what's compact about a hydrogen coulomb potential& wave function?
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]
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“My answer: compactness” Does that mean the answer in the physics community is currently unclear ?
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Somewhat, yes. It's unidirectional. We know that compactness => quantization, but as
@Quantensalat pointed out from our example of the hydrogen atom. Solutions to the S.E over non compact regions can still become quantized.
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Something like boundary conditions
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Boundaries.
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