Where is the derivation of the "Selection Rule" that stops Hydrogen from becoming a neutron and a photon?

Answer 1

Yes the rest mass of a neutron is at least 2 electron masses more than a proton, so as linwood l said there's insufficient energy for a light hydrogen atom by itself to become a neutron.

Something like this (K capture) is possible for an atom with a heavier nucleus but only with the involvement of at least one other lepton (electrons,positrons or neutrinos) in either the initial or final state. The total number:

electrons + electron neutrinos - [positrons + antineutrinos]

must be conserved. This is violated by e + p --> n + photon

A more basic way of seeing the same thing is angular momentum conservation. Electrons, protons and neutrons all have a spin of 1/2. Photons have spin 1. The orbital angular momentum of any particle is also always integral. So hydrogen (e+p) must have an integral angular momentum and (n + photon) must have a half integral total angular momentum. So the initial and final states would be mutually exclusive.