General Clebsch-Gordan coefficients

1. Jan 18, 2004

Sonty

I'm standing here looking at this problems for 1 hour already and don't know how to handle them. The problem is that the exam is tomorrow. I thought I had it in hand because I pretty much knew how to work my way through the CG coefficients in quantum mechanics. There you have the recurence relations, but here it's just general conditions.
Problem no.1: calculate the CG coefficient if S is an invariant operator.
there is no coherent solution attempt from me.
$$C^{\star}(\alpha\beta\gamma,ijk)=\frac{<\tilde{\phi}^\gamma_k|S^\alpha_i|\phi^\beta_j>} {<\tilde{\phi}^\gamma||S^\alpha||\phi^\beta>}$$
as S is invariant $$\alpha=0$$ and there is no more i necessary.
then what? introduce some $$T$$ and $$T^{-1}$$ and get what? Probably I'm not introducing these T's correctly because I have the feeling I'm running around in circles as I return to the same formula. I shoud get a damn number. What am I missing?

Problem no.2: Having a parity invariant potential V prove that $$<\phi|V|\psi>=0$$ if $$\psi$$ is even and $$\phi$$ is odd using WE theorem.
$$<\phi|V|\psi>=<\phi\pi^{-1}|\pi V \pi^{-1}|\pi\psi>=<-\phi|V|\psi>=-<\phi|V|\psi>=0$$
where does WE come in?

Please throw me a bone. Even a small one, but remember I'm cramming here.