When atoms are assembled into a condensed state, it is often the case that the outer valence electrons become delocalized, and are no longer associated with a given atom. The most obvious such case is a metal where the electrons are free to move, and can conduct electricity. An ionized classical plasma is another system of this sort, but, as we have seen, the electrons in solids must be treated with quantum theory. That is the subject of this chapter. Metallic liquids and glasses exist, but we will concentrate on metallic crystals. As we will see, the theory that we will develop will also apply to the valence electrons in semiconductors and insulators.

Since electrons in metals are free to move it is natural to think of them as a gas; the term electron gas is often used. The most extreme version of this idea is surprisingly useful, namely the idea of the free electron gas. In this idealization the electrons don't see the ions that they were detached from except in an average way, to neutralize their charge. Also, in this model the electrons are non-interacting, and act as if their Coulomb repulsion is not present.

We should say from the outset that both these assumptions appear to be totally unreasonable. The strength of the electron-electron interaction in Cu was estimated in Chapter 1. It turned out to be about 3 eV, which is the same order of magnitude as the energies that we will find in the next section.