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

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In summary, the conversation discusses using commutation relations and the ##SO(3)## generators in their operator form to calculate ##[J_x, \mathbf{r}]## and ##[J_x, \mathbf{p}]##. It also shows that ##[J_k, r^2]=0## using the angular momentum generators ##{\mathbf J} = {\mathbf r}\times{\mathbf p}##. The operator form of the ##J_k## is not explicitly stated, but it can be inferred from the provided matrices. It is also mentioned that the discussion is related to a textbook or lecture notes.
  • #1
Kara386
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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!
 
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  • #2
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).
 

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

1. What is SO(3)?

SO(3) refers to the special orthogonal group in three dimensions. It is a group of all the 3x3 rotation matrices that preserve the length of a vector and the orientation of a coordinate system.

2. What are generators in SO(3)?

In SO(3), generators refer to the infinitesimal elements of the group. They can be thought of as the "building blocks" for constructing the group elements through exponentiation.

3. How do you calculate the generators of SO(3)?

The generators of SO(3) can be calculated using the Lie algebra representation of the group, which involves using the commutation relations between the generators. These commutation relations are given by [J_k, J_l] = iε_klmJ_m, where ε_klm is the Levi-Civita symbol.

4. What is the significance of [J_k, r^2] in SO(3)?

[J_k, r^2] is a commutator that represents the angular momentum operator in SO(3). It is significant because it shows how the generators of the group are related to the physical quantity of angular momentum.

5. How is [J_k, r^2] used in physics?

[J_k, r^2] is used in physics to describe the rotational symmetry of physical systems. It is an important mathematical tool in quantum mechanics for understanding the properties of particles and their interactions.

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