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?
