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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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
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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But isn’t that just a necessary condition for normalization? Thanks for sharing these nice stuff
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Ah, I see what you’re saying. We usually say that it’s imposing the wave function not blow up at infinity is a condition for normalizABILITY.. that is the wave function must live in some Hilbert space
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