I Question about commuting operators

[Frank Einstein]

Hi everybody. I have a (I gess rather silly) question.

If I define [J_k,L_l]=iħΣ_mε_klmL_m, what would happen if I made [J, L]?. I gess it would be iħΣ_jε_iijL_i=0.

Can someone please confirm this?

Thanks for reading.

 

[blue_leaf77]

If $\mathbf J$ and $\mathbf L$ are both vector operators, how are we supposed to define $\mathbf J \mathbf L$ and $\mathbf L \mathbf J$?

 

[stevendaryl]

Hi everybody. I have a (I gess rather silly) question.

If I define [J_k,L_l]=iħΣ_mε_klmL_m, what would happen if I made [J, L]?. I gess it would be iħΣ_jε_iijL_i=0.

Can someone please confirm this?

Thanks for reading.

By [\boldsymbol{J}, \boldsymbol{L}, do you mean the commutator of the vectors? The definition of commutator is this:
[A,B] = AB - BA, so to make sense of a commutator, you must first have a notion of multiplication. What notion of multiplication of vectors do you mean?

I think the most straight-forward is the tensor product. If you have two vectors \boldsymbol{A} and \boldsymbol{B}, then you can define \boldsymbol{A}\boldsymbol{B} to the be the tensor \boldsymbol{T} with 9 components:

T_{ij} = A_i B_j

where i and j range from 1 to 3. In that case, the commutator [\boldsymbol{A},\boldsymbol{B}] would be just the tensor \boldsymbol{T}' with components T'_{ij} = T_{ij} - T_{ji}