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.