spin angular momentum

Spin is angular momentum that a particle simply has, built in, with nothing going around. An electron is not a tiny spinning ball -- take that picture literally and the surface would have to move faster than light -- yet it carries a fixed, intrinsic angular momentum and an associated magnetic moment, as real as any orbiting charge's. Spin is the property that makes electrons magnetic, that Stern and Gerlach caught splitting a beam into two, and that ultimately underlies the stability of matter through the Pauli principle.

Spin obeys the same angular-momentum algebra as orbital motion, but with one radical difference: its quantum number can be a half-integer. The magnitude is S^2 = s(s+1) hbar^2 and the projection is S_z = m_s hbar, but for an electron the spin quantum number is fixed at s = 1/2, so m_s can only be +1/2 ('spin up') or -1/2 ('spin down') -- exactly two states. Because s is half-integer, the states cannot be written as spherical harmonics; instead the spin lives in an abstract two-dimensional space, and its operators are represented by the Pauli matrices, S = (hbar/2) sigma. The associated magnetic moment is mu = -g_s (e/2m) S with a g-factor g_s very close to 2, which is why spin couples so strongly to magnetic fields.

The half-integer value has profound consequences. Particles with half-integer spin (electrons, protons, neutrons, quarks) are fermions, obey the Pauli exclusion principle, and build up the shell structure of atoms and the pressure that holds up white dwarfs; particles with integer spin (photons, the Higgs) are bosons and can pile into the same state, as in a laser or a Bose-Einstein condensate. This fermion/boson split is dictated by spin through the spin-statistics theorem, making spin one of the most consequential of all quantum numbers.