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 $T'_{ij} = T_{ij} - T_{ji}$