Relativistic coupling of internal and center of mass dynamics in classical and simple bound quantum mechanical systems

Einstein's famous equation $E_{\rm rest}=mc^2$ for the rest energy of a system with mass $m$ requires that the internal energy of the system be included in $m$. Pursuing this idea using Lagrangian and Hamiltonian dynamics yields a relativistic coupling between the center of mass motion and the internal dynamics of the system. Here we explore the consequences of this coupling, first classically, where we find that the dynamics of the system is time dilated when moving relative to another inertial frame. We then apply the dynamics to a quantum 2-level atom bound in a 1-dimensional infinite potential well, and show that the coupling produces collapses and revivals in quantum interference.