420 Quantum ideal gases
λ^3 ρ
g=
λ^3 ρ
g+a 2 ρ^2 −1
23 /^2
λ^6 ρ^2
g^2, (11.5.13)
or
a 2 =λ^6
23 /^2 g^2, (11.5.14)
from which
ζ≈λ^3 ρ
g+
λ^6
23 /^2 g^2ρ^2 , (11.5.15)and the equation of state becomes
P
kT=ρ+λ^3
25 /^2 gρ^2. (11.5.16)From the equation of state, we can read off the second virial coefficient
B 2 (T) =
λ^3
25 /^2 g≈ 0. 1768
λ^3
g> 0. (11.5.17)
Even at second order, we observe a nontrivial quantum effect, in particular, a second
virial coefficient with a nonzero value despite the absence of interactions among the
particles. The implication of eqn. (11.5.17) is that there is an effective“interaction”
among the particles as a result of the fermionic spin statistics. This “interaction” tends
to increase the pressure above the classical ideal gas result (B 2 (T)>0) and hence is
repulsive in nature. This result is a consequence of the Pauli exclusion principle: If
we imagine filling the energy levels, then since no two particles can occupy the same
quantum state, once the ground staten= (0, 0 ,0) is fully occupied by particles with
differentSˆzeigenvalues, the next particle must go into a higher energy state. The
result is an effective “repulsion” among the particles that pushes them into increasingly
higher energy states so as not to violate the Pauli principle.
If the third-order contribution is worked out, one finds (see Problem 11.1) that
a 3 =(
1
4
−
1
33 /^2
)
λ^9
g^3ζ=λ^3 ρ
g+
λ^6
23 /^2 g^2ρ^2 +(
1
4
−
1
33 /^2
)
λ^9
g^3ρ^3P
kT=ρ+λ^3
25 /^2 gρ^2 +λ^6
g^2(
1
8
−
2
35 /^2
)
ρ^3 , (11.5.18)so thatB 3 (T)<0. Since the third-order term is a second-order correction to the
ideal-gas equation of state, the fact thatB 3 (T)<0 is consistent with time-independent
perturbation theory, wherein the second-order correction lowers all of the energy levels.