ELECTRON-PROTON ATTRACTION: a simple, semi-classical analysis to avoid full scale QFT.
The exchange of the photon between the proton and the electron leads to attraction, only because the total energy of the electron is negative.
Let us consider the hydrogen atom for simplicity, and imagine the funnel-like shape of the electron energy. The total energy of the electron in the hydrogen atom at distance $r_0$ from the proton is
$E(r_0)=-\frac{e^2}{8\pi\epsilon_0 r_0}$.
If the distance $r_0$ is sufficiently short, then the electron will emit a photon which will be absorbed by the proton, and the amount of energy of the exchanged photon will be dictated by the uncertainty principle:
$\Delta E\Delta t=\hbar.$
But $\Delta t=\frac{r_0}{c}$ so that
$\Delta E r_0=\hbar c\rightarrow \Delta E= \frac{\hbar c}{r_0}$
So the new energy of the electron will be
$E_1=-\frac{e^2}{8\pi\epsilon_0 r_0}-\frac{\hbar c}{r_0}=-\frac{e^2+8\pi\epsilon_0\hbar c}{8\pi\epsilon_0r_0}$
or the equivalent amount of energy corresponding to some new position $r_1$
$-\frac{e^2}{8\pi\epsilon_0 r_1}=-\frac{e^2+8\pi\epsilon_0\hbar c}{8\pi\epsilon_0 r_0}$
from which we get $r_1$ in terms of $r_0$
$r_1=r_0\frac{e^2}{e^2+8\pi\epsilon_0\hbar c}<r_0$
Therefore the electron moves closer to the proton rather than farther from it (an attractive force.)
My main question is why don't they affect other photons (unless colliding), shouldn't there be an attraction or repulsion by the exchange force? Or is it because I would need a QFT treatment in that case?
Photons are quantum mechanical entities, so yes, it is a QFT case. Any interaction between two photons goes through exchange diagrams. Two photon interactions occur, with very low probability because the diagrams are box diagrams with at least four 1/137 couplings depressing the probability . For light frequencies this is a very small number . This probability grows with energy so even gamma gamma colliders are envisaged.
Best Answer
This is the lowest Feynman diagram showing the exchange of virtual photons in electron electron scattering |
Depending on the problem one would be studying with proton-proton elastic scattering, a similar diagram with "p" instead of "e" would be used for the electromagnetic interaction.
Considering that the protons are composed of charged quarks, for high enough energies of the proton scattering strong interactions will become important, even the quark structure..