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From the initial state of an electron and a proton in a box. I would like to find a reasonable hamiltonian, or way to describe the interaction that leads to the formation of a Hydrogen atom.

Here is the initial would like to go from.

For |e> and |p> the electron and proton in an eigenstate of a particle in a 1D box, at some $n_e$ and $n_p$ energy levels. (so $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, with $m$ the mass of the particle in consideration, and $L$ the length of the box). Then I define my initial state $$|\psi(0)> = |e>\otimes |p>$$ An electron and proton can bind to form a hydrogen atom (potentially excited at some energy level). So I would like the initial state to evolve to something like the following. $$|\psi(T)>=|a>\otimes|\gamma>$$ Where $|a>$ is a hydrogen atom state (delocalized in the box?), and $|\gamma>$ is a photon emitted by the process.

I understand the process could emit more than one photon or involve other particles. But I would like to consider, if possible a simplified process, as I would later like to add more protons and electrons to my box.

What is a good hamiltonian to describe my evolution to the atom state?

TheStressTensor
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    Hydrogen atom in 1D is somewhat problematic, see, e.g., this thread – Roger V. Mar 01 '23 at 10:57
  • I see. Is it important if I am not trying to detail the internal degrees of freedom of the hydrogen atom but just its formation or not? – TheStressTensor Mar 01 '23 at 13:03
  • It is not fully clear to me what you are trying to achieve. In the second quantized form it is quite easy to formulate a Hamiltonian for the process that you want to describe, but you seem to be interested in wave functions... Btw, working in a box does not seem like a good start either, if you are working with Coulomb potential. – Roger V. Mar 01 '23 at 13:28
  • If you want a model problem with a bound state formation, you might consider electron and proton in one dimension interacting via a delta-potential. In any case, the first thing you do is transforming to the center-of-mass system of coordinates (basically the proton coordinates, since $m_p/m_e\approx 1000$.) – Roger V. Mar 01 '23 at 14:22
  • Ok, if it easy to formulate a Hamiltonian for the process I want, could you tell me how? It is important that it be in a box, I am working with the restriction of making an atom from energy eigenstates. I do not understand why the delta potential would be good? My proton is as delocalized as my electron. – TheStressTensor Mar 01 '23 at 16:27
  • The Hamiltonian of a "formation of a hydrogen atom" is just the Hamiltonian of a hydrogen atom (possibly including its interaction with a quantized electromagnetic field, but you can probably treat the electromagnetic field as classical). You can treat the capture of an electron by a proton via time-dependent perturbation theory (or, perhaps, scattering theory), where the initial state is a plane wave (or, a scattering state of hydrogen, but that's more complicated) and the final state is a bound state of hydrogen. – march Mar 01 '23 at 16:33
  • I can imagine several different problems to solve, which would involve electron, proton and a bound state, so you need to be more specific about the problem that you are trying to solve. I could just write $H_{int}=\lambda A^\dagger\gamma^\dagger p e + h.c.$, where $A^\dagger$ is the operator creating a bound state (atom), $\gamma^\dagger$ creates a photon, whereas $p,e,\gamma$ are the annihilation operators of proton and electron. But I suspect that this is not the level of generalization that you are looking for. – Roger V. Mar 02 '23 at 09:16
  • Alright, I simplified the question. Let me know if you want some more details on what I am looking for. – TheStressTensor Mar 02 '23 at 16:14

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