How to Find Eigenfunctions of Lx for a Hydrogen Atom in a Specific State?

[wengsee]

The system is at the state of Φ=aY_11+bY_20 (a^2+b^2=1),please find the possible eigenfunctions of Lx and the relevant possibilities?
My solution: I have attempted to use the matrix mechanics to work out the exercise,but I should work out a 8-order matrix.
Firstly I use the Fmn=〈m︳F|n〉to work out the matrix Lx in the Hilbert space of Lz or
L^2,then use the Time-independent Schrödinger Equation ,namely (Lx)mn ψ=Lx ψ ,to work out its eigenvector|ψn〉.At last ,I use the 〈Φ︳ψn〉,we can figure out probability of the related eigenvalue.I have tried to solute it in this way ,but in the process ,we should work out a 8-order matrix .It is too difficult to deal with it .
Are there someone have easier way to solute the question?

 

[vela]

The system is at the state of Φ=aY_11+bY_20 (a^2+b^2=1), please find the possible eigenfunctions of Lx and the relevant possibilities?
My solution: I have attempted to use the matrix mechanics to work out the exercise,but I should work out a 8-order matrix.
Firstly I use the Fmn=〈m︳F|n〉to work out the matrix Lx in the Hilbert space of Lz or L^2, then use the Time-independent Schrödinger Equation, namely (Lx)mn ψ=Lx ψ, to work out its eigenvector |ψn〉.

You have the right equation, but that's not the Schrödinger equation.

At last, I use the 〈Φ︳ψn〉, we can figure out probability of the related eigenvalue. I have tried to solute it in this way, but in the process, we should work out a 8-order matrix. It is too difficult to deal with it.
Are there someone have easier way to solute the question?

Your approach will work, though, as you found out, it seems rather unwieldy. One thing you can do is treat the l=1 and l=2 cases separately.

Another approach might be to consider how the spherical harmonics transform under rotations.

 

[wengsee]

Yes,The equation is just the eigenvalue equation of Lx.
As you say ,the another approach is" consider how the spherical harmonics transform under rotations. ".But in my opinion,it is not so easy to do as you expected.