In this paper we develop the nonlinear Maxwell equations that describe the coupling of the quantum mechanical Heisenberg representation electromagnetic field operator to the excitations of a uniform many-particle system. It is assumed that these excitations can be treated as harmonic, in the sense that one can neglect correlations between them. We give an iterative method for solving these equations appropriate when one or more strong beams are incident on the plasma and produce weaker scattered radiation. Applying the lowest iteration to the optical mixing of two beams by a plasma, we find a result in agreement with that of Kroll, Ron, and Rostoker. We then solve for the scattering of one light beam by another in a plasma, a process proposed by DuBois and Gilinsky, and by Platzman, Tzoar, and Buchsbaum, and find a result far too small to be observed. Finally, we study the stimulated Raman scattering from an electron plasma and find that, although it would be difficult to observe with present technology, it is the most likely of these effects to be observed in plasmas in the near future.