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A quantum harmonic oscillator, or a quantum model of the hydrogen atom, have discrete energy spectrum. However, both models involve a potential, which means that the system is not isolated. One could think that in principle, the external environment responsible for the potential could be integrated into the model, so as to obtain an isolated system again. In the case of the hydrogen atom, this would be the wave function of nucleus+electron and the photon they could exchange. My questions are the following:

1/ is this idea correct? Is it possible to have a quantum model of nucleus+electron as an isolated system, without external potential, or more generally, to eliminate potentials by considering larger systems?

2/ if this is the case, would the energy spectrum still be discrete in such a larger system? Intuitively I would say no: an isolated system must have continuous energy spectrum but I'm not sure this intuition is correct.

3/ if this intuition is correct, isn't there a puzzle in the fact that the energy spectrum becomes discrete when the nucleus is represented as a potential rather than as a quantum system? Shouldn't both representations give the same result?

4/ if the intuition of (2) is not correct, where does discreteness "comes from" in an isolated complex system? Does it come from spliting the system in parts and having an interaction hamiltonian with discrete spectrum between these parts for example?

Quentin Ruyant
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  • Related: https://physics.stackexchange.com/q/39208/2451 and links therein. – Qmechanic Dec 23 '17 at 11:55
  • Thanks. I had read it before posting my question but it didn't really answer it, since boundary conditions are apparently involved for retrieving discreteness according to the answers, so the puzzle remains: why couldn't these boundary conditions be incorporated into the model, and then would we still have discreteness? – Quentin Ruyant Dec 23 '17 at 12:05
  • either classically, the potential between two charges, or quantum mechanically the potential is between the two charges, it is not external. – anna v Dec 23 '17 at 12:16
  • @annaV in Newtonian gravitation, in a model of free fall for example, we can introduce a potential associated with the Earth, but this is because the Earth is not part of the model. We could in principle come up with a larger model containing the Earth, and we would have gravitational forces between two objects instead of a linear potential applying to the object in free fall. Isn't it the same in quantum mechanics? – Quentin Ruyant Dec 23 '17 at 12:52
  • You are wrong. The gravitational potential of the earth and apple system is a 1/r potential, except that the mass of the earth is enormous with respect to the mass of the apple and one can use the approximation that leads to f=mg. instead of f~m1m2/r^2 . https://en.wikipedia.org/wiki/Gravity#Newton's_theory_of_gravitation (The potential is the same with a 1/r form.) https://en.wikipedia.org/wiki/Gravity#Equations_for_a_falling_body_near_the_surface_of_the_Earth – anna v Dec 23 '17 at 14:50
  • @annaV ok my point is that when incorporating the earth you can calculate the potential instead of putting it by hand in the equation. But all examples of discrete energy spectrum involve a potential that is put by hand in Schrödinger equation. So my question is: can we calculate it instead by incorporating the wave function of the source, and then, do we get a discrete spectrum? – Quentin Ruyant Dec 23 '17 at 23:57
  • The wave function at the source, if by source you mean the proton in a single hydrogen ( or an electron ) will be something like a plane wave or a wavepacket. Without the interaction proton-electron there are no spectra because no boundary conditions are imposed on these plane waves. Introducing the potential in the Shrodinger equation changes the boundary conditions and the spectrum appears. This answer of mine https://physics.stackexchange.com/questions/375930/is-there-an-objective-asymmetry-between-a-collapsed-and-un-collapsed-wave-functi/375978#375978 might help. – anna v Dec 24 '17 at 05:07
  • @annaV thanks. You say that there are no spectra without interaction but I thought that a free isolated electron (a wave packet) had a continuous energy spectrum, is this wrong? Formally, will the electron-proton interaction take the form of an interaction hamiltonian? – Quentin Ruyant Dec 24 '17 at 14:49
  • yes free single particles are described by wavepackets. Then two wave packets can scatter off each other and if the energy is low enough they are caught in the potential between them at the energy levels there. This happens with not stable systems too, as in e+e- scattering. see http://pdg.lbl.gov/2017/hadronic-xsections/rpp2016-sigma_R_ee_plots.pdf the resonance spectrum – anna v Dec 24 '17 at 17:17

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