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I am looking for a reference or derivation of the non-relativistic bound states of hydrogen in an electromagnetic field that include Zeeman effect, Stark shift, and Lamb shift. I am looking for a full QED treatment. I have seen Scully and Zubairy Quantum Optics but they only provide a rough approximation.

I am finding that those who are doing QED are mostly interested in high-energy particle physics, whereas I am interested in atomic physics and the terms relevant to corrections of the spectral line energies (resonances in QFT parlance).

I have used the Dirac equation in the low energy limit to arrive at terms for relativistic correction, spin-orbit interaction, and Darwin term, what I would like to be able to do is write out a derivation that includes the electron self-interaction term; that is a derivation with a more explicit treatment than Scully of the quantum vacuum for the Lamb shift.

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  • $\begingroup$ I don't understand this question. What's a non-relativistic QED treatment? Why do you single out the Lamb shift, which is smaller than the (relativistic) fine-structure terms? $\endgroup$
    – Emilio Pisanty
    Apr 12, 2017 at 8:23
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    $\begingroup$ That said, the NIST energy levels database points to this paper as its reference for hydrogen. That paper has a very readable and very complete overview in its introduction, and it presumably has a pretty complete set of references to the state of the art of theoretical calculations of the hydrogen spectrum as of 2010, with suitable references to midway milestones as to each of the conceptual steps involved. It's unclear how far down that rabbit hole you want to go, though. $\endgroup$
    – Emilio Pisanty
    Apr 12, 2017 at 9:01
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    $\begingroup$ @EmilioPisanty: by non-relativistic I mean that there is no chance of electron/positron pair creation. I single out the Lamb shift because it occurs at non-relativistics energies. I started out by looking at the QM solution of the energy levels for the Hydrogen atom then I proceeded to the Dirac equation which naturally gives spin-orbit coupling term and so on. I now want to look at how to derive Lamb shift term and so on which can not be derived from Dirac equation but affect bound state energy levels. $\endgroup$
    – vivian
    Apr 12, 2017 at 9:42
  • $\begingroup$ OK, though that could probably go into the question to sharpen what you're asking. It's still not clear what you mean by "full QED treatment", though. If you want an exact solution with all QED terms, that's not feasible - all you can hope for is an accurate-enough perturbation theory. In addition, the Kramida paper I linked to above strongly suggests that theory currently lags experiment, so a "full" treatment is probably a cutting-edge QED calculation and it's still not good enough. So, again - you should specify how deep down the rabbit hole you want to go. If what you want is ... $\endgroup$
    – Emilio Pisanty
    Apr 12, 2017 at 9:46
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    $\begingroup$ Regarding the Lamb shift, the Scully section looks pretty adequate to me. It's approximate, sure, but as I said, every treatment is approximate to some level. Are you after a description of the Lamb shift with a more explicit treatment of the quantum vacuum? That could be a way to sharpen the question. (In any case, though, do read the first few sections of the Kramida paper - it has a lot of informative background which should help provide context.) $\endgroup$
    – Emilio Pisanty
    Apr 12, 2017 at 10:37

2 Answers 2

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Let us state first the idea behind QED. QED stands for quantum electrodynamics and it has two main components that are merged. One the one hand special relativity (SR) on the other quantum mechanics. The first one concerns very high speeds or high energies, the second one pertains very small things. If you have a Hydrogen atom just sitting there there is no need to include (SR) effects. Same as, if your system is to big, there is no need to use quantum mechanics.

With that cleared out, you should understand the different origins of the effects you are speaking of. The Lamb shift has to do with the vacuum polarization (or electron self-energy), or if you want the "dressing" of the photon propagator. As can be seen in most books of quantum field theory (QFT), it is related to the Uehling term correction to the Coulomb potential. We could call this an intrinsic effect.

The Zeeman and Stark effects are different in origin. They appear through external fields. This means they will not pop out of QED without modifying anything else. You should be asking then how to model external fields within QED and you will have to look for effective field theories (EFTs).

In the context of EFTs, one essentially finds an effective field theory by some method/approximation. There are several ways to speak about effective theories, but I believe you might be interested in expanding your theory around external backgrounds.

I hope there are enough keywords to point you in some useful direction.

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Sections 2-8 and 4-7 of Sakurai, J. J. (1967) Advanced Quantum Mechanics give derivations of the Lamb Shift and electron self-interaction from the perspective of quantum mechanics (Section 2) and quantum field theory (Section 4).

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