94 Microcanonical ensemble
The partition function isΩ(E) =
E 0
h∫∞
−∞dp∫∞
−∞dx δ(
p^2
2 m+
1
2
kx^2 −E)
. (3.6.3)
In order to evaluate the integral in eqn. (3.6.3), we first introducea change of variables
p ̃=p
√
2 mx ̃=√
k
2x, (3.6.4)so that the partition function can be written as
Ω(E) =
E 0
h√
m
k∫∞
−∞d ̃p∫∞
−∞d ̃x δ(
̃p^2 + ̃x^2 −E)
. (3.6.5)
Recall from Section 1.3, however, that
√
k/m=ωis just the fundamental frequency
of the oscillator. The partition function then becomes
Ω(E) =
E 0
hω∫∞
−∞d ̃p∫∞
−∞d ̃x δ(
p ̃^2 + ̃x^2 −E)
. (3.6.6)
Theδ-function requires that ̃p^2 + ̃x^2 =E, which defines a circle in the scaled ( ̃p, ̃x)
phase space. Therefore, it is natural to introduce polar coordinates in the form
p ̃=√
Iωcosθx ̃=√
Iωsinθ. (3.6.7)Here, the usual “radial” coordinate has been expressed as
√
Iω. The new coordinates
(I,θ) are known asaction-anglevariables. They are chosen such that the Jacobian is
simply a constant,ω, so that the partition function becomes
Ω(E) =
E 0
h∫ 2 π0dθ∫∞
0dI δ(Iω−E). (3.6.8)In action-angle variables, the harmonic Hamiltonian has the rather simple formH=
Iω. If one were to derive Hamilton’s equations in terms of action-angle variables, the
result would be simply,θ ̇=∂H/∂I=ωandI ̇=−∂H/∂θ= 0 so that the actionIis a
constantI(0) for all time, andθ=ωt+θ(0). The constancy of the action is consistent
with energy conservation;I∝E. The angle then gives the oscillatory time dependence
ofxandp. In eqn. (3.6.8), the angular integration can be performed directlyto yield
Ω(E) =
2 πE 0
h∫∞
0dI δ(Iω−E). (3.6.9)Changing the action variable toI′=Iω, we obtain
Ω(E) =
E 0
̄hω∫∞
0dI′δ(I′−E), (3.6.10)