PROBABILITY
particles can be distributed among allRquantum states of the system, withniparticles in
theith level, is given by
W{ni}=
N!
n 1 !n 2 !···nR!∏R
i=1wi for distinguishable particles,∏Ri=1wi for indistinguishable particles. (30.37)It therefore remains only for us to find the appropriate expression forwiin each of the
cases (i)–(iv) above.
Case(i). If there is no restriction on the number of particles in each quantum state,
then in theith energy level each particle can reside in any of thegidegenerate quantum
states. Thus, if the particles are distinguishable then the number of distinct arrangements
is simplywi=gnii. Thus, from (30.37),
W{ni}=N!
n 1 !n 2 !···nR!∏R
i=1gnii=N!∏R
i=1gnii
ni!.
Such a system of particles (for example atomsor molecules in a classical gas) is said to
obey Maxwell–Boltzmann statistics.
Case(ii). If the particles are indistinguishable and there is no restriction on the number
in each state then, from (30.36), the number of distinct arrangements of theniparticles
among thegistates in theith energy level is
wi=(ni+gi−1)!
ni!(gi−1)!.
Substituting this expression in (30.37), we obtain
W{ni}=∏R
i=1(ni+gi−1)!
ni!(gi−1)!.
Such a system of particles (for example a gas of photons) is said to obey Bose–Einstein
statistics.
Case(iii). If a maximum of one particle can reside in each of thegidegenerate quantum
states in theith energy level then the number of particles in each state is either 0 or 1.
Since the particles are indistinguishable,wiis equal to the number of distinct arrangements
in whichnistates are occupied andgi−nistates are unoccupied; this is given by
wi=giCni=gi!
ni!(gi−ni)!.
Thus, from (30.37), we have
W{ni}=∏R
i=1gi!
ni!(gi−ni)!.
Such a system is said to obey Fermi–Diracstatistics, and an example is provided by an
electron gas.
Case(iv). Again, the number of particles in each state is either 0 or 1. If the particles
are distinguishable, however, each arrangement identified in case (iii) can be reordered in
ni! different ways, so that
wi=giPni=gi!
(gi−ni)!