Compactness and Quantization

Asked Jan 15 '19 at 01:24

Active Jan 15 '19 at 05:53

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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?

edited Jan 15 '19 at 05:53

Qmechanic

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asked Jan 15 '19 at 01:24

InertialObserver

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What's wrong with the 1D quantum oscillator? – Cosmas Zachos Jan 15 '19 at 01:37
Nothing. I suppose that I'm just bad at coming up with examples – InertialObserver Jan 15 '19 at 01:38
Another example is morse oscillator. – Sunyam Jan 15 '19 at 04:28
Possible duplicates: Reason for the discreteness arising in quantum mechanics? and links therein. – Qmechanic Jan 15 '19 at 05:52
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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. – Cosmas Zachos Jan 16 '19 at 21:01
Perhaps the oscillator is not a perfect example, since it only has bound states, without continuum scattering states! For every energy, the domain of allowable xs is restricted by the potential to a compact segment between turning points, and this never stops no matter the Energy! In a square well, you see better how outside the well, with unlimited domain, you are in the continuum, but, inside it (limited domain), you are discrete. – Cosmas Zachos Jan 16 '19 at 22:30

Compactness and Quantization

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