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## Homework Statement

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?).

## Homework 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

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. Thanks.