# I Question about commuting operators

1. Aug 7, 2016

### Frank Einstein

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

If I define [Jk,Ll]=iħΣmεklmLm, what would happen if I made [J, L]?. I gess it would be iħΣjεiijLi=0.

2. Aug 7, 2016

### blue_leaf77

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

3. Aug 7, 2016

### stevendaryl

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