Rotation operators

cragar

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

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.

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.

Staff Emeritus
Homework Helper
$s= $left (begin{array}{cc}<+z|+x>& <+z|-x>\\ <-z|+x>& <-z|-x> \end{array}\right)$$