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I understand that there are no states between $n$ and $n{\pm}1$, so for a particle to get from state $n$ to $n\pm{1}$ takes $1$ step. Also depending on the potential operator function $V$ a particle may jump from a bound state to a scattered state or from a scattered state to a bound state. So for instance a hydrogen atom could absorb a photon with enough energy for the electron to no longer be bound to the nucleus. Similarly an electron nucleus system that starts in a scattered state could emit a photon with enough energy for the electron to become bound to the nucleus.

I was wondering if it's possible for a particle to jump from state n to state $n\pm{m}$ with $m>1$ without crossing through every integer state in between. For instance if a hydrogen atom was in state $n=7$ could it get to state $n=1$ in $1$ step or would it necessarily take $6$ steps to get to $n=1$ assuming the that state remains bound at every step?

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Yes. See the named hydrogen spectral series:

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The idea that the atomic energy levels are discrete and that electrons jump instantaneously from one to another is a relic of the pre-quantum Bohr model.

Atomic transitions in a more accurate model are very complicated because they involve also the electromagnetic field (or whatever carries away the energy). They are continuous, with the system passing though intermediate states that don't resemble undergraduate-QM orbitals. And in particular, there is no requirement that the path from the initial to the final state go through states that look like undergraduate-QM orbitals of intermediate energy.

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  • $\begingroup$ They are continuous, with the system passing though intermediate states that don't resemble undergraduate-QM orbitals could you please add a reference (paper or textbook)? I find this topic fascinating and I would like to look a bit further into it. Thank you $\endgroup$
    – Prallax
    Sep 24, 2021 at 18:24
  • $\begingroup$ @Prallax this statement just means that intermediate states are superpositions of eigenstates, rather than exact eigenstates. For a start, you may be interested in time-dependent perturbation theory. $\endgroup$
    – Ruslan
    Sep 24, 2021 at 21:49

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