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General Clebsch-Gordan coefficients

by Sonty
Tags: clebschgordan, coefficients
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Sonty
#1
Jan18-04, 08:15 AM
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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.
I start with Wigner-Eckart to get:
[tex]
C^{\star}(\alpha\beta\gamma,ijk)=\frac{<\tilde{\phi}^\gamma_k|S^\alpha_ i|\phi^\beta_j>}
{<\tilde{\phi}^\gamma||S^\alpha||\phi^\beta>}
[/tex]
as S is invariant [tex]\alpha=0[/tex] and there is no more i necessary.
then what? introduce some [tex]T[/tex] and [tex]T^{-1}[/tex] 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 [tex]<\phi|V|\psi>=0[/tex] if [tex]\psi[/tex] is even and [tex]\phi[/tex] is odd using WE theorem.
[tex]
<\phi|V|\psi>=<\phi\pi^{-1}|\pi V \pi^{-1}|\pi\psi>=<-\phi|V|\psi>=-<\phi|V|\psi>=0
[/tex]
where does WE come in?

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