522
10 Quantum Statistics
The mean occupation numbers n k (( k ) thus provide us with a means of calculating
averages of quantities that depend upon the energies k of the quantum particles. The
simplest such calculation is that of the internal energy, which is given by
U(T , V , λ) ≡
k
n k (( k )) k =
k
λλ k e −ββ k
1 ± λe −ββ k
.
(10.2.23)
As for our calculation of the grand potential , T , λ), we shall replace the
summation over k by an integration over :
U(T , V , μ) =
(2I a + 1)V
3 (T )
2k B T
√
π
∞
0
x
3
2
e x−α ± 1
dx ,
(10.2.24)
so that a comparison between Eqs. (10.2.15) and (10.2.24), recalling also that =
−P V , shows us that for both BE and FD gases we have
P V =
2
3 U ,
(10.2.25)
which is the same relation that we obtained for a classical gas.
Example 10.2 Equation of state for structureless particles in a gravitational field.
We have seen in Example 3.4 of Chap. 3 that the density-of-states for structureless particles subject to a gravitational field characterized by an acceleration g is
given by
(x, η) =
4πVβ
3η
2m
h 2 β
3
2
x
3
2 − (x − η)
3
2 H(η/x)
,
in which x ≡ ββ, η = βmgH , and H(y) is the Heaviside function. The effect of the
gravitational field is parameterized by η. Equation (10.2.14a) for the grand potential
, V , μ) = −P V then gives the equation of state as
P V = ±k B T
∞
0
(x, η) ln(1 ± e
α−x ) dx
(α ≡ βμ), so that P V is given by
P V = ±
4V
√
π
3ηη 3 k B T
∞
0
x
3
2 − (x − η)
3
2 H(η/x)
ln(1 ± e
α−x ) dx .
The integral appearing in the first term, namely
I (α) =
∞
0
x
3
2 ln(1 ± e
α−x ) dx