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General ClebschGordan coefficients 
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#1
Jan1804, 08:15 AM

P: 96

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 WignerEckart to get: [tex] C^{\star}(\alpha\beta\gamma,ijk)=\frac{<\tilde{\phi}^\gamma_kS^\alpha_ i\phi^\beta_j>} {<\tilde{\phi}^\gammaS^\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]<\phiV\psi>=0[/tex] if [tex]\psi[/tex] is even and [tex]\phi[/tex] is odd using WE theorem. [tex] <\phiV\psi>=<\phi\pi^{1}\pi V \pi^{1}\pi\psi>=<\phiV\psi>=<\phiV\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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