6. Macroscopic Systems


Abstract

Macroscopic systems involve extremely large numbers of particles. Such systems are very hard to analyze exactly in quantum mechanics. An exception is a system of noninteracting particles stuck in a rectangular box. This chapter therefore starts with an examination of that model. For a model of this type, the system energy eigenfunctions are found to be products of single-particle states.

One thing that becomes quickly obvious is that macroscopic system normally involve a gigantic number of single-particle states. It is unrealistic to tabulate them each individually. Instead, average statistics about the states are derived. The primary of these is the so-called density of states. It is the number of single-particle states per unit energy range.

But knowing the number of states is not enough by itself. Information is also needed on how many particles are in these states. Fortunately, it turns out to be possible to derive the average number of particles per state. This number depends on whether it is a system of bosons, like photons, or a system of fermions, like electrons. For bosons, the number of particles is given by the so-called Bose-Einstein distribution, while for electrons it is given by the so-called Fermi-Dirac distribution. Either distribution can be simplified to the so-called Maxwell-Boltzmann distribution under conditions in which the average number of particles per state is much less than one.

Each distribution depends on both the temperature and on a so-called chemical potential. Physically, temperature differences promote the diffusion of thermal energy, heat, from hot to cold. Similarly, differences in chemical potential promote the diffusion of particles from high chemical potential to low.

At first, systems of identical bosons are studied. Bosons behave quite strangely at very low temperatures. Even for a nonzero temperature, a finite fraction of them may stay in the single-particle state of lowest energy. That behavior is called Bose-Einstein condensation. Bosons also show a lack of low-energy global energy eigenfunctions.

A first discussion of electromagnetic radiation, including light, will be given. The discussed radiation is the one that occurs under conditions of thermal equilibrium, and is called blackbody radiation.

Next, systems of electrons are covered. It is found that electrons in typical macroscopic systems have vast amounts of kinetic energy even at absolute zero temperature. It is this kinetic energy that is responsible for the volume of solids and liquids and their resistance to compression. The electrons are normally confined to a solid despite all their kinetic energy. But at some point, they may escape in a process called thermionic emission.

The electrical conduction of metals can be explained using the simple model of noninteracting electrons. However, electrical insulators and semiconductors cannot. It turns out that these can be explained by including a simple model of the forces on the electrons.

Then semiconductors are discussed, including applications such as diodes, transistors, solar cells, light-emitting diodes, solid state refrigeration, thermocouples, and thermoelectric generators. A somewhat more general discussion of optical issues is also given.




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