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Some quick QM questions

  1. Nov 9, 2008 #1
    I just had a couple quick questions while I'm reviewing for a test...

    If two operators commute, what can we say about their eigenfunctions?

    x commutes with pz, correct?

    I understand that the energy for a hydrogen-like atom depends on n according to the equation ((-z^2 μe^4)/(2(4πϵ_o )^2 ℏ^2 ) 1/n^2 ), but why does it not depend on l also? Isn't a 3p electron supposed to have more energy than a 3s electron? Or is that only in many electron atoms?

    Thanks, any help appreciated
  2. jcsd
  3. Nov 9, 2008 #2


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    If two [hermitian] operators commute, then there exists a common eigenbasis for the two operators.
    You are correct. Since the Hydrogen atom has a purely Coulomb potential, the energy levels are indeed degenerate with respect to l. However, as you correctly note, this is not the case with many-electron atoms.
  4. Nov 9, 2008 #3
    Actually, even in hydrogen the 3p and 3s (or 2p and 2s) levels are not quite degenerate because of QED effects: the Lamb shift of 2p vs 2s being the most famous example.

    But leaving out QED effects the levels with the same n and different l are indeed degenerate in H: this has to do with a symmetry that the H atom has, but no other: the Laplace-Runge-Lenz vector is conserved for an exact 1/r potential.
  5. Nov 12, 2008 #4


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    Staff: Mentor

    Attention mna skt!

    mna skt, I've moved your homework question to a new thread in one of the homework forums... click the following link to go to it:


    Everybody else, please carry on. I'll try to remember to delete this post in a few days.
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