〈 Berger | Dillon 〉

what's compact about a hydrogen coulomb potential& wave function?

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

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

But isn’t that just a necessary condition for normalization? Thanks for sharing these nice stuff

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

I was going to say "finite boundary conditions" as a more descriptive alternative to what I assumed you meant by comoactness. Just apply periodic ones to the atom and you get your radial solutions.