Here is a simple explanation: They can't fall in due to Heisenberg's uncertainty. For that, I'd need you to accept as an axiom[1] (unprovable truth) that the following principle is true:
ΔpΔx (uncertainty of momentum times uncertainty of position) is roughly equal (or greater) to some constant, we'll call it H.
Due to this, if one uncertainty grows smaller, the other grows larger.
If the electron is located in the nucleus, its position (Δx) would be much more narrow than if it was around the nucleus. Since the uncertainty of Δx goes down, Δp must go up co compensate. Turns out, this Δp is enough to give it an enough momentum to overcome attractive force.
But then comes a smart observer, and says, but what if an electron managed to lodge itself exactly into the center of a proton. Since Coulomb's law says F= q1q2/r^2, and r is 0[2], that's Infinity! You can't escape infinite charge attractiveness!
To that, you can notice that as r and Δx approaches 0, Δp also approaches infinity, so it will have more chance to escape before that happens. But in some rare cases, it will interact and form a neutron, with some energy being emitted as a neutrino.
[1] It's not an axiom, it's an observation derived from experiments. However, why is the matter behaving like that out of physics wheelhouse. It can tell you a lot about laws, but very little WHY are laws like that. So it might as well be an axiom.
[2] This is, of course, assuming that space is infinitely divisible, which is yet to be confirmed or denied.
I have heard this too and still feel that there's more to this that is critically missed.
>> For that, I'd need you to accept as an axiom (unprovable truth) that the following principle is true: ΔpΔx is roughly equal (or greater) to some constant, we'll call it H.
As far as I know, Heisenberg's principle isn't exactly an axiom. It can largely be derived!
A waveform in time and its Fourier transform have such a relationship, which is entirely mathematical. More confined a time waveform is, more spread out the frequency spectrum is, and vice versa.
Now de Broglie principle linked momentum of a particle to its frequency! That makes the trick. So far, in Newtonian mechanics, momentum just related to position via derivative of position, i.e., velocity. Both were time waveforms.
But with de Broglie principal, momentum (also?) links to frequency domain, while the position remains in time domain. Now the Schrodinger's equation yields the said relation between position and momentum.
What I'll write next fits better in another comment box, so will do that.
How can something be an axiom and at the same time has a mathematical derivation well understood from Quantum Mechanics [1] and are supported by experimental observations ?
The idea of axioms in mathematics is not quite the same as it is with physics.
Physicists don't need the type of mathematical rigor such as the set theory axioms to build up all of mathematics. In fact, by keeping in mind the multitudes of paths from which one part of physics could be related or derived from another part, you are exploring the possibly undiscovered laws in nature.
It will only be relevant to axiomatically define physics (from a set of basic axioms to reach all laws) after we confirmed to have discovered all of physics. That day hasn't come yet - there's no grand unified theory so far, and there's still stuff that we dont know surely.
> How can something be an axiom and at the same time has a mathematical derivation well understood from Quantum Mechanics
See my notes, it's not an axiom, just take it for granted for the explanation to work. You can derive it from experiments like squeezing light, or you can derive it from other things as you mentioned.
Due to this, if one uncertainty grows smaller, the other grows larger.
If the electron is located in the nucleus, its position (Δx) would be much more narrow than if it was around the nucleus. Since the uncertainty of Δx goes down, Δp must go up co compensate. Turns out, this Δp is enough to give it an enough momentum to overcome attractive force.
But then comes a smart observer, and says, but what if an electron managed to lodge itself exactly into the center of a proton. Since Coulomb's law says F= q1q2/r^2, and r is 0[2], that's Infinity! You can't escape infinite charge attractiveness!
To that, you can notice that as r and Δx approaches 0, Δp also approaches infinity, so it will have more chance to escape before that happens. But in some rare cases, it will interact and form a neutron, with some energy being emitted as a neutrino.
[1] It's not an axiom, it's an observation derived from experiments. However, why is the matter behaving like that out of physics wheelhouse. It can tell you a lot about laws, but very little WHY are laws like that. So it might as well be an axiom.
[2] This is, of course, assuming that space is infinitely divisible, which is yet to be confirmed or denied.