How Do You Find the Energies and Wavefunctions for a Particle on a Sphere?

[land]

OK, I have that a particle of mass m is moving on the surface of a sphere of radius R but is otherwise free. The Hamiltonian is H = L^2/(2mR^2). All I have to do is find the energies and wavefunctions of the stationary states...

this seems like it should be really easy, but I am struggling mightily for some reason. To be honest I don't even know how to get started.. I know eigenfunctions of Lz are also eigenfunctions of L^2. I know L^2 operating on $$Y^m_l$$ is $$\hbar^2l(l+1)Y^m_l$$. I know once I get one stationary state I should be able to get the rest by operating the raising and lowering operators on it. But I just don't know how to get there. :(

Thanks as always for the help.

 

[land]

Anybody have a clue how to go about this? I feel like I should be able to do this but I just can't. I've looked at it for hours and don't know how to begin.. nothing I've tried works. sigh. thanks for the help :)

 

[land]

Anybody? This should be a straightforward problem, which is what makes it so frustrating :(