# Angular Momentum for Particle on a Hoop

1. Jan 2, 2012

### thelonious

1. The problem statement, all variables and given/known data

A particle of mass m is constrained to a circular loop of radius R in the x-y plane. The particle's position is given by the angle $\varphi$, measured with respect to the x axis. Given $\Psi$($\varphi$,0), what is the probability that a measurement of the z-component of the angular momentum will yield +$\hbar$ at t=0?

2. Relevant equations

The classical Hamiltonian: H=Lz2 / 2mR2
The wave function: $\Psi$($\varphi$,0)=(1/√2$\pi$)(cos$\varphi$-sin$\varphi$)

3. The attempt at a solution

I need to find the eigenfunction of Lz with eigenvalue +$\hbar$.
Once I have this eigenfunction, the probability is given by P=|<+$\hbar$|$\Psi$>|2

My question: how do I begin to solve for this eigenfunction of Lz?

2. Jan 4, 2012

### vela

Staff Emeritus
You need to solve the eigenvalue equation Lzφ=λφ. Do you know the representation of Lz so you can write down the differential equation?