Angular Momentum for Particle on a Hoop

[thelonious]

Homework Statement

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?

Homework Equations

The classical Hamiltonian: H=L_z² / 2mR²
The wave function: $\Psi$($\varphi$,0)=(1/√2$\pi$)(cos$\varphi$-sin$\varphi$)

The Attempt at a Solution

I need to find the eigenfunction of L_z 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 L_z?

 

[vela]

You need to solve the eigenvalue equation L_zφ=λφ. Do you know the representation of L_z so you can write down the differential equation?