What set of 2x2 matrices represents spin-1/2 observables?

Pauli matrices are the set of 2×2 matrices used to represent spin-1/2 observables in quantum mechanics.

The three standard matrices are conventionally called σx, σy, and σz. They are Hermitian, so they can represent measurable quantities, and they describe spin projections along the three spatial axes. The physical spin operators are Sᵢ = ℏσᵢ/2, giving measurement outcomes of +ℏ/2 or −ℏ/2.

Together with the 2×2 identity matrix, the Pauli matrices form a basis for all 2×2 Hermitian matrices. That makes them useful far beyond basic spin measurements: they describe qubit states, polarization, magnetic resonance, and many calculations involving two-level quantum systems.

They are named after Wolfgang Pauli, who introduced them in connection with his theory of electron spin. They should not be confused with Dirac matrices, which are larger matrices used in the relativistic Dirac equation. Gell-Mann matrices similarly generalize the idea to three-dimensional internal spaces associated with SU(3).