# Probability Density Function for a Pendulum?

1. Oct 2, 2012

### bencmier

What is the probability density equation as a function of angle for a simple pendulum using the small angle approximation?

2. Oct 2, 2012

### dipole

Do you mean to treat the situation quantum mechanically? If so, then assume a potential:

$$U(\phi) = mglsin(\phi) ≈ mgl\phi$$

Where l is the fixed distance from the origin this thing oscillates at. Then your Hamiltonian is given by:

$$H = -\frac{\hbar^2}{2ml^2}\frac{d^2}{d\phi^2} + mgl\phi$$

Solve this with some boundary condition and you'll know.