146 Canonical ensemble
which is just the cube of eqn. (4.5.1). Using eqn. (4.5.4), we obtain the partition
function as
Q(N,V,T) =1
N!
(
L
λ) 3 N
=
VN
N!λ^3 N. (4.5.12)
From eqn. (4.5.12), the thermodynamics can now be derived. Using eqn. (4.3.23) to
obtain the pressure yields
P=kT∂
∂V
ln[
VN
N!λ^3 N]
=NkT
∂lnV
∂V=
NkT
V, (4.5.13)
which we recognize as the ideal gas equation of state. Similarly, the energy is given by
E=−
∂
∂βln[
VN
N!λ^3 N]
= 3N
∂lnλ
∂β=
3 N
β=
3 N
2 β=
3
2
NkT, (4.5.14)which follows from the fact thatλ=
√
βh^2 / 2 πmand is the expected result from
the Virial theorem. From eqn. (4.5.14), it follows that the heat capacity at constant
volume is
CV=(
∂E
∂T
)
=
3
2
Nk. (4.5.15)Note that if we multiply and divide byN 0 , Avogadro’s number, we obtain
CV=
3
2
N
N 0
N 0 k=3
2
nR, (4.5.16)wherenis the number of moles of gas andRis the gas constant. Dividing by the
number of moles yields the expected result for themolarheat capacitycV= 3R/2.
4.5.2 The harmonic oscillator and the harmonic bath
We begin by considering a one-dimensional harmonic oscillator of massmand fre-
quencyωfor which the Hamiltonian is
H=
p^2
2 m+
1
2
mω^2 x^2. (4.5.17)The canonical partition function becomes
Q(β) =1
h∫
dpdxe−β(p(^2) / 2 m+mω (^2) x (^2) /2)
=
1
h∫∞
−∞dpe−βp(^2) / 2 m
∫L
0dxe−βmω(^2) x (^2) / 2
. (4.5.18)
Although the coordinate integration is restricted to the physical box containing the
oscillator, we will assume that the width of the distribution exp(−mω^2 x^2 /2) is very
small compared to the size of the (macroscopic) container so thatwe can perform