The previous sections were chiefly concerned with Hamiltonians under the long-wavelength approximation and for a single photonic mode to more clearly demonstrate the relations between different gauges and representations. However, to accurately reproduce the results of experiments such as those with Fabry-Pérot cavities, 1-7,9,10,12,87-98 these approximations are no longer adequate. In this manner, we must use generalized Hamiltonians to model such systems, so building on the formalism introduced in the previous sections, we use this section as a practical example of creating Hamiltonians that represent more complex systems. In doing so, many modes coupled to many molecules are considered, and we partially relax the long-wavelength approximation such that Â is no longer spatially invariant while the matter interactions are still approximated as dipoles. Such a Hamiltonian is necessary to describe many molecules coupled to a Fabry-Pérot cavity, as depicted in Fig. . In that situation, we explicitly consider a 1-D array of molecules. Several useful review articles related to this topic can be found in Ref. 100.