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OK so when we looked at the Zeeman effect, we used the states

|n, j, l, m>

where these are the quantum numbers associated with H, J^2, L^2 and Jz respectively.

We derived the perturbation Hamiltonian as

h = (eB/2m)(Lz + 2Sz)

Then you can work out the energy shifts for, the state with n=2, j=1/2 and l=0.

BUT what confuses me is that J^2 doesn't commute with the peturbed Hamiltonian. So the eigenstates of the peturbed Hamiltonian do not have well-defined j. But as we know all the possible values of m of the peturbed states (as they are the same as the possible values of the unpeturbed states), doesn't that imply a value of j=m(max)?

I'm not putting this across well but I can't think how else to describe the difficulty I'm having.

Thanks for reading and hopefully replying!