Eigenstate of $S_x^2$ and measurement How could I go about finding eigenstates of a given operator, namely $S_x^2$ in basis with good $S^2$ and $S_z$, that is $|s, m\rangle$?
I had the idea of writing it as a sum $4S_x^2=S_+^2+S_-^2+2S_+S_-$.
I know I have to solve eigenstate problem $$S_+^2+S_-^2+2S_+S_-|\psi\rangle=4\lambda|\psi\rangle.$$ I know also to write a general wavefunction in form $|\psi\rangle =a|11\rangle+b|10\rangle+c|1-1\rangle$, then check how it acts upon this given wavefunction, but I am unsure how to proceed further.
I also have to measure outcomes on spin 1 state, that is $|\psi\rangle =a|11\rangle+b|10\rangle+c|1-1\rangle$ consecutively with $S_x^2, S_y^2$ and $S_z^2$. I assume I have to find eigenstates of $S_y^2$ as well in order to complete the measurement. I already know that $S_z^2$ is trivially $S_x^2|s m\rangle=\hbar^2m^2|s m\rangle$.
 A: I can give you the strategy without telling you the answer.
The strategy is to know how the operator acts on any state of the form $|s,m\rangle$ and thus on any superposition of such states over different values of $m$. If you already know the action of $S_x$ on any state $|s,m\rangle$ then you can skip ahead. If not, for an operator like $S_x$ or $S_x^2$, it is indeed easier to write things in terms of the raising and lowering operators $S_\pm$. From some definition or by the commutation relation with $S_x$, you should know that $$S_\pm|s,m\rangle\propto |s,m\pm 1\rangle.$$ This is an important starting point! Then you must find the normalization constant (another standard exercise) and you can proceed.
One now ostensibly knows the relation
$$S_\pm |s,m\rangle=\mathcal{N}_\pm(s,m)|s,m\pm1\rangle.$$ One can use this to write $S_x^2\sum_m\psi_m|s,m\rangle$ for any amplitudes $\psi_m$. Then we indeed just solve the eigenvalue problem
$$S_x^2\sum_m\psi_m|s,m\rangle=\lambda \sum_m\psi_m|s,m\rangle$$ by using orthonormality of the $S_z$ eigenstates to write the $2s+1$ coupled linear equations
$$\lambda \psi_n=\langle s,n|S_x^2\sum_m \psi_m|s,m\rangle.$$
Of course, a more specific strategy involves knowing or guessing the eigenstates of $S_x$ and then using those to construct eigenstates of $f(S_x)$ for any function $f$, but that's the shortcut.
