Zero dimensional points do not take up space, so then wouldn't everything in the universe be literally empty? Or is there something that I'm missing?
Answer
Although it's commonly said that fundamental particles are point particles you need to be clear what this means. To measure the size of the particle to within some experimental error $d$ requires the use of a probe with a wavelength of $\lambda=d$ or less i.e. with an energy of greater than around $hc/\lambda$. When we say particles are pointlike we mean that no matter how high the energy of your probe, or how small its wavelength, you will never measure a particle radius greater than your experimental limit $d$. That is the particle will always appear pointlike no matter how precise your experiment is.
But this does not mean that the particles are actually zero dimensional, infinite density, dots whizzing around. An elementary particle does not have a position in the way we think of a macroscopic object as having a position. It is always delocalised to some extent, i.e., it exists across a region of some non-zero volume. More precisely the probability of finding the particle is non-zero anywhere within that region.
So an atom is not empty space. The usual analogy is that it is a fuzzy blob, and actually that's a not a bad metaphor. If we take any small volume $\mathrm dV$ within the atom then the probability of finding the electron in that region is given by:
$$ P = \int \psi^*\psi\,\mathrm dV $$
where $\psi$ is the wavefunction describing the electron in the atom. And since this probability is just the charge density that means the charge density varies smoothly throughout the atom.
It is important to be clear that this is not just some time average due to the electron whizzing about the atom very fast. It is not the case that the electron has a precise position in the atom and our probability is some time average. The electron genuinely has no position in the usual macroscopic sense.
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