Representing spin operators in alternate basis

[pondzo]

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

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.

 

[pondzo]

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?

 

[blue_leaf77]

the basis vectors $\{|\frac{1}{2},\frac{1}{2}\rangle \equiv |\alpha\rangle, |\frac{1}{2},\frac{-1}{2}\rangle\equiv |\beta\rangle\}$ t

Which eigenvectors are they, $S_x$, $S_y$, or $S_z$?

However, I'm not sure what the eigenvectors (basis vectors) of SxSxS_x are.

Find out $S_x$ in matrix form in the basis of the eigenvectors of $S_z$ (this is the most common form found in any literature) then find its eigenvalues as well as its eigenvectors (in the basis of the eigenvectors of $S_z$).

On top of that, the actions of the operators i have supplied would no longer apply in a different basis, would they?

The operator equations you wrote above are the action on an eigenvector of $S_z$. Moreover, the forms of those equation are in the basis-free, operator forms, therefore no matter which basis you chose the form of the above equations will not change. However, if you redefine the notation $|s,m_s\rangle$ to be the eigenvector of $S_x$, then those equations will indeed look different.

I would prefer to do it via this method rather than using a similarity transform if possible. Thanks.

It's not clear to me which method you were referring to. To be honest, in matrix form I know no other way to transform a matrix from one basis into another one unless using the usual similarity transformation.

 

[rude man]

This belongs in the advanced physics forum IMO.