Pauli Matrices: Calculating Expression

[Amok]

Hey guys,

I was wondering how to get the expression for pauli matrices. I know that for one electron:

$$S_i = \frac{\hbar}{2} \sigma_i$$

But I also know that you can get to the above expression by explicitly calculating the matrix elements of the Sz, Sx and Sy operators (in the basis generated by Sz and S and composed of two vectors) by using a few rules about angular momentum operators, I just don't remember how exactly. Anyone can help?

 

[jfy4]

See this thread I gave a brief run through here. But it's available in Sakurai also.

https://www.physicsforums.com/showthread.php?t=535524

 

[Amok]

Thanks, but what equations should you use to find the x and y matrices? The z one is the easiest.

 

[jfy4]

You can write $S_{x}$ and $S_{y}$ just like $S_{z}$.
$$S_{x}=\frac{\hbar}{2}|\uparrow\rangle \langle \uparrow |-\frac{\hbar}{2}|\downarrow \rangle \langle \downarrow |$$
Then if you hit this from both sides with z-basis eigenstates you can evaluate the inner products as $1/\sqrt{2}$ or $i/\sqrt{2}$ etc...