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

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To find the energies and wavefunctions for a particle on a sphere, the Hamiltonian is given as H = L^2/(2mR^2). The eigenfunctions of Lz are also eigenfunctions of L^2, with L^2 operating on Y^m_l yielding the eigenvalue \hbar^2l(l+1)Y^m_l. The challenge lies in determining the stationary states, starting from one known state and using raising and lowering operators to find others. Despite the straightforward nature of the problem, the individual is struggling to initiate the process and has not found a successful approach after extensive effort. Assistance is sought to clarify the initial steps in solving this quantum mechanics problem.
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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.
 
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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 :)
 
Anybody? This should be a straightforward problem, which is what makes it so frustrating :(
 
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