One of the principal applications of quantum theory in physics is in the interpretation of spectroscopies. Spectroscopy is one of the key tools for learning about condensed matter on the microscopic scale. In this chapter we are going to consider the theory of spectroscopy on a quantum mechanical level with particular emphasis on effects that are intrinsically relativistic. This boils down to a study of the interaction of the electrons in the material with incident photons.

A bit of quantum field theory is unavoidable in this chapter. The field theory here is as elementary as it gets. Furthermore, it is conceptually easy and is introduced as a natural extension of the relativistic quantum theory already covered in this book, so it should present no difficulty.

If you scan through this chapter the mathematics appears pretty daunting, (so what's new). Don't worry, it could be worse, at least we make plenty of use of the Dirac notation introduced in chapter 2. Many of the equations in this chapter would be horrendous without it. In the first sections we discuss some properties of the photon and quantization of the electromagnetic field. Then we come to the backbone of the chapter, time-dependent perturbation theory, from which the Golden Rule for transition rates is derived to second order.