The Pauli exclusion principle

Left to minimize energy alone, every electron in an atom would seem to want to fall into the single lowest orbital, since that is the configuration of lowest total energy. Real atoms plainly do not do this — a carbon atom's six electrons spread across multiple orbitals, not all six crammed into the lowest one — and the reason is the Pauli exclusion principle: no two electrons in the same atom can occupy an identical quantum state, meaning no two can share the same full set of quantum numbers (n, l, ml, ms) at once.

The principle applies specifically to fermions — the class of particles that includes electrons, protons, and neutrons — and does not apply to the other broad class of particles, bosons (which includes photons), which can and routinely do pile into the same state without restriction, as in a laser beam. For electrons in an atom, the exclusion principle means that once an orbital's spatial quantum numbers (n, l, ml) are fixed, it can hold at most two electrons, and only because the spin quantum number ms provides one remaining way to differ — one electron spin-up, the other spin-down. A third electron cannot join that same orbital in any spin state without duplicating an already-occupied set of all four numbers, so it is forced into a higher orbital instead. This is exactly the mechanism, applied orbital by orbital, that builds up the shell structure of the periodic table: electrons fill orbitals from lowest energy upward, two at a time, precisely because no further pair can fit into an already-full orbital.

Beyond atomic structure, the exclusion principle is the fundamental reason ordinary matter has bulk and resists compression at all: the electrons in a solid cannot all be squeezed into the lowest available states, so pushing atoms closer together requires forcing electrons into higher-energy states, which costs energy and produces the effective repulsive pressure — called electron degeneracy pressure in its most extreme form — that keeps everyday solids from collapsing, and that ultimately supports white dwarf stars against their own gravity.