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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?

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    $\begingroup$ Hydrogen atom in 1D is somewhat problematic, see, e.g., this thread $\endgroup$
    – Roger Vadim
    Mar 1 at 10:57
  • $\begingroup$ 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? $\endgroup$
    – Marin Girard
    Mar 1 at 13:03
  • $\begingroup$ 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. $\endgroup$
    – Roger Vadim
    Mar 1 at 13:28
  • $\begingroup$ 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 $m_p/m_e\approx 1000$.) $\endgroup$
    – Roger Vadim
    Mar 1 at 14:22
  • $\begingroup$ 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. $\endgroup$
    – Marin Girard
    Mar 1 at 16:27

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