time reverse process, the associative desorption, zero-point effects cannot only quantitatively but even qualitatively affect the chemical processes and rates. Our calculations (treating electrons as well as H nuclei quantum mechanically) establish the importance of additional zero-point effects generated by the H-2 surface interaction and how energy of the H-H stretch vibration is transferred into these and vice versa.