# Electrostatic force and virtual photon exchange

## Main Question or Discussion Point

Hello,
although being graduated in physics (but not being an expert on QED) I find myself embarassed in trying to explain in very simple terms how the electrostatic force arise from the exchange of virtual photon. Especially when it comes to actually reduce all calculations to the expected result F = k q^2/ r^2
Myself I fall in the trap of a lot of complexities about series expansions in term of powers of 1/137, etc. etc. but without being able to sum up all of those fine concepts into a simple formula such as:
F = dp/dt
where:
dp = average momentum exchanged / each virtual photon exchange
1/dt = average number of virtual photon exchanged / unit of time

Considering only the static case, and limiting to the first term (i.e.: single photon exchange), and accepting to sacrifice some of the formal rigour, if it could be possible to obtain simple approximated expressions for dp and dt in terms of 1/r^2 and fine structure constant, that would also help a lot when trying to popularise at least the fundamental concepts.
Oddly, all my efforts to find this in text books have failed.
Is somebody interested in helping to derive such simple approximated expressions for dp and dt ?

Thanks for the help

Luca

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CarlB
Homework Helper
Maybe the problem is in using perturbation theory to create a bound state rather than to perturb a free state (or an already bound state).

The chemists seem to approach this sort of thing by way of density matrices. I'm not sure how to set those up in terms of QED. It seems like QED is inherently linear, while density matrices are second order.

Carl

Thank you Carl
What I am really looking for is a description of the fundamental mechanisms at play, and that shall be suitable for understanding from high school kids (at least the bright ones in the last year), making use of plain algebra only, and without the need for calculus.
I would already know how to introduce the quantum concepts of probability amplitudes, interference effects, and non-locality, in a relatively simple and discursive way, accessible to high schools kids.
However, when I try to write down simple algebraic expressions starting from (Feynman diagrams approach):
- the expression for the amplitude that particle_1 emits a virtual photon
- the expression for the amplitude that said photon travels to the distance of particle_2
- the expression for the amplitude that said photon is adsorbed by particle_2
then, I cannot find in any text book how to get from there to a simple (although approximated to the single photon exchange ... and maybe not very rigorously ...)
F = dp/dt, which would be easily understandable to last year high school students if we could find simple algebraic expressions for:
dp = average momentum exchanged at each virtual photon exchange
1/dt = average number of virtual photon exchanged / unit of time

Luca

Physics Monkey
Homework Helper
Luca,

You will find that in A. Zee's book "Quantum Field Theory in a Nutshell" he 'derives' the energy of two charges by page 30 using QFT methods. Unfortunately, he uses plenty of calculus and the path integral so it's hardly appropriate for high school. It boils down to the fact that the photon propagator (~1/k^2) is just the fourier transform of the energy (~1/r) up to some factors. I don't think this helps you much considering you target audience, but if you're interested check out Zee's book or reply here and I can show the details. I don't know of any way to get the result with just algebra, I think it's just a little more complicated than that, but I will keep trying to find/cook up something.

Thanks, Monkey