Einstein-De Broglie relations on the lattice

Historically the starting point of wave mechanics is the Planck and Einstein-de Broglie relations for the energy and momentum of a particle, where the momentum is connected to the group velocity of the wave packet. We translate the arguments given by de Broglie to the case of a wave defined on the grid points of a space-time lattice and explore the physical consequences such as integral period, wave length, discrete energy, momentum and rest mass.