Quantum mechanics - Find S_x and S_y

[Graham87]

Homework Statement

Lecture slide (see photo)

Relevant Equations

See second last row on picture

I have a lecture slide that shows how to find S_x and S_y. I get all the steps except the last row.
Where did 1/2 come from? I think my linear algebra needs polishing.

Thanks!

 

[PeroK]

Have you tried to use $\hat S_{\pm} = \hat S_x \pm i\hat S_y$ to solve for $\hat S_x$ and $\hat S_y$?

I suggest that if you do the algebra yourself, you'll see where the $\frac 1 2$ arises.

 

[Graham87]

Have you tried to use $\hat S_{\pm} = \hat S_x \pm i\hat S_y$ to solve for $\hat S_x$ and $\hat S_y$?

I suggest that if you do the algebra yourself, you'll see where the $\frac 1 2$ arises.

Thanks. I have, and I get something like this:
I think what confuses me is the algebra.

 

[PeroK]

$$S_+ + S_- = (S_x + iS_y) + (S_x - iS_y) = ?$$

 

[malawi_glenn]

It is basically just a system of equations
$A = x + \mathrm{i}y$
$B = x -\mathrm{i}y$
solve for $x$ and $y$ in terms of $A$ and $B$

 

[Graham87]

Aha thanks a lot! Got it!

 

[malawi_glenn]

and then when you are done with that, insert the matrices, and find the final matrix representation of those operators.