〈 Berger | Dillon 〉

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.

What do you mean by periodicity in the radial solutions?

On a very basic Bohr-y level, the orbit is a circle, so the wf is subject to a periodic boundary condition. At a distance 2 pi r, the wf must repeat to avoid destructively interfering itself away. Wavelengths work that satisfy this AND the wavelength describes the corresponding E