Sonty
Jan18-04, 09:15 AM
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:
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.
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:
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.