We have the initial orbital angular momentum state in the x basis as |l,ml>x=|1,1>x. We are asked to find the column vector in the z-basis that represents the initial orbital angular momentum of the above state. It then says "hint: use an eigenvalue equation".
I feel like this is a simple change of basis from the x basis to the z basis with the Lx operator. However, I am not convinced this is quite the steps I need to take.
The Attempt at a Solution
To start I applied the Lx operator (the 3x3 operator that I think is responsible for changing basis) it is equal to half the angular momentum ladder operators added with each other. This only gives me one column vector where there should be 3, I think, since there should be some probability with each value of the quantum number m possible for the z component.
Another attempt was to try to find the eigenvalues of the Lx operator and to apply the 3 eigenvalues found as the probability to each vector but this didn't pan out so well. So here I am wondering how I might approach this problem in a different way. Any idea is appreciated.