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An electron (negative charge) and a positron (positive charge) annihilate in pure energy. For an electron and a proton it is all similar, the different charge, the mutual attraction and the emission of EM radiation.

What is the mechanism by which an electron and a proton do not annihilate, but stop their approach at some distance?

To confine the answers, please do not refer to statements such as that the calculated energy levels are stable states. This is a consequence of the initial observations; the corresponding calculations only reflect the observations.

HolgerFiedler
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    You seem to think that the electron and a proton are each other's antiparticle. They are not, and so they do not annihilate. For an answer to your question see https://physics.stackexchange.com/questions/106020/how-exactly-do-protons-and-electrons-interact-with-each-other – hdhondt Jun 26 '20 at 10:08
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    @hdhondt Wrong interpretation of what I think. please stay with what is written. second comment: the answer refers to calculations. I want to know the mechanism of the barrier. – HolgerFiedler Jun 26 '20 at 10:09
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    Why should they annihilate? And why do you say "stop their approach at some distance"? An s orbital electron has a non-zero probability of being located inside the nucleus. – PM 2Ring Jun 26 '20 at 10:28
  • Also note that a proton & electron can react (via the weak force) to produce a neutron & a neutrino. – PM 2Ring Jun 26 '20 at 10:30
  • @PM2Ring “Why should...”. For an electron and a proton it is all similar, the different charge, the mutual attraction and the emission of EM radiation. – HolgerFiedler Jun 26 '20 at 10:59

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A particle can only annihilate with its own antiparticle, not just any other oppositely charged particle.

Another way to look at this is that there are many quantities that are conserved in electromagnetic interactions, which would not be conserved in an electron-proton annihilation. For example, baryon number and lepton number would not be conserved.

DavidH
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  • To be fair, reactions between composite matter & antimatter particles that aren't exact opposites are sometimes referred to as annihilation. Eg, neutron + antiproton, or proton + antineutron. – PM 2Ring Jun 26 '20 at 10:34
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This is a surprising formulation: "What is the mechanism by which an electron and a proton do not annihilate?". There is no mechanism and hence protons and electron do not annihilate.

my2cts
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As has been already said, the proton is not the anti-particle of the electron. Now, in your post, you say that electron and positron annihilate. Actually what we call "an electron" is a special excitation of what we call "electron/positron field". When you solve the Dirac equation for hydrogenoïds it is very clear that what we call an electron is in fact not just the two first components of the field. So "negative excitation" annihilates with its positive counterpart. There has never been any reason for the electron to annihilates with the proton. Particles do not annihilate because they have the opposite charge. They annihilate because they are particle/anti-particle counterpart (think of the neutrinos: no charge but annihilates with the anti-neutrino).

Jeanbaptiste Roux
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More a comment than an answer.

For me, the only difference between an electron-proton interaction and an electron-positron annihilation is the mass difference for the first interaction. Only in a perfect charge equilibrium - opposite charge AND equal masses - does annihilation take place.

The mechanism behind it can only be modelled with the next ("standard") particle on the next smaller particle level. Since I do not see any practical application, it is pure science. But isn't it the freedom of science to think about something that seems to have no application?

HolgerFiedler
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    Protons & positrons have the same EM charge, but they differ in other quantum numbers, not just mass, principally baryon number, and lepton number. Protons feel the strong nuclear force, positrons do not. FWIW, Dirac originally thought that protons & electrons may be each other's antiparticle, but it was quickly realised that that wouldn't work. – PM 2Ring Jun 26 '20 at 11:46
  • Also see https://physics.stackexchange.com/q/451323/123208 – PM 2Ring Jun 26 '20 at 12:53