What Are the Possible Values of L_x in a Quantum System with l=1?

[bznm]

Homework Statement

I've a physical system with $l=1$ and I have to calculate the values I can obtain if I measure $L_x$ and their probability.

Homework Equations

I know that:

- the values I can obtain are $\ m=0, \pm 1$
- $\displaystyle L_x=\frac{L_+ + L_-}{2}$
- $L_x|1, m>_x=\hbar m |1, m>_x$

The Attempt at a Solution

But I can't understand, for example, why I should obtain
$|1, 1>_x=\frac{1}{2}[|1,1>+\sqrt{2}|1,0>+|1,-1>]$

(I have obtained $|1, 1>_x=\frac{\sqrt{2}}{2} |1,0>$)

Can I have some hints?

 

[blue_leaf77]

But I can't understand, for example, why I should obtain
$|1, 1>_x=\frac{1}{2}[|1,1>+\sqrt{2}|1,0>+|1,-1>]$

The three kets on the RHS are the eigenstates of $L_z$. You are asked to represent the eigenstate of $L_x$, $|1,1\rangle_x$ in terms of eigenstates of $L_z$.

I've a physical system with l=1l=1 and I have to calculate the values I can obtain if I measure LxL_x and their probability.

You have to know precisely the state associated to your system. From what you are asked to do above, it seems like your system is already in one of the eigenstates of $L_x$ and you are asked to calculate the probability of finding the state in the eigenstates of $L_z$.

(I have obtained $|1, 1>_x=\frac{\sqrt{2}}{2} |1,0>$)

Can I have some hints?

It's obvious that the right handside is not normalized while the LHS is. How did you obtain this?