We develop a fully quantum mechanical formalism to calculate ionic transition states in solids and determine diffusion constants for H in Si and Li and Fe in LiFePO4. The formalism is quantitative and does not involve empirical parameters. From the quantum mechanical treatment we recover some quantities known from classical theory, e.g. the temperature dependent diffusion constant reflects the activation energy at high T. At low temperature however we discover a constant diffusion rate linked to the ionic tunneling. Tunneling and hopping rates are considered on an equal footing and result from the same formalism. We apply the quantum mechanical formalism to the diffusion of H in Si and discover the influence of the zero point energy of the diffusive species in the potential well. For LiFePO4 we shed some light on the importance of the cross channel diffusion probably intermediated by Fe anti-site vacancies in the understanding of the experimentally observed diffusion constant. This work opens up the possibility to study quantitatively diffusion e.g. in potential electrode materials for Li-ion batteries. (C) 2016 Elsevier B.V. All rights reserved.