- #1

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Here Taylor makes the choice to use a basis of total angular momentum eigenvectors instead of using the simple tensor product given in the first equation above (6.47). I understand that this is because ##j## is conserved in this case, so the ##|E, l,s, j, m\rangle## states consist of good quantum numbers which are clearly convenient. However in the way it is worded by Taylor I think he may also imply that we should use these states such that equation 6.48 holds, i.e. such that we may apply the Wigner-Eckart theorem to the scattering matrix ##S##. In my understanding of the Wigner-Eckart theorem this is not a requirement, and we could just as well have written something like, $$\langle l', s', m_{l'}, m_{s'}|S|l, s, m_{l}, m_{s} \rangle = \delta_{ll'} \delta_{m_l m_{l'}}\frac{\langle l' ,s',m_{s'}||S||l, s, m_s\rangle }{\sqrt{2l+1}}$$ where I have now made the decision to not couple ##l## and ##s## but rather apply Wigner-Eckart to the ##l## states only. I realize that this is probably not the most convenient choice in this case, but I am just wondering if its still correct. The reason I am asking is that I have to set up a scattering theory in a much more complicated system with additional nuclear spin and an external magnetic field, such that the total angular momentum is not a good quantum number. I would however still like to apply the Wigner-Eckart theorem to the scattering matrix if possible.