What Are the Generators of the 4D Irreducible Representation of SO(3)?

[negru]

Homework Statement

Find the generators of the four dimensional irreducible representation of SO(3), such that J_3 is diagonal.

The Attempt at a Solution

I know how to get the rest if I know J_3, by using ladder operators. But what is J_3?
For a 3d representation it's diagonal with 1,0-1, in 4d would it be 3/2, 1/2, -1/2 -3/2? Wasn't orbital angular momentum supposed to be of integer values, or did I forget QM so bad?

Thanks for any help

-------
nevermind, it is obviously not necessarily orbital angular momentum so that works..

 

[Jhero]

Hello,

Thank you for your post. It is great to see someone actively engaging in understanding the four dimensional irreducible representation of SO(3). You are correct, J_3 is the diagonal generator of the representation. However, it is not necessarily related to orbital angular momentum. In fact, J_3 can represent any type of angular momentum, including spin or orbital angular momentum.

To find the generators of the four dimensional irreducible representation of SO(3), you can use the ladder operators as you mentioned. These operators are defined by the commutation relations [J+,J-]=2J_3 and [J_3,J±]=±J±. Using these relations, you can find the generators J_+, J_-, and J_3.

I hope this helps. Please let me know if you have any further questions or need clarification. Keep up the good work!