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Quantum Physics
Does the Wigner-Eckart theorem require good quantum numbers?
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[QUOTE="HAYAO, post: 6440410, member: 558519"] I'm not a physicist and not very knowledgeable in quantum mechanics so I also want to understand this. Since I don't have the textbook I don't know the ultimate goal of this section, but from what I read here, the author is attempting to prove that the rotational invariance means that scattering operator commutes with total angular momentum. From my understanding, total angular momentum operator is a generator of rotation operator of the entire system (including both orbital and spin) so that's what I speculate the author focused on. So I don't really see the point in attempting to evaluate [itex]S[/itex] without total angular momentum. If you only consider the [itex]l[/itex] states, then you would only be rotating the spatial position of electrons without spin. So like you said, this would be inappropriate in this section. Wigner-Eckart theorem does require good quantum numbers, but if you are ignoring spin-orbit coupling [itex]l[/itex], [itex]s[/itex], [itex]j[/itex], and [itex]m[/itex] are all "good" quantum numbers. I'm assuming if you do the calculations, they should all commute with the scattering operator; since the author says "Just as in the scattering of spinless particles...", the previous section must've been about commutation relationship of [itex]l[/itex] operator with [itex]S[/itex], so at least we have one example there probably, and another one in the end of this section. Even if you include spin-orbit coupling, [itex]j[/itex] and [itex]m[/itex] are still "good" quantum numbers, so I don't see a problem in the author's statement. If you must include spin-orbit coupling, only [itex]j[/itex] and [itex]m[/itex] are going to be good quantum numbers, but if you want to start with the basis of L and S eigenstate in the L-S coupling scheme, and then you would have to apply spin-orbit coupling operator to it. The resulting wavevectors are going to be expressed with the linear combinations of these basis (mixes the states with same [itex]j[/itex]). If you are going to apply nuclear spin and external magnetic field, I'm completely clueless. If they are relatively small interactions, then you might be able to apply perturbation theory. In the evaluation of the transition dipole moments of lanthanides, crystal field is important but it is small enough to be treated as perturbation. In this case, we start with basis of L and S angular momentum eigenstates in L-S coupling scheme for 4f-electrons without spin-orbit coupling. Then we apply electrostatic interaction and spin-orbit coupling. Finally, we use these 4f states with crystal field operator in perturbation that admixes 4f5d states. I'm hoping that you can do something similar in your system if your external field can be treated as a perturbation. (If I'm making a stupid error here, then please point it out. It really helps me understand.) [/QUOTE]
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Does the Wigner-Eckart theorem require good quantum numbers?
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