Possible outcomes of angular momentum state

[EricTheWizard]

Homework Statement

A particle is in the state $\psi = R(r)(\sqrt{\frac{1}{3}}Y_{11} + i\sqrt{\frac{2}{3}}Y_{10})$]. If a measurement of the x component of angular momentum is made, what are the possible outcomes and what are the probabilites of each?

Homework Equations

$$L_{\pm}Y_{lm}=\sqrt{l(l+1)-m(m \pm 1)}Y_{l(m\pm 1)}$$
$$L_x = \frac{1}{2}(L_+ + L_-)$$
$$\psi = \sum \alpha_{lm} Y_{lm}$$

The Attempt at a Solution

I understand how to get the expectation value of $L_x$ for the entire wavefunction through the inner product $\langle \psi |L_x| \psi \rangle$ and how to get the Fourier coefficients for the state probabilities, but I don't see how to get the "possible outcomes". Expectation values of individual eigenstates $\langle Y_{lm} |L_x| L_{lm} \rangle$ are always equal to 0, so I don't see how you can measure any outcome but 0 for definite eigenstates. Shouldn't the only outcome be the expectation value of the entire wavefunction?

 

[vela]

You need to expand the state in terms of the eigenfunctions of L_x.

 

[EricTheWizard]

You need to expand the state in terms of the eigenfunctions of L_x.

Could you explain this a bit more? I was under the impression that there were no $L_x Y_{lm}$ eigenstates because the effect of the operator on the spherical harmonics is to raise and lower the "m" index, a la $L_x Y_{lm} = \frac{1}{2}(L_+ +L_-)Y_{lm} = \frac{\hbar}{2}(\sqrt{l(l+1)-m(m+1)}Y_{l(m+1)}+\sqrt{l(l+1)-m(m-1)}Y_{l(m-1)})$, changing the basis vectors. Taking the expectation value of the entire wavefunction only leads to 0 as well, so I'm starting to think that that's it.

 

[vela]

That's right. The spherical harmonics are not eigenstates. You have to find linear combinations which are eigenstates of L_x.

 

[EricTheWizard]

Is it valid to just take x as equivalent to the z direction (since the coordinates are arbitrary anyways) and define $L_x Y_{lm} = m \hbar Y_{lm}$ and just solve it that way?

 

[vela]

No, it's not. The wave function as written implies a coordinate system, and the problem is asking you questions with respect to this coordinate system.

Find the matrix representation of L_x, and then find the eigenvalues and eigenvectors of that matrix.