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# Illustrative Example

Consider a very simple gas made up of two identical particles. Suppose that each particle can be in one of three possible quantum states, . Let us enumerate the possible states of the whole gas according to Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics, respectively.

For the case of Maxwell-Boltzmann statistics, the two particles are considered to be distinguishable. Let us denote them and . Furthermore, any number of particles can occupy the same quantum state. The possible different states of the gas are shown in Table 8.1. There are clearly nine distinct states.

Table 8.1: Two particles distributed amongst three states according to Maxwell-Boltzmann statistics.
 1 2 3                           For the case of Bose-Einstein statistics, the two particles are considered to be indistinguishable. Let us denote them both as . Furthermore, any number of particles can occupy the same quantum state. The possible different states of the gas are shown in Table 8.2. There are clearly six distinct states.

Table 8.2: Two particles distributed amongst three states according to Bose-Einstein statistics.
 1 2 3                  Finally, for the case of Fermi-Dirac statistics, the two particles are considered to be indistinguishable. Let us again denote them both as . Furthermore, no more than one particle can occupy a given quantum state. The possible different states of the gas are shown in Table 8.3. There are clearly only three distinct states.

Table 8.3: Two particles distributed amongst three states according to Fermi-Dirac statistics.
 1 2 3         It follows, from the previous example, that Fermi-Dirac (FD) statistics are more restrictive (i.e., there are less possible states of the system) than Bose-Einstein (BE) statistics, which are, in turn, more restrictive than Maxwell-Boltzmann (MB) statistics. Let (8.12)

For the case under investigation,  (8.13)  (8.14)  (8.15)

We conclude that in Bose-Einstein statistics there is a greater relative tendency for particles to cluster in the same state than in classical (i.e., Maxwell-Boltzmann) statistics. On the other hand, in Fermi-Dirac statistics there is a lesser tendency for particles to cluster in the same state than in classical statistics.   Next: Formulation of Statistical Problem Up: Quantum Statistics Previous: Symmetry Requirements in Quantum
Richard Fitzpatrick 2016-01-25