Statistical Mechanics - Physics at Oregon State University

38 CHAPTER 2. THE CANONICAL ENSEMBLE
as expected. The system is in the ground state, which is non-degenerate. For
large values of the temperature we get
T → ∞ : U ≈ 2kBT, S ≈ 2kB − 2kB log( ω
) (2.65)
kBT
This makes sense too. When the temperature is large, the details of the quantum
levels are unimportant and the internal energy should be proportional to kBT .
and the entropy should
The number of quantum states at energy kBT is kBT
ω
be proportional to log( kBT
ω ).
The heat capacity is also easy to get:
C(T ) = ω d ω (ω)2
coth( ) =
dT 2kBT 2kBT 2
The heat capacity takes the simple limiting forms
1
sinh 2 ( ω
2kBT )
(2.66)
T → 0 : C ≈ 0 (2.67)
T → ∞ : C ≈ 2kB
(2.68)
In a later chapter we will derive the result that a classical harmonic oscillator
in D dimensions has an internal energy of the form DkBT . If the temperature is
large, our quantum system will mainly be in states with large quantum numbers,
and the correspondence principle tells us that such a system behaves like a
classical system. Our calculated value of the internal energy in the limit T → ∞
is consistent with this idea.
In order to introduce other extensive variables, we have to make assumptions
for ω. Suppose the frequency ω is a linear function of volume, ω = αV , with
α > 0. In that case we have F(T,V) and the pressure is given by
pV = −ω coth( ω
) (2.69)
kBT
This pressure is negative! This is an immediate consequence of our relation
between frequency and volume, which implies that the energy increases with
increasing volume, opposite to the common situation in a gas of volume V.
The previous formula also shows that the pressure is directly related to the
temperature, as it should be when there are only two sets of conjugate variables.
The limiting values of the pressure are
T → 0 : pV ≈ −ω (2.70)
T → ∞ : pV ≈ −2kBT (2.71)
We can compare these values with the formula for the pressure we derived before
∂U
p = −
∂V
∂S
+ T
∂V
(2.72)
T
T