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I've been given the following problem, and I'm quite lost with it.

Let $L_1$, $L_2$, and $L_3$ denote the abstract o(3) algebras. You are given that $\vec{A} = (A_1, A_2, A_3)$ and $\vec{B} = (B_1, B_2, B_3)$ transform as vector operators of o(3).

Show that $[L_j, \vec{A} \cdot \vec{B}] = 0$

I know that $L_j = \varepsilon_{jlm} q_l p_m$, and I can obviously determine the dot product, but I'm not sure where to go from there.

I do, however, know that $\vec{A} = \frac{1}{Ze^{2}\mu}(\vec{L} \times \vec{p}) + (\frac{1}{r})\vec{r}$, but I'm not sure how to integrate that into this problem.

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You don't need to know the specific form of $\vec{A}$ or representation of $\vec{L}$, only that $\vec{A},\vec{B}$ transform like vectors. That tells you the commutators $[L_j, A_i]$ and $[L_j, B_i]$, and from that you can compute $[L_j, \vec{A}\cdot\vec{B}]$. Also note the homework tag applies even if this is not assigned coursework - it's close enough. :) –  Michael Brown Sep 18 '13 at 8:31
@MichaelBrown I just saw this; I probably shouldn't have written my response given this comment. My apologies. –  joshphysics Sep 18 '13 at 8:42
@joshphysics No worries. The only reason I didn't post the commutators is that I can never remember the sign. :) –  Michael Brown Sep 18 '13 at 9:03

1 Answer 1

up vote 1 down vote accepted

A collection $\{V_1, V_2, V_3\}$ of operators on a vector space $V$ is called an $\mathfrak{o}(3)$ vector operator with respect to a representation $\rho$ of $\mathfrak{o}(3)$ acting on $V$ provided \begin{align} [V_i, L_j] = i\epsilon_{ijk}V_k \end{align} where $L_1, L_2, L_3$ are the representatives of the standard basis on $\mathfrak{o}(3)$. This means, in your case, that merely knowing that $A_i$ and $B_i$ are the components of vector operators tells you that \begin{align} [A_i, L_j]=i\epsilon_{ijk}A_k, \qquad [B_i, L_j]=i\epsilon_{ijk}B_k \end{align} These commutation relations alone are sufficient to demonstrate the desired result using commutator identities.

As an tangential note, you will probably find the Wikipedia page on tensor operators to be generally, conceptually helpful for understanding this stuff. Also, a while back I asked the following (as of yet unanswered in my opinion) question on physics.SE dealing with how to generalize and formalize the notion of tensor operators in a less basis-dependent way than how I defined them at the beginning of this answer. In case you're interested in math:

Tensor Operators

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I know this is probably really stretching my luck, but how do I go about applying those commutation relations?? Should I be using $[A, BC] = [A, B]C + B[A, C]$?? –  Jack Sep 18 '13 at 10:07
@Jack Cryptic general advice: if you have an idea for how to proceed in a problem, try that idea and see if it works :) –  joshphysics Sep 18 '13 at 10:29

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