580
10 Quantum Statistics
Because we may treat n s and n s as statistically independent when s = s , we obtain
n s n s = n s n s for s = s , and only the terms for which s = s remain in the double
sum, thereby giving
N 2 − N
2 =
s
n s (1 + n s )
(10.6.52a)
for the variance in N (i.e., in the thermodynamic N). We may also express the
variance in N in terms of the photon state energies s as
N 2 − N
2 =
s
e ββ s
(e ββ s − 1) 2 ,
(10.6.52b)
from which it is also clear that the variance in N is a function of temperature alone.
Evaluation of the sum over photon states on the right-hand side of Eq. (10.6.52b)
gives the variance explicitly as
N 2 − N
2 =
8π 3 k 3
B
3h 3 c 3 V T
3
which, by employing Eq. (10.6.52b), may finally be written in terms of N as
N 2 − N
2 =
π 2
6ζ(3)
N .
(10.6.53)
This expression may, in turn, be utilized to establish that the relative rms deviation
σ N /N , with σ N defined via σ 2
N
≡ [N 2 − N
2 ]
1
2 , thus behaves as
σ N
N
=
1
N
π
√
6ζ(3)
1.170
N
.
(10.6.54)
10.7 Density Matrix and Ensemble Averages
In the derivation of the Liouville Equation given in Sect. 7.5.1, we introduced the
classical density of phase-space points, designated by f (N ) (p, q), associated with
a system of N particles. The equivalent quantity in a fully quantum mechanical
description is referred to as the density matrix operator, and is traditionally
designated by ρ. The form taken by ρ is most readily illustrated using Dirac
notation, in which the members of a set of quantum states {φ|i = 1, . . . , } are
designated {|i; i = 1, . . . , }, with |i known as a ‘ket’. The adjoint, |i † , of a ket