So we obtain the perturbation Hamiltonian H as something proportional to(adsbygoogle = window.adsbygoogle || []).push({}); S.L/r^{3}and the first order energy shift is then the expectation value of this perturbation Hamiltonianin the state that is being perturbed.

So let a general gross structure state that we are perturbing be |n l m_{l}s m_{s}>. Finding the expectation of 1/r^{3}is fine using this state. However we must recast the other part asS.L=(J^{2}-L^{2}-S^{2})/2. This is where I'm losing what's going on with the maths and books don't tend to bother explaining it. So the state I have chosen is not an eigenket of J^{2}. What I believe is happening is that we express the above state as a linear combination of eigenkets of the form |n j m_{j}l, s> using the techniques of addition of angular momentum (because then we have it in terms of eigenkets of the squared operators which makes life easier). However this doesn't give the correct answer because the linear combination gives multiple terms whereas there should only be one.

Can anybody see where I have made a mathematical error? Thank you :)

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# Basis for Spin Orbit Coupling

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