Angular Momentum Eigenfunctions for Bead on a Wire

[thelonious]

Homework Statement

A bead of mass m on a circular ring has the wave function Acos$\stackrel{2}{}$θ.
Find expectation value, eigenfunctions & eigenvalues.

Homework Equations

The differential operator for the angular momentum is L = $\hbar$/i ($\partial$/$\partial$θ).

The Attempt at a Solution

I have found that the expectation value is zero, and now I would like to solve for eigenvalues and eigenfunctions. I understand that the eigenfunctions must have the form Ce^-iθ/$\hbar$ ... but I don't know where to go from here. How can I begin to solve for these eigenfunctions?

 

[vela]

Homework Statement

A bead of mass m on a circular ring has the wave function Acos$\stackrel{2}{}$θ.
Find expectation value, eigenfunctions & eigenvalues.

Homework Equations

The differential operator for the angular momentum is L = $\hbar$/i ($\partial$/$\partial$θ).

The Attempt at a Solution

I have found that the expectation value is zero, and now I would like to solve for eigenvalues and eigenfunctions. I understand that the eigenfunctions must have the form Ce^-iθ/$\hbar$ ... but I don't know where to go from here. How can I begin to solve for these eigenfunctions?

How did you come up with that form for the eigenfunction? It's not quite right.

 

[thelonious]

It is my understanding that since the eigenfunction will have the form L|λ> = α|λ> (where |λ> denotes the eigenvector and α its eigenvalue), and the operator L contains a partial with respect to θ, the eigenfunction must include e^iθ. Is this incorrect? I do not know how to solve for the exact eigenfunctions, but I expect that they will have this form.

 

[vela]

It's close, but the function should depend on the eigenvalue a.

You want to solve the differential equation
$$\frac{\hbar}{i}\frac{\partial \psi}{\partial \theta} = a\psi$$to find the eigenfunctions. The boundary conditions will then tell you what the eigenvalues are.

 

[thelonious]

For the eigenfunctions, I have found:

L$\Psi$=2A$\hbar$icos$\phi$sin$\phi$

The boundary conditions:

$\Psi$($\phi$,0) = $\Psi$($\phi+2\pi$,0)

Can I find the eigenvalues with these equations?

 

[vela]

Forget about the given wave function for right now. You want to solve the differential equation and apply the boundary conditions to find the eigenvalues and eigenfunctions.

Once you have those, then you want to write the given wave function in terms of those eigenfunctions.