# Rotation operators

1. Sep 21, 2012

### cragar

1. The problem statement, all variables and given/known data
Determine the matrix representation of the rotation operator
$R(\phi k)$ using the states |+z> and |-z> as a basis. Using your matrix representation verify that $R^{\dagger}R=1$
3. The attempt at a solution
Do I need to write $R| \psi>$ in terms of a matrix.
If I have $|\psi>=a|+z>+b|-z>$
Then do I just operate R on $\psi$ and then write this in terms of a matrix.
are these related to the Pauli spin matrices

2. Sep 22, 2012

### vela

Staff Emeritus
I'm not sure exactly what you had in mind, but it's probably not the most straightforward way to solve this problem.
Yes. Remember that the spin operators $\hat{S}_x$, $\hat{S}_y$, and $\hat{S}_z$ are generators of rotations. (This is definitely covered in your textbook.) Use that fact to calculate R.

3. Sep 22, 2012

### cragar

ok thanks for your help. My book gives the matrix for R and it is in the z basis.
And I took $R^{\dagger}R$ and it equaled one. But If the matrix wasn't in the
z basis would I use the roatation matrix to get the answer.
I would take $S^{\dagger}RS$ and this would give the correct R for the problem.

4. Sep 23, 2012

### vela

Staff Emeritus
I don't understand what you're asking. Well, I sorta do, but I'd like you to clarify your question. What is S? How did you find S?

5. Sep 23, 2012

### cragar

$s= $left (begin{array}{cc}<+z|+x>& <+z|-x>\\ <-z|+x>& <-z|-x> \end{array}\right)$$
where the bras are what basis I am going to and the kets are the basis that I was in. I don't really know how S is derived though

Last edited: Sep 23, 2012
6. Sep 23, 2012

### vela

Staff Emeritus
Yes, that's the regular method you use to change bases. You're doing the same thing you learned in linear algebra. It's just the notation that's different.