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Suppose we have an electron and proton that are separate... not forming an atom... At some point when bringing them together, they form a hydrogen atom.

What is the dividing line between atom and non-atom and what exactly is the distinction? Is there a sharp difference in the nature of the interaction between an electron and proton when they form and atom and when they don't? What's the nature of this distinction?

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    $\begingroup$ Allowable electron energy states in a hydrogen atom go right up to the continuum limit. So, the only distinction is whether the electron is bound or not to the proton. $\endgroup$ – Jon Custer Jul 7 '17 at 16:48
  • $\begingroup$ How do we distinguish an electron that's bound to a proton and one that's not. Is the boundedness a measurable property? $\endgroup$ – Ameet Sharma Jul 7 '17 at 17:00
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The electron is defined to be "bound" to the proton if its total energy (kinetic + potential) is negative. In this case there is a maximum distance away from the proton that it can reach. It's very similar to planetary orbits - the electron needs to reach "escape velocity" (corresponding to zero energy) in order to, well, escape.

An atom is defined to be a nucleus and bound electrons. If you give the electron in a hydrogen atom the "ionization energy" for it to escape arbitrarily far from the proton, then the atom falls apart into a nucleus and a "loose" electron.

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A hydrogen atom is formally formed when an electron is sitting in an energy level of the corresponding potential.

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A stable hydrogen atom is when the electron is sitting at the n=1 ground level. If it is in any other level the atom is unstable, and the electron will cascade down to the ground level. At the ionization energy, where n theoretically is infinity,a free electron nearing a free proton will cascade down with electromagnetic radiaton to the ground level. Whether it will be caught, or a slight change in the energy will send it off and away from the proton is a matter of probabilities.

In cosmology it is an important state when free electrons and protons start forming hydrogen atoms. It is a thermodynamic phenomenon, and is described here

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  • $\begingroup$ So if the energy is something other than these levels, is the electron free? $\endgroup$ – Ameet Sharma Jul 7 '17 at 17:45
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    $\begingroup$ @AmeetSharma: if the electron is bound (ie the energy is below the ionization energy) then the energy can't be other than these levels. $\endgroup$ – tfb Jul 7 '17 at 17:48
  • $\begingroup$ The electron approaching the proton should be at rest to it, at the ionization level, then it can be caught. Other wise it will scatter off, electron proton scattering. It is the photons that need to have within a width the correct energy to raise an electron to a higher level, photon-hydrogen scattering. The electron just needs to have zero momentum with respect to the available hydrogen ionisation level ( within the widths of the solution) $\endgroup$ – anna v Jul 7 '17 at 18:28

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