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So I was thinking today about when observables become "quantized", and came to the conclusion that every instance of quantization I've ever come across has come about from solving the Schrödinger equation over a compact region.

This seemed to hold for everything I could think of which exception to the radial solution to the hydrogen atom. The region over which it is solved is $r\in [0, \infty)$ (the angular regions are compact). Also, the Simple Harmonic Oscillator is an example. (I'm sure there are more)

Question: Is there a more general notion of "quantization" that will work in all cases? That is, a bidirectional "if and only if" statement. And if not, why isn't that a huge problem with our theory?

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  • $\begingroup$ What's wrong with the 1D quantum oscillator? $\endgroup$ Jan 15 '19 at 1:37
  • $\begingroup$ Nothing. I suppose that I'm just bad at coming up with examples $\endgroup$ Jan 15 '19 at 1:38
  • $\begingroup$ Another example is morse oscillator. $\endgroup$
    – Sunyam
    Jan 15 '19 at 4:28
  • $\begingroup$ Possible duplicates: Reason for the discreteness arising in quantum mechanics? and links therein. $\endgroup$
    – Qmechanic
    Jan 15 '19 at 5:52
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    $\begingroup$ Ah... in view of the reminder of the near duplicate... the oscillator in phase space is restricted to a (compact!) circle, indeed... In some sense, the periodicity of the motion restricts time to one period. The "region" in the question is more general than I thought. $\endgroup$ Jan 16 '19 at 21:01

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