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10 Quantum Statistics
(particles possessing an intrinsic integer or zero spin), the N-body wavefunction
must be symmetric to the interchange of two identical bosons. The statistics obeyed
by these two basic types of particle are quite different, and are known, respectively,
as Fermi–Dirac (FD) and Bose–Einstein (BE) statistics.
Fermi–Dirac and Bose–Einstein statistics are, in fact, the only exact statistics,
and are hence the only appropriate statistics to use, especially for the treatment of
indistinguishable species. Of course, classical (or Maxwell–Boltzmann) statistics
applies for systems of distinguishable species, and we have argued heuristically in
Chap. 3 (Sect. 3.2.3) that, as first proposed by Gibbs, the canonical partition function
Z N (T , V ) for N indistinguishable particles can be obtained from that for N distinguishable particles via division by N!. Indeed, we shall see in Sect. 10.3 that both the
exact Fermi–Dirac and Bose–Einstein quantum-statistical partition functions reduce
under appropriate conditions to the Gibbs version of the Maxwell–Boltzmann
canonical partition function for a system of N indistinguishable particles.
Fermions
• A fundamental QM symmetry requirement for fermions is that the total wavefunction T for an N-particle system be antisymmetric under the interchange of
any two of the identical fermions, viz.,
T (· · · i · · · j · · · ) = − T (· · · j · · · i · · · ) ;
this does not lead to a new state of the gas, so that the particles must be genuinely
indistinguishable in enumerating the distinct states of the gas.
• Suppose now that two particles i and j , both in the same single-particle state s,
are interchanged, then necessarily
T (· · · i · · · j · · · ) = T (· · · j · · · i · · · ) .
Because both this statement and the previous one can be true simultaneously only
if T ≡ 0, this implies that there are no states of a fermi gas for which two or
more particles are in the same single-particle state. This is the origin of the Pauli
Exclusion Principle.
Bosons
• A fundamental QM symmetry requirement for bosons is that the total wavefunction T for an N-particle system be symmetric under the interchange of any two
of the identical bosons, viz.,
T (· · · i · · · j · · · ) = T (· · · j · · · i · · · ) ,
and we see that in this case, there is no restriction on how many bosons may be
in any given one-particle state.
10 Quantum Statistics
(particles possessing an intrinsic integer or zero spin), the N-body wavefunction
must be symmetric to the interchange of two identical bosons. The statistics obeyed
by these two basic types of particle are quite different, and are known, respectively,
as Fermi–Dirac (FD) and Bose–Einstein (BE) statistics.
Fermi–Dirac and Bose–Einstein statistics are, in fact, the only exact statistics,
and are hence the only appropriate statistics to use, especially for the treatment of
indistinguishable species. Of course, classical (or Maxwell–Boltzmann) statistics
applies for systems of distinguishable species, and we have argued heuristically in
Chap. 3 (Sect. 3.2.3) that, as first proposed by Gibbs, the canonical partition function
Z N (T , V ) for N indistinguishable particles can be obtained from that for N distinguishable particles via division by N!. Indeed, we shall see in Sect. 10.3 that both the
exact Fermi–Dirac and Bose–Einstein quantum-statistical partition functions reduce
under appropriate conditions to the Gibbs version of the Maxwell–Boltzmann
canonical partition function for a system of N indistinguishable particles.
Fermions
• A fundamental QM symmetry requirement for fermions is that the total wavefunction T for an N-particle system be antisymmetric under the interchange of
any two of the identical fermions, viz.,
T (· · · i · · · j · · · ) = − T (· · · j · · · i · · · ) ;
this does not lead to a new state of the gas, so that the particles must be genuinely
indistinguishable in enumerating the distinct states of the gas.
• Suppose now that two particles i and j , both in the same single-particle state s,
are interchanged, then necessarily
T (· · · i · · · j · · · ) = T (· · · j · · · i · · · ) .
Because both this statement and the previous one can be true simultaneously only
if T ≡ 0, this implies that there are no states of a fermi gas for which two or
more particles are in the same single-particle state. This is the origin of the Pauli
Exclusion Principle.
Bosons
• A fundamental QM symmetry requirement for bosons is that the total wavefunction T for an N-particle system be symmetric under the interchange of any two
of the identical bosons, viz.,
T (· · · i · · · j · · · ) = T (· · · j · · · i · · · ) ,
and we see that in this case, there is no restriction on how many bosons may be
in any given one-particle state.