Spin-½

Spin-½ is an intrinsic angular-momentum state with quantum number one-half. Its states form a two-dimensional representation of SU(2), the double cover of the rotation group.

Pauli matrices: Three matrices that represent spin components for a spin-½ system and obey angular-momentum commutation relations. They encode the three spin components as operators on its two-dimensional state space.
Angular momentum: A physical quantity measuring rotational motion or its quantum-mechanical counterpart. Spin is an intrinsic form of angular momentum, distinct from orbital motion.
Electron: A negatively charged elementary particle with spin one-half. The electron is the standard physical example of a spin-½ particle.
Spin-1: An intrinsic angular-momentum state with quantum number one and a three-dimensional irreducible rotation representation. Unlike spin-½, spin-1 has three spin-component outcomes along any chosen axis.
Spinor: A mathematical object that transforms under a representation of a rotation group's double cover. A spin-½ state transforms as a two-component spinor rather than an ordinary vector.
Quantum state: A mathematical description that encodes the possible outcomes and probabilities of a quantum system. A spin-½ system's pure states occupy a two-dimensional complex vector space.
Nuclear magnetic resonance: A spectroscopy technique that uses transitions between nuclear spin states in a magnetic field. Spin-½ nuclei such as hydrogen produce two-level transitions used in NMR.
Spin-0: An intrinsic angular-momentum state with quantum number zero and no spin-component degrees of freedom. It shows the contrast between nonzero half-integer spin and a rotationally invariant scalar state.
Stern–Gerlach experiment: An experiment that separates particles by projecting their magnetic moments through a nonuniform magnetic field. Its discrete beam splitting reveals two possible outcomes for a spin component.
Rotation group SO(3): The group of proper rotations in three-dimensional Euclidean space. Ordinary spatial rotations act through SO(3), whose representations do not capture spinors directly.

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