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Physics
Quantum Physics
Does the Wigner-Eckart theorem require good quantum numbers?
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[QUOTE="Decimal, post: 6440470, member: 625746"] Thank you for the response HAYAO, I have since discussed this problem with my supervisors as well. Your point on the spin orbit coupling was actually very helpful, I had (stupidly) not considered yet that this is the actual reason ##l## and ##s## can no longer be considered good quantum numbers and should be coupled. Taylor also doesn't mention this explicitly, so I had assumed it is a more general property of angular momentum theory rather than a consequence of spin orbit coupling. In the literature on the actual problem I am studying I can only assume then that there is an implicit assumption that the spin orbit coupling is ignored. Let me give an overview of the actual problem. We have two particles ##j## of electronic spin ##\mathbf{s}_j## and nuclear spin ##\mathbf{i}_j##, which scatter in a magnetic field. The total Hamiltonian reads, $$H = H^0 + \sum_j \left[H_j^{\mathrm{hf}} + H_j^{\mathrm{Z}} \right] + V.$$ Here ##H_j^{\mathrm{hf}} \sim \mathbf{s}_j \cdot \mathbf{i}_j## denotes the hyperfine interaction and ## H_j^{\mathrm{Z}} \sim \mathbf{s}_j \cdot \mathbf{B} + \mathbf{i}_j \cdot \mathbf{B}## is the Zeeman coupling to the magnetic field. Let us for now ignore the potential ##V##. At zero magnetic field eigenstates of the Hamiltonian are then formed as ##\ket{\chi} = \ket{f_1, m_{f_1}} \ket{f_2, m_{f_2}}##, where ##\mathbf{f}_j = \mathbf{s}_j + \mathbf{i}_j## is the total spin angular momentum. At nonzero magnetic field ##f_j## is no longer a good quantum number and these basis states mix. Let us then call the total eigenstates of the Hamiltonian (still without ##V##) ##\ket{\alpha}##, which will be superpositions of ##\ket{\chi}## states with magnetic field dependent expansion coefficients. The question is now how to form a partial wave expansion of this problem. If there exists no spin orbit coupling I would think that writing something like this for the scattering matrix is correct, $$\langle l', m_{l'}, \alpha' | S | l, m_l, \alpha \rangle = \delta_{l'l} s_{l' \alpha', l \alpha}^l.$$ Here I have neglected the kinetic energy terms since it is not so important for this problem. I think this obeys Wigner-Eckart since the orbital angular momentum is conserved, and ##S## will commute with ##L^2## and ##L_z##. If the interaction ##V## is included I think this still holds as long as the potential is spherically symmetric, but I don't have direct proof for this. If spin-orbit coupling is included as you said the problem seems to become very difficult. I think at that point the partial wave expansion is possible in theory but doesn't really make sense anymore practically. Any spherical symmetry is broken and the scattering matrix is not diagonal in any quantum number, so why would you even go to the effort. I suppose that in literature the neglect of spin orbit coupling is usually implicit, but I have not yet found any text that looks at this problem rigorously. It seems like most authors just assume that an expansion in terms of partial waves is possible without justifying what this means for the scattering matrix. [/QUOTE]
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Does the Wigner-Eckart theorem require good quantum numbers?
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