1. The problem statement, all variables and given/known data
I want to find the matrix representation of the $\hat{S}_x,\hat{S}_y,\hat{S}_z$ and $\hat{S}^2$ operators in the $S_x$ basis (is it more correct to say the $x$ basis, $S_x$ basis or the $\hat{S}_x$ basis?).
2. Relevant equations
$$\hat{S}^2|s,m_s\rangle=s(s+1)\hbar^2|s,m_s\rangle$$
$$\hat{S}_z|s,m_s\rangle=m_s\hbar|s,m_s\rangle$$
$$\hat{S}_x|s,m_s\rangle=\frac{1}{2}(\hat{S}_++\hat{S}_-)|s,m_s\rangle$$
$$\hat{S}_y|s,m_s\rangle=\frac{1}{2i}(\hat{S}_+-\hat{S}_-)|s,m_s\rangle$$
$$\hat{S}_{\pm}|s,m_s\rangle=\sqrt{s(s+1)-m_s(m_s \pm 1)}\hbar|s,m_s \pm 1\rangle$$
3. The attempt at a solution
Finding these in the $S_z$ basis is simple enough. All I need to do is investigate how the basis vectors $\{|\frac{1}{2},\frac{1}{2}\rangle \equiv |\alpha\rangle, |\frac{1}{2},\frac{-1}{2}\rangle\equiv |\beta\rangle\}$ transform under the action of the operator i wish to represent. Then the columns of the matrix become the image of the basis vectors under the operation. However, I'm not sure what the eigenvectors (basis vectors) of $S_x$ are. On top of that, the actions of the operators i have supplied would no longer apply in a different basis, would they? I would prefer to do it via this method rather than using a similarity transform if possible.
Would I do it by defining the operators as follows?
$$\hat{S}^2|s,m_s\rangle=s(s+1)\hbar^2|s,m_s\rangle$$
$$\hat{S}_x|s,m_s\rangle=m_s\hbar|s,m_s\rangle$$
$$\hat{S}_+=\hat{S}_y+i\hat{S}_z$$
$$\hat{S}_-=\hat{S}_y-i\hat{S}_z$$
Which both imply that:
$$\hat{S}_y=\frac{1}{2}(\hat{S}_++\hat{S}_-)$$
$$\hat{S}_z=\frac{1}{2}(\hat{S}_+-\hat{S}_-)$$
And with the basis vectors of $S_y$, $|\alpha\rangle$ and $|\beta\rangle$ defined as before?
Best Answer
The solution is quite simple, and based on 2 observations -
The solution is to make a permutation of the known operators, i.e $S_x \rightarrow S_z$, $S_z \rightarrow S_y$, $S_y \rightarrow S_x$.
$$S_x = {\hbar \over 2}\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$$ $$S_z = {\hbar \over 2}\begin{pmatrix} 0 & -i \\ i & 0\end{pmatrix}$$ $$S_y = {\hbar \over 2}\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}$$ The $S^2$ operator, being rotationaly invariant (proportional to unit matrix), remains unchanged $$S^2 = {3 \hbar^2 \over 4}\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$$
Another way to solve this is to look at this as purely algebraic problem - find the unitary matrix which diagonalizes $S_x$ and apply it to all the remaining matrices (changing representation basis of the matrix).