# Rotation operators

## Homework Statement

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 $R^{\dagger}R=1$

## 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

## Answers and Replies

Related Advanced Physics Homework Help News on Phys.org
vela
Staff Emeritus
Homework Helper

## Homework Statement

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 $R^{\dagger}R=1$

## 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.
I'm not sure exactly what you had in mind, but it's probably not the most straightforward way to solve this problem.
are these related to the Pauli spin matrices
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.

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.

vela
Staff Emeritus
Homework Helper
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?

$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:
vela
Staff Emeritus