8: Quantum Statistical Mechanics
- Page ID
- 32042
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One of the important lessons of quantum mechanics is that there is no a priori meaning to qualities of any system, no independent reality, aside from what can be defined operationally in terms of observations. Thus we cannot speak of this electron (or photon, or any other particle) versus that electron (or photon, or any other particle). We can only say that there is one particle with a certain set of values for observables and there is another, perhaps with a different set of values for observables. This basic identity of particles affects the counting of states and hence leads to distributions different from the Maxwell-Boltzmann distribution we have discussed. This is the essential refinement due to quantum statistics.
- 8.1: Prelude to Quantum Statistical Mechanics
- There are two kinds of particles from the point of view of statistics, bosons and fermions. The corresponding statistical distributions are called the Bose-Einstein distribution and the Fermi-Dirac distribution. Bosons have the property that one can have any number of particles in a given quantum state, while fermions obey the Pauli exclusion principle which allows a maximum of only one particle per quantum state. Any species of particles can be put into one of these two categories.
- 8.2: Bose-Einstein Distribution
- We will now consider the derivation of the distribution function for free bosons carrying out the counting of states along the lines of what we did for the Maxwell-Boltzmann distribution. Let us start by considering how we obtained the binomial distribution. We considered a number of particles and how they can be distributed among, say, K boxes. As the simplest case of this, consider two particles and two boxes. The ways in which we can distribute them are as shown below.
- 8.3: Fermi-Dirac Distribution
- The counting of distinct arrangements for fermions is even simpler than for the Bose-Einstein case, since each state can have an occupation number of either zero or 1. Thus consider g states with n particles to be distributed among them. There are n states which are singly occupied and these can be chosen in g!(n!(g−n)!) ways.
- 8.4: Applications of the Bose-Einstein Distribution
- We shall now consider some simple applications of quantum statistics, focusing in this section on the Bose-Einstein distribution.
- 8.5: Applications of the Fermi-Dirac Distribution
- We now consider some applications of the Fermi-Dirac distribution (8.2.5). It is useful to start by examining the behavior of this function as the temperature goes to zero. Thus all states below a certain value, which is the zero-temperature value of the chemical potential, are filled with one fermion each. All states above this value are empty. This is a highly quantum state. The value of ϵ for the highest filled state is called the Fermi level.