# Quantum mechanics - Find S_x and S_y

• Graham87

#### 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!

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.

vanhees71 and malawi_glenn
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.

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

vanhees71, Graham87 and 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 alot! Got it!

vanhees71 and malawi_glenn
and then when you are done with that, insert the matrices, and find the final matrix representation of those operators.

vanhees71 and Graham87