Up until now, we have only considered light propagating essentially in vacuum; this has been used as the basis to describe the effects of reflection, refraction, and transmission in a medium. These essentially passive phenomena, alone, serve as a reminder that nothing is achieved with light, or learned about it, without material interaction. However, a more extensive realm of science centers upon light actively coupling with matter - and many of the systems, in which the photonic aspects of light are most clearly manifest, concern matter in the form of atoms, molecules, or nanoparticles.
A quantum analysis of light–matter interactions typically begins with the system Hamiltonian H, the energy operator of the system. In basic quantum mechanics, the "system" simply comprises matter, and the state wavefunctions are secured as eigenstates of the corresponding Hamiltonian Hmat. A quantum formulation of the radiation requires the additional inclusion of a radiation Hamiltonian Hrad. Here, it becomes appropriate to use the representation of quantum electrodynamics (QED): however, because the charges that comprise matter move at significantly sub-luminal speeds in almost all optics and photonics applications, the full relativistic form of QED is unnecessary. To complete the full Hamiltonian for systems in which light and matter are not isolated from each other, an interaction Hamiltonian Hint is also required. The system Hamiltonian can thus be decomposed into three parts: