In the previous chapter we have seen a number of important outcomes from the application of quantum mechanics to electronic states in solids. The key outcomes are: (i) Depending upon the physical system (i.e., the potential energy profile) certain energy levels are allowed and other energies are forbidden. This means an electron in the system can only occupy the allowed energy levels, (ii) In some systems the allowed energies form a continuous band extending over a range of energies, (iii) In allowed bands near the edge of the bands it is possible to describe electrons by an effective mass and an effective equation of motion that looks similar to Newton's equation. In this chapter we will examine several categories of solids and see how the outcomes listed above impact their electronic properties.
The solution of the Schrodinger equation for a particular system is just the first step in being able to understand and manipulate the behavior of the system. The second step in the problem is to obtain information on the distribution of particles in the allowed states (energy levels). Finally, we use quantum mechanics to understand the response of the particles to an external disturbance. In the next section we will discuss how electrons are distributed in allowed states.
OCCUPATION OF STATES: DISTRIBUTION FUNCTION
Let us say we have solved the Schrodinger equation for electrons or Maxwell's equation for photons and we have a certain number of particles. How will the particles distribute among the allowed states? To answer this question we need to use statistical physics; in particular, quantum statistical physics.