Pendulum & Partition Function Problem

[emc201]

1. A pendulum of mass m hangs from a weightless string of length l
The string makes an angle θ with the vertical

Find
(i) <θ>
(ii) <θ^2>
(iii) <v>
(iiii) <v^2>

Homework Equations

The Hamiltonian in terms of θ and the angular momentum L=

H= L^2/2ml^2 + mgl(1-cosθ)

The Attempt at a Solution

I have the Hamiltonian derived for the pendulum
I am unsure how to derive the partition function in terms of the angular momentum and θ from the equation
Z=e^(-βH) where H is the Hamiltonian and β=1/T

 

[jbsteele]

where T is the temperature. From there I believe I can use the definition of the mean value for a system to solve for each of the required values. (i) <θ> = (1/Z) ∫_0^2π e^(-βH) θ dθ (ii) <θ^2> = (1/Z) ∫_0^2π e^(-βH) (θ^2) dθ (iii) <v> = (1/Z) ∫_0^2π e^(-βH) v dθ (iv) <v^2> = (1/Z) ∫_0^2π e^(-βH) (v^2) dθ