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The Hydrogen Atom

  1. Nov 24, 2006 #1
    Hi,

    I had a few questions about the most well behaved of atoms.

    1) Solving the Schroedinger equation gives us a basis (say the energy eigenstates) such that all possible wavefunctions can be written as a linear combination of these stationery solutions. This means that the system is not confined to these stationery solutions but to their combinations, which are not necessarily eigenstates of the (time-independent) Hamiltonian.

    Then why are the lines observed in the spectra are only those corresponding to transitions between the eigenstates labeled with (n,l,m,s)? Does the electron in a H atom only occupy these states as Bohr would have believed?

    2)In a H atom, the energy depends only on n, not on l. Then why do we observe distinct lines for different l values of initial and final states?

    3)The usual solution for the H atom is obtained by treating the nucleus as a classical point charge and utilising the classical Coulomb potential. Is it possible to obtain a completely quantum solution of the H atom? If so, what equations would be used?

    Thanks for your help.

    Molu
     
  2. jcsd
  3. Nov 24, 2006 #2

    OlderDan

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    Some comments included in the quote area
     
  4. Nov 27, 2006 #3
    2)That link is about spin splitting. I'm talking about different lines for different value of l. Like 2s and 2p.

    3)But the coulomb potential is from CED. Wouldn't it be changed in QED?

    Thanks for the help.

    Molu
     
    Last edited: Nov 27, 2006
  5. Nov 27, 2006 #4

    OlderDan

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  6. Nov 28, 2006 #5

    dextercioby

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    The Dirac equation has a full exact solution is one treats the electrostatic field b/w the pointlike electron & proton classically. Corrections to account for the finite (i.e. nonzero) size of the nucleus (in H-like ions) have been made. Also, there's a perturbative resolution of this problem in QED, where, of course, the em interaction is treated QM-ally. See the textbooks of Lifschitz, Pitayevski and Berestetskii, Jauch and Rohrlich or Akhiezer and Berstetskii.

    Daniel.

    Da
     
  7. Nov 28, 2006 #6
    That is more-or-less what I was looking for, thank you.

    I know that the H atom admits closed-form solutions in both the non-relativistic (Schroedinger) model and the semi-relativistic (Dirac) model. In QED is it again exactly solvable or do we use perturbation theory or some other approximating technique? Thanks.

    Molu
     
  8. Nov 28, 2006 #7

    dextercioby

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    Perturbation theory and see Akhiezer's book for the formalism for tackling bound states in QED.

    Daniel.
     
  9. Nov 29, 2006 #8
    So the H atom is not exactly solvable in QED?
     
  10. Nov 29, 2006 #9

    dextercioby

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    No. See for instance the first volume of Weinberg, if you don't have Akhiezer.

    Daniel.
     
  11. Nov 30, 2006 #10
    I doubt I could make head or tail of a QFT textbook (even if I managed to find one somehow) without first properly learning non-relativistic quantum mechanics.
     
  12. Nov 30, 2006 #11

    dextercioby

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    That's true. Mumbles to himself: <<I doubt he could make head or tail of a non-relativistic quantum mechanics textbook (even if he managed to find one somehow) without first properly learning mathematics>>.

    Daniel.
     
  13. Dec 1, 2006 #12
    I protest. I can make both head and tail of a non-relativistic quantum mechanics textbook (Griffiths, to be specific), though I may not be able to savour the unifying insights provided by a deeper knowledge of the underlying mathematics like fourier analysis.

    Molu
     
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