Spin and angular momentum in quantum mechanics

Electrons carry angular momentum that has nothing to do with orbiting anything. It is intrinsic — as much a fixed property of an electron as its charge or mass — and it is called spin. The name invites a picture of a small ball rotating, and that picture fails immediately: an electron has no measured size to rotate, and reproducing the observed angular momentum with a classical sphere would need its surface moving faster than light. Spin is its own thing, with no classical counterpart.

What makes it strange is that it is quantised in half-integer units and, for an electron, has only two possible measurement outcomes. Choose any axis at all, measure the spin along it, and you get +ħ/2 or −ħ/2 — never anything between, and never a value that depends on how the electron was prepared. The Stern–Gerlach experiment shows this directly: a beam of silver atoms sent through a non-uniform magnetic field splits into exactly two beams, where a classical magnetic moment would have smeared into a continuous band.

Spin is what completes the account of atomic structure. It supplies a fourth quantum number alongside n, l and m, and the Pauli exclusion principle says no two electrons in an atom may share all four. That does not forbid sharing an orbital — it forbids sharing a full quantum state. So each orbital holds exactly two electrons, one of each spin, and no more. Fill the orbitals under that rule and the periodic table's row lengths, the chemistry of the noble gases and the reason matter takes up space all follow.