In chapter two, the interaction potential between the atom and field was written as
The semiclassical model assumed that was classical, linearly polarized radiation. In order to generate a quantum mechanical model, eq:j1 is written as
where g is the atom--field coupling constant, and are the raising and lowering operators for the field, and and are the raising and lowering operators for the atom. The latter two operators were not introduced in chapter two because a different route of derivation was utilized. The complete hamiltonian for the atom--field system is the hamiltonian for the field plus the hamiltonian for the atom and the interaction potential:
where provides a positive one if it operates on an atom in the upper state and a negative one if it operates on an atom in the lower state. eq:j3 neglects the zero point energy of the quantum field and defines the upper atomic state with energy and the lower with energy . In chapter two, the upper state had an energy of , and the lower state had zero energy. The total energy remains in either case because energy is arbitrary to an additive constant. Although eq:j3 seems like a different hamiltonian from the one analyzed in the semiclassical theory, they are equivalent.