Rotating the angular momentum eigenket by an arbitrary angle

[Icaro Lorran]

Homework Statement

What is the best way to evaluate the rotation of a angular momentum eigenket ($| l,m \rangle$) around an Cartesian axis?

2. The attempt at a solution
I've tried to use the method in https://www.physicsforums.com/threads/can-the-operator-exp-i-pi-l_x-h-be-faced-as-parity.841885/ but it gets too complicated for an arbitrary angle.

 

[mark726]

Hello,

There are a few different ways to approach this problem, depending on what exactly you are looking to evaluate. One approach is to use the standard rotation operators, which are the exponential of the angular momentum operators ($L_x, L_y, L_z$) multiplied by the rotation angle.

For example, to evaluate the rotation of an angular momentum eigenket ($|l,m\rangle$) around the x-axis, you can use the operator $e^{-i\phi L_x}$, where $\phi$ is the rotation angle. This will give you a new eigenket with the same quantum numbers ($l,m$) but rotated by the angle $\phi$ around the x-axis.

If you are looking to evaluate the expectation value of the angular momentum around a Cartesian axis, you can use the corresponding component of the angular momentum operator ($L_x, L_y, L_z$) and calculate the expectation value using the eigenket ($|l,m\rangle$). This will give you the average value of the angular momentum around that axis.

Alternatively, you can also use the Wigner D-functions to calculate the rotation matrix for an arbitrary angle and then use it to rotate the eigenket. This method may be more complicated, but it allows for rotations at any angle.

I hope this helps. Let me know if you have any further questions.