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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what's compact about a hydrogen coulomb potential& wave function?
That's a nice counterexample.. If I had to make my statement more precise.. I would have said "Solutions to the Schrödinger equation over compact regions".. Which, for the hydrogen atom is true for the θ, φ directions, but as you point out the radial part is over [0, \infty]
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Now that I think about it, it's rather counterintuitive (to me) that the radial solutions get quantized as well.. That's quite an amazing fact
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