526
10 Quantum Statistics
Thus, for λ 1, the Maxwell–Boltzmann mean occupation number n
MB
k
can be
written as
n
MB
k
=
N
z
e
−ββ k ,
(10.3.5a)
from which we see that the classical probability p MB
k
is related to the mean
occupation number via
p
MB
k
≡
e −ββ k
z
=
n
MB
k
N
.
(10.3.5b)
In this sense, for both FD and BE statistics, the mean occupation number n k reduces
to the classical probability p MB
k .
We note, in passing, that as the average number of particles, N, and the average
energy E, are given, respectively, by
N =
k
n k
k
λe
−ββ k
(10.3.6)
and
E
k
λλ k e
−ββ k ,
(10.3.7)
the average energy per particle, ≡ E/N, becomes
=
k
λλ k e
−ββ k
k
λe
−ββ k
=
1
z
k
k e
−ββ k ,
(10.3.8)
and the absolute activity λ (equivalently, the chemical potential μ) no longer plays
a role in the statistical expressions.
In this same limit, the equation of state becomes
P V = ±k B T
k
(±)λe
−ββ k .
(10.3.9)
Upon utilizing Eq. (10.2.11) for and employing the approximation ln(1 + x) x
for x 1, we may also express P V as
P V = λk B T
k
e
−ββ k
= k B T λ z .
(10.3.10)