I've been given the following problem, and I'm quite lost with it.
Let L1, L2, and L3 denote the abstract o(3) algebras. You are given that →A=(A1,A2,A3) and →B=(B1,B2,B3) transform as vector operators of o(3).
Show that [Lj,→A⋅→B]=0
I know that Lj=εjlmqlpm, and I can obviously determine the dot product, but I'm not sure where to go from there.
I do, however, know that →A=1Ze2μ(→L×→p)+(1r)→r, but I'm not sure how to integrate that into this problem.
Answer
A collection {V1,V2,V3} of operators on a vector space V is called an o(3) vector operator with respect to a representation ρ of o(3) acting on V provided [Vi,Lj]=iϵijkVk
As a 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 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, here is that question:
No comments:
Post a Comment