We know that not all quantum systems are quantized. For example, the free particle has a continuous spectrum of momenta. Fun question of the day: If you had to sum up in one phrase when a particles observables become quantized, what would it be? My answer: Compactness.
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I was going to say "finite boundary conditions" as a more descriptive alternative to what I assumed you meant by comoactness. Just apply periodic ones to the atom and you get your radial solutions.
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On a very basic Bohr-y level, the orbit is a circle, so the wf is subject to a periodic boundary condition. At a distance 2 pi r, the wf must repeat to avoid destructively interfering itself away. Wavelengths work that satisfy this AND the wavelength describes the corresponding E
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I agree that the angular part of the Hamiltonian is quantized since it’s solved over a compact region, but the radial solutions that describe the radial probabilities are something you can’t get from the Bohr model or come from periodicity
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