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Up to now, we have assumed that the number of particles,
, contained
in a given system is a fixed
number. This is a reasonable assumption if the particles possess nonzero
mass, because we are not generally considering
relativistic systems in this course (i.e., we are assuming that the particle energies are much less than their restmass energies).
However, this assumption breaks down for the case of photons, which are
zeromass bosons. In fact, photons enclosed in a container
of volume
, maintained at temperature
, can readily be absorbed
or emitted by the walls. Thus, for the special case of a gas of
photons, there is no requirement that limits the total number of particles.
It follows, from the previous discussion, that photons obey a simplified
form of BoseEinstein statistics in which there is an unspecified
total number of particles. This type of statistics is called photon
statistics.
Consider the expression (8.21). For the case of photons, the
numbers
assume all values
for each
,
without any further restriction. It follows that the sums
in the
numerator and denominator are identical and, therefore, cancel. Hence,
Equation (8.21) reduces to

(8.35) 
However, the previous expression can be rewritten

(8.36) 
Now, the sum on the righthand side of the previous equation is an infinite
geometric series, which can easily be evaluated. In fact,

(8.37) 
(See Exercise 1.)
Thus, Equation (8.36) gives

(8.38) 
or

(8.39) 
This is known as the Planck distribution, after the German physicist
Max Planck who first proposed it in 1900 on purely empirical grounds.
Equation (8.20) can be integrated to
give

(8.40) 
where use has been made of Equation (8.39).
Next: BoseEinstein Statistics
Up: Quantum Statistics
Previous: FermiDirac Statistics
Richard Fitzpatrick
20160125