$SO(3)$ generators

1. Jan 26, 2017

Kara386

1. The problem statement, all variables and given/known data
Using the commutation relations $[x,p_x] = i\hbar$ etc, together with the $SO(3)$ generators $J_k (k=x,y,z)$ in their operator form to calculate $[J_x, \mathbf{r}]$ and $[J_x, \mathbf{p}]$ where $r = (x, y, z)$ and $p = (p_x, p_y, p_z)$.

Then show that $[J_k, r^2]=0$.
2. Relevant equations

3. The attempt at a solution
I can't find the operator forms anywhere. I have looked on the internet and in textbooks, but nowhere does it specifically state that a particular form is the 'operator' form. Is it just these matrices:
$\left( \begin{array}{ccc} 0& 0 & 0 \\ 0& 0 & 1 \\ 0& -1 & 0 \end{array} \right)$
$\left( \begin{array}{ccc} 0& 0 & -1 \\ 0& 0 & 0 \\ 1& & 0 \end{array} \right)$
$\left( \begin{array}{ccc} 0& 1 & 0 \\ -1& 0 & 0 \\ 0& 0 & 0 \end{array} \right)$
Even if that's the case how could I show $[J_k, r^2]=0$? $J_k$ could be any one of the three. Do I have to show it for all of them?
Any help is much appreciated!

Last edited: Jan 26, 2017
2. Jan 27, 2017

strangerep

What textbook or lecture notes are you using? (I'd have thought the operator form of the $J_k$ would have been provided.)
I'm guessing you're meant to use the angular momentum generators ${\mathbf J} = {\mathbf r}\times{\mathbf p}$. In component form, this is $J_i = \epsilon_{ijk} r_j p_k$ (using implicit summation over repeated indices).