Spin Angular Momentum

Spin angular momentum is the intrinsic angular momentum possessed by particles and quantum systems, distinct from angular momentum associated with motion through space. In quantum mechanics, spin is characterized by a quantum number and operators describing measurable components. An electron, for example, has spin quantum number $s=\tfrac12$, regardless of its orbital motion. Although its name suggests rotation, spin is not the literal turning of a miniature solid object; it has no complete counterpart in classical mechanics. (ocw.mit.edu)

Quantum numbers and measurement

For a system with definite spin quantum number $s$, the squared spin angular momentum and its component along a chosen axis satisfy

Here $\hbar=h/(2\pi)$, where $h$ is the Planck constant. The projection quantum number takes the values $m_s=-s,-s+1,\ldots,s$. Thus a spin-$s$ system has $2s+1$ possible projections along an axis. The quantum angular-momentum magnitude is $\hbar\sqrt{s(s+1)}$, not simply $s\hbar$. For spin one-half, the magnitude is $\sqrt3\,\hbar/2$, while a component measurement yields only $+\hbar/2$ or $-\hbar/2$. (ocw.mit.edu)

These two outcomes are conventionally called “spin up” and “spin down.” They are defined relative to the measurement axis, not an absolute direction in space. A state definite along one axis can be a quantum superposition of the two states defined along another axis. Consequently, preparing spin up along $z$ does not make measurements along every other axis predictable. (ocw.mit.edu)

Operators and rotations

The spin components obey the angular-momentum commutation relations,

with corresponding cyclic relations. Because different components do not commute, they cannot generally have simultaneously sharp values. However, $\hat S^2$ commutes with each component, allowing states labeled by total spin and one projection. For spin one-half,

where the $\sigma_i$ are the three Pauli matrices, acting on a two-dimensional Hilbert space. (ocw.mit.edu)

A general normalized spin-one-half state can be written

The Born rule assigns probabilities $|\alpha|^2$ and $|\beta|^2$ to the two outcomes along the chosen axis. This two-state structure also supplies a natural realization of a qubit. (ocw.mit.edu)

Spin operators generate rotations of these internal quantum states. Under a full $2\pi$ rotation, a half-integer-spin state vector acquires a minus sign; a $4\pi$ rotation restores the original vector. An overall sign alone does not alter measurement probabilities, so this is not a claim that an isolated particle becomes visibly different after one revolution. (ocw.mit.edu)

Experimental development

The Stern–Gerlach experiment of 1922 passed a beam of silver atoms through a nonuniform magnetic field. The beam separated into two components rather than showing the continuous spread expected from classically distributed magnetic-moment orientations. Initially interpreted through older atomic models, the splitting is now understood primarily through the spin magnetic moment of silver’s unpaired electron. The experiment preceded the explicit electron-spin hypothesis. (arxiv.org)

George Uhlenbeck and Samuel Goudsmit proposed electron spin in 1925. It helped explain features of atomic spectroscopy, including spectral splitting in magnetic fields. Spin subsequently became part of the quantum description of electrons rather than remaining a mechanical model of a rotating charge distribution. (doi.org)

Magnetic interactions and angular-momentum coupling

Electron spin carries a magnetic dipole moment, approximately described by

where $e$ is the positive elementary charge and $g_e$ is close to two. The negative sign makes the electron’s magnetic moment opposite to its spin. Its interaction energy in a field is $-\boldsymbol{\mu}_s\cdot\mathbf B$, producing spin-dependent level splitting. Precision measurements reveal an anomalous magnetic moment beyond the simplest relativistic prediction. (arxiv.org)

Spin combines with orbital angular momentum according to quantum addition rules. For one particle, $\mathbf J=\mathbf L+\mathbf S$, with allowed total quantum numbers $j=|l-s|,\ldots,l+s$. Spin–orbit coupling links these degrees of freedom and contributes to atomic fine structure. Two spin-one-half systems can combine into a spin-zero singlet or a spin-one triplet; total angular momentum therefore depends on the combined state, not merely on adding numerical spin magnitudes. (ocw.mit.edu)

Particle classification and applications

In ordinary relativistic quantum field theory, integer-spin particles are bosons, whereas half-integer-spin particles are fermions. This spin–statistics connection underlies their different many-particle behavior. Identical fermions obey the Pauli exclusion principle. Electrons and quarks have spin one-half; photons have spin one; the Higgs boson has spin zero. (atlas-public.web.cern.ch)

Spin-dependent energy levels enable electron paramagnetic resonance, in which an oscillating field drives transitions between electron-spin states. Resonance methods investigate material properties and control spin-based qubits. Nuclear spin likewise underlies nuclear magnetic resonance and magnetic resonance imaging. In spintronics, electronic devices exploit spin and magnetization alongside charge, including the use of spin torque to switch magnetic memory elements. (nist.gov)