Spin and Polarization: Understanding the Spin Operator

[peterpang1994]

Recently, I read an article about spin and polarization (http://www.mathpages.com/rr/s9-04/9-04.htm) , but I don't understand how the spin operator defined, can anyone give so help? any help would be great!

 

[jfy4]

Consider if we have a complete set of vectors in 2D characterized experimentally by spin-up and spin-down in the z direction:
$$\mathbb{I}=|\uparrow\rangle \langle \uparrow |+|\downarrow\rangle \langle \downarrow |$$
Then experimentally we find we can only observe two different numbers corresponding to two different physical situations, namely we measure something and it is spining up, or spining down with a number value for its angular momentum,
$$S_{z}=\frac{\hbar}{2}|\uparrow\rangle \langle \uparrow |-\frac{\hbar}{2}|\downarrow\rangle \langle \downarrow |$$
Now let's consider the different possibilities for measuring these values along the z-axis, this is given by
$$\sum_{n,m=1}^{2}\langle n|S_{z}|m\rangle$$
where by we examine the different situations
$$\langle \uparrow |S_{z}|\uparrow\rangle$$
$$\langle \uparrow |S_{z}|\downarrow \rangle$$
$$\langle \downarrow |S_{z}|\uparrow \rangle$$
$$\langle \downarrow |S_{z}|\downarrow \rangle$$
These can be combined into a single object and the inner products can be evaluated explicitly to give
$$\begin{pmatrix} \langle \uparrow |S_{z}|\uparrow\rangle & \langle \uparrow |S_{z}|\downarrow \rangle \\ \langle \downarrow |S_{z}|\uparrow \rangle & \langle \downarrow |S_{z}|\downarrow \rangle \end{pmatrix}=\frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}=\frac{\hbar}{2}\sigma_{z}$$
This is one component of the spin matrices. a similar method yields the other two.

 

[peterpang1994]

thank you very much. Is that the operator you suggested in only focused on the electron in a hydrogen atom? And I am still having problems on the total orbital angular momentum and the total spin.

 

[jfy4]

It's for the z-component of the total spin. Once one has constructed all three components they can calculate $S^2=\mathbf{S}\cdot \mathbf{S}$ which is the total spin operator (squared). This operator commutes with all three components of the spin.