# Zeemen Effet - having trouble understanding

## Main Question or Discussion Point

Please help!

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!

## Answers and Replies

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Matterwave
Science Advisor
Gold Member
Uh, I'm not exactly sure what your question is, but the eigenstates change as well (for the ones which the energies have changed).

So, you will get a linear superposition of your previous states as your new eigenstates.

For example, if you have |a> and |b> which were degenerate and then they split off into 2 non-degenerate energies, Ea and Eb, then you have new eigenstates corresponding which will be some superposition of |a> and |b>.

I don't know if that's what you're asking though...