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## An 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 (MB) 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 Tab. 6.

Table 6: Two particles distributed amongst three states according to Maxwell-Boltzmann statistics.
 1 2 3

There are clearly 9 distinct states.

For the case of Bose-Einstein (BE) 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 Tab. 7.

Table 7: Two particles distributed amongst three states according to Bose-Einstein statistics.
 1 2 3

There are clearly 6 distinct states.

Finally, for the case of Fermi-Dirac (FD) 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 Tab. 8.

Table 8: Two particles distributed amongst three states according to Fermi-Dirac statistics.
 1 2 3

There are clearly only 3 distinct states.

It follows, from the above example, that Fermi-Dirac statistics are more restrictive (i.e., there are less possible states of the system) than Bose-Einstein statistics, which are, in turn, more restrictive than Maxwell-Boltzmann statistics. Let

 (575)

For the case under investigation,
 (576) (577) (578)

We conclude that in Bose-Einstein statistics there is a greater relative tendency for particles to cluster in the same state than in classical statistics. On the other hand, in Fermi-Dirac statistics there is less tendency for particles to cluster in the same state than in classical statistics.

Next: Formulation of the statistical Up: Quantum statistics Previous: Symmetry requirements in quantum
Richard Fitzpatrick 2006-02-02