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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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]
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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Is it because we impose normalization? So for the hydrogen atom we insist that the polynomial in the series solution must terminate at some highest order, and the same thing for Hermite polynomial in the harmonic oscillator
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The fact that the series terminates isn’t normalization.. the fact that the series terminates comes from the boundary conditions at infinity
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