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10 Quantum Statistics
for each of the three types of statistics that we have been discussing. That is, we
shall calculate the quantity
r ≡
probability that the 2 particles are found in same state
probability that the 2 particles are found in different states
We obtain the answers:
r MB =
3/9
6/9
=
1
2
,
r FD =
0/3
3/3
= 0 ,
r BE =
3/6
3/6
= 1 ,
from which we may deduce that the BE case has a greater tendency for particles to
bunch up, while the FD case has a greater tendency for particles to be in different
states.
Literal enumeration of the number of microstates accessible to systems consisting of large numbers of Bose–Einstein or Fermi–Dirac particles along the lines that
we have just considered for two such particles in Example 10.1 would quickly
become a formidable task. We may, however, utilize combinatoric formulae to
determine the numbers of possible BE or FD microstates accessible to such multiparticle systems. The number of microstates obtained for n b bosons distributed over
n s states is given by
N BE =
(n b + n s − 1)!
n b ! (n s − n b )!
,
(10.1.1a)
while the number of microstates obtained for n f fermions distributed over n s states
is given by
N FD =
n s !
n f ! (n s − n f )!
.
(10.1.1b)
For comparison, we note that the expression for the number of microstates available
to a system of distinguishable particles corresponding to classical Maxwell–
Boltzmann statistics, with n s states available to n mb particles, is given by
N MB = (n s )
n mb .
(10.1.1c)
Let us now consider directly the effect on enumeration of the many-body states
available to a macroscopic system of particles. The energy for a given state is