# Formation of a hydrogen atom from a proton and electron - detailed quantitative description

Suppose a region of empty space, to which we add one proton and one electron, initially separated by a distance on the order of centimeters, and as close to "at rest" in the center-of-mass frame as we can practically achieve.

They will of course attract each other and combine into a hydrogen atom, emitting at least one photon in the process. I'd like to understand this process in detail. Specifically: for an initial separation this large, it seems likely that the system will not emit just one photon and go directly to the 1H ground state, because that photon would be very energetic. Instead there would be some number of bremsstrahlung interactions, followed by atomic orbital transitions, before reaching the ground state.

• How does one model this quantitatively?
• What would the typical number of emitted photons be, and what would their wavelengths be?
• How small can you make the initial separation before the most probable outcome is just one emitted photon?
• How does one draw the line between "bremsstrahlung" and "atomic orbital transition" in a system like this?
• What is the (ballpark) probability of something else happening, such as a weak interaction producing a neutron? (That's the only possible "something else" I know of. Are there others?)
• I like this question, but I have a feeling it might be too broad. You might want to ask something more specific, or divide it into a couple of different questions. – Javier Feb 26 '16 at 18:40
• What could possibly make you think that Quantum Mechanics turns off at some distance? In cold regions of space Rydberg atoms can be quite large, but they are still Quantum. – Timaeus Feb 26 '16 at 19:04
• @Timaeus Is it seriously the case that classical electrodynamics is still a poor approximation for macroscopic initial separations in this hypothetical, or are you just being pedantic? – zwol Feb 26 '16 at 19:11
• @zwol You claim you want to start with them having a known fixed position and a known fixed momentum. Do you see the problem? That would be like in Newtonian mechanics saying you want a particle to start out with an initial force and an initial acceleration that defy $F=ma$ you decided to break a law of physics as your alleged initial condition. The true initial conditions are totally different. They involve an initial state. In Newtonian mechanics you aren't free to adjust your acceleration. In QM you aren't free to adjust your velocity. – Timaeus Feb 26 '16 at 19:27
• @Timaeus I would think that the initial state here is something like a product of two localized wavepackets, one for the electron and one for the photon – Mitchell Porter Feb 27 '16 at 18:32

In non-relativistic theory, one has the Schroedinger equation. The radiation due to system is NOT accounted for in the standard form of this equation. The equation conserves $\langle H \rangle$. The $\psi$ function will change in time but won't get to a point where it resembles atom of Bohr size. It will diffuse and change, but it will get bigger rather than smaller.