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I was wondering about the Hamiltonian description of the classical hydrogen atom (two point particles interacting through a Coulumb potential).

In particular, I want to know if the fact that accelerated charges radiate (Larmor's formula) can be derived from the Hamilton's equation of the system.

If you can provide bibliography for the discussion it would be great.

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In particular, I want to know if the fact that accelerated charges radiate (Larmor's formula) can be derived from the Hamilton's equation of the system.

If the Hamilton's equation include the electric and magnetic fields as dynamical, then yes, it should be do-able... However, if you are just including the electrostatic interaction between the electron and proton and no vector field then I don't think it would be possible...

As for a reference, have you checked Jackson?

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  • $\begingroup$ Oh, I see! But why do you think the electrostatic interaction is the only interaction considered for the "quantum" case? $\endgroup$ – dapias Apr 25 '15 at 1:57
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    $\begingroup$ @dapias The presence of a dynamical electromagnetic field only leads to small corrections in the energies and eigenstates of the (quantum) hydrogen atom. Therefore it is sufficient to ignore them if you just want to get an idea of what the energies and eigenstates look like. However, you do need to include the dynamical fields if you want to understand transitions between these eigenstates and the corresponding emission of photons (e.g. spontaneous emission). $\endgroup$ – Mark Mitchison Apr 25 '15 at 2:29
  • $\begingroup$ @MarkMitchison, have you a reference in which this that you mention is done? $\endgroup$ – dapias Apr 25 '15 at 4:44
  • $\begingroup$ @dapias The interaction of atoms with the electromagnetic field is a standard problem in quantum optics. It will be treated in, for example, Scully & Zubairy, Breuer & Petruccione or Gardiner & Zoller. I expect you also can find a wealth of free notes on the subject online with some creative Googling. $\endgroup$ – Mark Mitchison Apr 25 '15 at 11:47
  • $\begingroup$ In my opinion, the best reference for this is Heitler's "Quantum Theory of Radiation". It's an oldie but a goodie. $\endgroup$ – hft Apr 25 '15 at 16:53

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