Suppose we have two particles, 1 and 2, and two possible states for such particles, α and β. Quantum physics assigns an amplitude, a, for every possible state and process in nature. If we put particle 1 into state α and particle 2 into state β, the corresponding amplitude is a_α(1)a_β(2).

But there is a basic concept of quantum physics: when two or more situations are completely indistinguishable, the various possibilities combine algebraically. And there is another basic concept: for fermions, the combinations have minus signs, while for bosons they have plus signs. Thus if particles 1 and 2 are absolutely identical fermions, the actual amplitude for the situation in which we are interested is

Since there is no distinction between particles 1 and 2, the two possibilities cannot be distinguished. And this is why “solid” matter appears to be solid! The atoms in the chair seat have electrons in the same states as atoms in you. If the atoms in the chair seat could penetrate into the atoms of you, even slightly, there would be regions of space where two identical electrons are in the same state. What is the probability that this can happen, that two electrons can both be in state α, for example?

An atom is a very abstract thing, a probability distribution for the electrons, and a probability distribution for the quarks and gluons that make up the nucleus.

Atoms interact somewhat as if they were spongey foam balls... they can get just so close and no closer... they stack up against one another in a crystal like grapefruits in a display at the grocery store.

Because... this kind of overlap of the two different probability amplitudes has a probability of zero. Atoms cannot overlap or pass through one another, because they are made of identical fermions, namely electrons and quarks.

Model of table salt crystal

Actual STM image of the surface of a metal