quantum mechanics - Electron shells in atoms: What causes them to exist as they do?

I have seen similar posts, but I haven't seen what seems to be a clear and direct answer.

Why do only a certain number of electrons occupy each shell? Why are the shells arranged in certain distances from the nucleus? Why don't electrons just collapse into the nucleus or fly away?

It seems there are lots of equations and theories that describe HOW electrons behave (pauli exclusion principle), predictions about WHERE they may be located (Schrödinger equation, uncertainty principle), etc. But hard to find the WHY and/or causality behind these descriptive properties. What is it about the nucleus and the electrons that causes them to attract/repel in the form of these shells at regular intervals and numbers of electrons per shell?

Please be patient with me, new to this forum and just an amateur fan of physics.

Answer

Any answer based on analogies rather than mathematics is going to be misleading, so please bear this in mind when you read this.

Most of us will have discovered that if you tie one end of a rope to a wall and wave the other you can get standing waves on it like this:

Depending on how fast you wave the end of the rope you can get half a wave (A), one wave (B), one and a half waves (C), and so on. But you can't have 3/5 of a wave or 4.4328425 waves. You can only have a half integral number of waves. The number of waves is quantised.

This is basically why electron energies in an atom are quantised. You've probably heard that electrons behave as waves as well as particles. Well if you're trying to cram an electron into a confined space you'll only be able to do so if the electron wavelength fits neatly into the space. This is a lot more complicated than just waving a rope because an atom is a 3D object so you have 3D waves. However take for example the first three $s$ wavefunctions, which are spherically symmetric, and look how they vary with distance - you get (these are for a hydrogen atom) $^1$:

Unlike the rope the waves aren't all the same size and length because the potential around a hydrogen atom varies with distance, however you can see a general similarity with the first three modes of the rope.

And that's basically it. Energy increases with decreasing wavelength, so the "half wave" $1s$ level has a lower energy than the "one wave" $2s$ level, and the $2s$ has a lower energy than the "one and a half wave" $3s$ level.

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_{$^1$ the graphs are actually the electron probability distribution $P(r) = \psi\psi^*4\pi r^2$. I did try plotting the wavefunction, but it was less visually effective.}