(a) Denote the two states with \(\varepsilon_0\) = 0 by A and B and the states with \(\varepsilon\) and 2\(\varepsilon\) by 1 and 2 respectively.
(1) The system can have the following configurations if the particles obey fermi statistics: ,
Configuration: (A, B) (A, 1) (B, 1) (A, 2) (B, 2) (1, 2)
Energy: \(0\,\varepsilon\,\varepsilon\,2\varepsilon\,2\varepsilon\,3\varepsilon\)
Thus the partition function is \(Z = 1 + 2e^{-\varepsilon} + 2e^{-2\varepsilon} + e^{-3\varepsilon}
\),
and the mean energy is \(\bar \varepsilon = (2\varepsilon e^{-\varepsilon} + 4\varepsilon e^{-2\varepsilon} + 3\varepsilon e^{-3\varepsilon})/Z\)
(2) If the particles obey Bose statistics, in addition to the above states, the following configurations are also possible:
Configuration: (A, A) (B, B) (1, 1) (2, 2)
Energy: \(0\,0\,2\varepsilon\, 4\varepsilon\)
Hence the partition function and average energy are
(3) for destinguisable particles obeying Boltzmann statistics, more configurations are possible. These are (B, A), (1, A), (1, B), (2, A), (2, B) and (2, 1). Thus we have
(b) Fermions and bosons can be treated as Boltzmann particles when the number of particles is much less than the number of energy levels, for then the exchange effect can be neglected.
