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.


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*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?)

 A: Your question falls into an area of physics that is hard to understand - evolution of a system in time according to quantum-theoretic AND relativistic theory of matter.
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.
In a relativistic theory, there are many approaches and equations, but I do not know if there is some preferred way to model the process you imagine. I have never seen a paper that would try to simulate formation of bound state in time.
On a general level in QFT, one has the action principle for the quantum fields of electron and proton. One can derive equations restricting the fields from this principle and try to extract description of the system in time. But how to set up initial conditions for those fields that correspond to your picture of two particles 1m apart and how to visualize the evolving quantum fields, I do not know.
There is also a so-called Bethe-Salpeter equation that deals with bound states in a way close to QFT, but from the papers on it I've seen I got the impression that it is hard to find solutions and the authors themselves did not think it gives more insight beyond what can be extracted from the non-relativistic equations of Breit type with relativistic corrections (these still do not account for retardation and radiation).
To anyone reading this, if you know of some paper on this, please link it in a comment or post an answer, I would like to read it.
