Calculating SO(3) Generators and [J_k, r^2]

[Kara386]

Homework Statement

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##.

Homework Equations

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 anyone of the three. Do I have to show it for all of them?
Any help is much appreciated!

 

[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).