# Hydrogen atom in QED

## Main Question or Discussion Point

I have a fairly straightforward question: how does one formulate the problem of hydrogen atom with quantum field theoretical treatment?

I understand that one can just take Uehling potential and find approximately the bound states' energies and wave functions, but it would not make electron field second quantized! So how to formulate and solve the problem with all fields involved (photonic and electron) second quantized?

Thanks.

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tom.stoer
The usual approach is to split the field into a classical part solving the classical e.o.m. and a fluctuation

$$\phi = \phi_\text{class.} + \tilde{\phi}$$

Then 2nd quantization is applied to the fluctuation. Introducing creation and annihilation operators requires distorted waves, so you don't use a plane wave basis but solutions of the modified wave equation in the classical background.

Usually perturbation theory is used. It starts with the 2nd order b/c the 1st order vanishes due to the e.o.m. (this is due to the Euler-Lagrange eq. for the classical field). In addition matrix elements for 1st order terms vanish.

Now you can apply the standard mechanism, expand the Hamiltonian H or any other operator e.g. to 2nd order and calculate the corrections for the matrix elements

Thanks for the explanation. Might it be you can give me a reference?

Bill_K
bound states problem are somewhat conveniently treated using bethe-salpeter eqn.
The Bethe-Salpeter Equation is so inconvenient that even Bethe and Salpeter themselves do not use it.

tom.stoer
Thanks for the explanation. Might it be you can give me a reference?
As a starting point any QM II textbook with relativistic QM, introduction to field quantization and especially Lamb-shift calculation will do. I don't remember exactly, but I guess that Messiah, Cohen-Tannoudji or Bjorken and Drell will contain a chapter.

Bill_K
Messiah vol II gives an exact solution of the Dirac Equation for the hydrogen atom. Bjorken and Drell vol I gives a brief discussion of the Lamb shift in the first quantized framework, but they don't ever come back to it in vol II

The best treatment by far is in the ancient book "Theory of Photons and Electrons" by Jauch and Rohrlich, which devotes two full chapters to it - Theory of the External Field, including both scattering and bound states, followed by External Field Problems, which includes Coulomb scattering and Delbruck scattering, and a very detailed treatment of the Lamb Shift.

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dextercioby
I will check the Jauch and Rohrlich's book.

I am wondering, can't we just expand the wave function operator in terms of hydrogen bound state eigenfunctions and take the emerging coefficients as new creation and annihilation operators, then redefine new commutations, propagators, etc, and then through wick theorem obtain contractions and everything else? Is there any book or article that went in this direction?

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tom.stoer
Yes, you can do this as well, but keep in mind that especially for the Lamb-shift the task is different. It's not about a refinement in the fermionic sector but about a QFT treatment of the el.-mag. field. Here the starting point is not a solution of a Schrödinger-like problem with eigenstates b/c the field is not quantized at all. So you want to quantize it w/o having any eigenstates as a basis to start with.

Usually you do both: expand all fields using either the coupling constant alpha or hbar as small parameter and take all corrections up to a certain order into account. If you already have the solution of an e.o.m. to start with you will use it, of course.

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The Bethe-Salpeter Equation is so inconvenient that even Bethe and Salpeter themselves do not use it.
this is not true.The success of bethe salpeter was proved for positronium hyperfine splitting case for deriving the α5 corrections which includes two photon virtual annihilation.It is not treated in the book because it is rather advanced and very few treatments are done with it.

tom.stoer
this is not true.The success of bethe salpeter was proved for positronium hyperfine splitting case for deriving the α5 corrections which includes two photon virtual annihilation.It is not treated in the book because it is rather advanced and very few treatments are done with it.
What I have seen so far is that from the BS eq. you can't expect new conceptual insights, but it may be useful for selected problems.

Staff Emeritus
2019 Award
After three years? Not surprising.

After three years? Not surprising.
Do you know what was the article about? Or who was the author?

George Jones
Staff Emeritus
Gold Member
Do you know what was the article about? Or who was the author?
The poster who gave the link, DrDu, is still active at Physics Forums. You could send a Physics Forums message to DrDu.

The poster who gave the link, DrDu, is still active at Physics Forums. You could send a Physics Forums message to DrDu.
Well I thought that maybe other people could be interested in this topic so it would be nice if we upload the file again.

But you are right I saw he is still active, but I cannot find a button to "send message".

Are you able to send him a message?

George Jones
Staff Emeritus
Gold Member
Well I thought that maybe other people could be interested in this topic so it would be nice if we upload the file again.

But you are right I saw he is still active, but I cannot find a button to "send message".

Are you able to send him a message?
Mouse-over "INBOX" at the top of the Physics Forums webpage, and the click on "Start a New Conservation" at the bottom of the drop-down menu.

jonjacson
Mouse-over "INBOX" at the top of the Physics Forums webpage, and the click on "Start a New Conservation" at the bottom of the drop-down menu.

I get an error message:

"You may not start a conversation with the following recipients: DrDu."

Apparently I can't send messages to him.

DrDu