This is not true in quantum physics. This is a big misunderstanding among many physics students that they live with for many years because they first learn about the atom using the planetary model of atom. In the planetary model the atom looks mostly empty. But this view of the atom becomes nonsense after learning QFT.
The space within an atom is filled with the wavefunctions of the electrons. If you still ask "but the electron is still going to be somewhere", I can only say that our intuition about classical physics fails in the quantum physics.
According to QFT, the quantum fields permeate the space everywhere, but the field itself can be in a "vacuum state", which means it has the lowest possible energy. (But unlike in classical physics, the lowest possible energy is not zero.)
A wall of electrons against anther wall of electrons, same charge, no luck moving through with only our muscles' strength. But if suddenly at least one of those walls can move through electrons it finds mostly vacuum in front of itself. I think this is what happens in that material.
The visual we draw in our mind that the atom is mostly vacuum inside is due to taking our intuition about classical physical world and misapplying it into the world inside an atom where classical physics intuition is not relevant.
In quantum physics, an atom is not mostly vacuum. It is filled with wavefunctions of the electrons.
people say this casually... and fail to define what a wave-function is physically.
It sounds like a non-sense answer.
if there is matter there, in the form of a 'wave function', what is the matter of a 'wave function' (other than just the electron zipping about)?
no: a wave function is a mathematical construct we use to predict where an electron might appear... (well, not exactly the wave function, but we get the probability of an electron's appearance by squaring the wave function).
A wave function (probably), doesn't 'physically' exist. it's just a useful model to predict probabilities...
waves exist only in a medium of expressive material - if that material is a single electron, there isn't magically more material there once we use a wave function to predict it's location - it's still only one electron - it's position predicted by the square of the wave function - but that doesn't mean the wave function is a physical entity.
And it's not like the electron is 'going faster than light' and 'blurring frames in reality'... it's just the electron there - the squaring wavefunction is just a way to predict it's location: just because it (seems to) works doesn't mean it physically exists.
but if it DOES exist physically (in a concrete way) - I would love an explanation or link to that proof - as that would be news to me.
fundamentally this might just be a semantics issue on the word physical
What you're asking for doesn't currently exist. At least not any proven ones. There's a dozen new philosophical interpretations of quantum mechanics every year but they usually make no new predictions, so there is no way to test if the interpretation is correct.
Most (all?*) interpretations only differ in defining the "wavefunction collapse" and what happens before it. But since measuring anything involves collapsing the wavefunction, it's impossible to measure what happens before that.
We do know, from real-world experiments, that particles cannot actually be moving point-like objects. The most famous example is the double slit experiment where a single particle can cause wavelike interference with itself.
But also we know it isn't exactly a classical wave. We can only measure it as a single point, and it arrives in discrete events, not a continuous transfer of energy like a wave.
So "wavefunction" and the rest of QM lingo is what we have. We don't fully know what those are, but we also know that being just a point or wave in the style of classical physics cannot be correct.
* If an interpretation does make a new prediction that is measurable, I'm not sure if it's considered just an interpretation anymore.
Note that you are demanding something impossible: you are demanding a "physical" explanation, but what you seem mean by "physical" is "something that I can intuitively understand with my preconceived notions". But nature doesn't care whether the reality fits your intuition or not!
The wave-function – a model that fits the data – is defined mathematically, and by Occam's razor, anything "fluff" added to it that makes it "easier to grok", makes it _further_ away from an actual explanation of reality.
This appears to be true on the human scale, but if you "punch the wall" (accelerate hydrogen and carbon into silicon and aluminum) at say 1/10th the speed of light, there will be penetration well past the surface.
More obviously if you think about how easily neutrons interact compared to protons, neutrons routinely go right through people without noticing them at all because they aren't net charged so the coulomb interactions that we are familiar with in "normal" matter don't apply.
The distinction that matters, however, is the extent to which these particles interact with each other through their fields. The "size" of a nucleus has little to do with it.
Rather, what we think of as a "solid" has very different definition than what we would intuitively think. But in the end, that's just semantics. (According to the linked prof. Moriarty, the "boundary" of a solid is the point in space where quantum degeneracy pressure is in balance with van der Waals forces – so there is a well-defined boundary!)
Also, I think that "point particles" are a rather illusory concept. More like, interactions between particles are highly localized when observed by a macroscopic classical observer.
"The diameter of the nucleus is in the range of 1.70 fm (1.70×10−15 m ...", from:
https://en.m.wikipedia.org/wiki/Atomic_nucleus