Spin (physics)

Spin is a fundamental type of angular momentum that elementary particles possess inherently, and it is also carried by composite particles like hadrons, atomic nuclei, and atoms. This property is quantized, and properly describing how spin interacts requires the frameworks of relativistic quantum mechanics or quantum field theory. Evidence for electron spin came from experiments such as the Stern–Gerlach experiment, where silver atoms showed two distinct, discrete angular momentum states even though they had no orbital angular momentum.

The relativistic spin–statistics theorem links the quantization of electron spin to the Pauli exclusion principle: if a particle obeys exclusion, it must have half-integer spin, and if it has half-integer spin, it must obey exclusion. Mathematically, spin is represented as a vector for some particles (like photons) and as a spinor for others (like electrons). Spinors share some traits with vectors—they have fixed magnitudes and change under rotations—but they behave in an unconventional "direction." All elementary particles of the same type share the same magnitude of spin angular momentum, though its direction can vary; this is captured by assigning each particle a spin quantum number.

In SI units, spin has the same dimensions as classical angular momentum (N·m·s, J·s, or kg·m²·s⁻¹). In quantum mechanics, both angular momentum and spin angular momentum take on discrete values that are multiples of the Planck constant. In practice, spin is often expressed as a dimensionless spin quantum number, obtained by dividing the spin angular momentum by the reduced Planck constant ħ. Frequently, the term "spin" is used to refer to this quantum number itself.

**Models**

**Rotating charged mass** Early models imagined the electron as a rotating charged mass, but this idea fails under scrutiny. The required spatial distribution would exceed the electron's radius limits, and the rotation speed would surpass the speed of light. In the Standard Model, fundamental particles are considered point-like, exerting effects through their surrounding fields. Any spin model based on mass rotation would need to be consistent with that picture.

**Pauli's "classically non-describable two-valuedness"** Wolfgang Pauli, a key figure in spin's history, initially rejected the notion that the "degree of freedom" he introduced to explain experiments had anything to do with rotation. He called it a "classically non-describable two-valuedness." Later, he acknowledged its connection to angular momentum but insisted on treating spin as an abstract property. This approach enabled Pauli to prove the Pauli exclusion principle, now known as the spin–statistics theorem. In retrospect, his insistence and proof style launched the modern particle-physics era, where abstract quantum properties derived from symmetries dominate, and concrete interpretations became secondary.

**Circulation of classical fields** The first classical model for spin involved a small rigid particle rotating about an axis, as the everyday word suggests. However, angular momentum can also be computed from a classical field. Using Frederik Belinfante's method for calculating field angular momentum, Hans C. Ohanian showed that "spin is essentially a wave property ... generated by a circulating flow of charge in the wave field of the electron." This same concept applies to gravity waves in water: "spin is generated by subwavelength circular motion of water particles." Unlike classical field circulation, which allows continuous angular momentum values, quantum wavefields permit only discrete values. Thus, energy transfer to or from spin states always occurs in fixed quantum steps, and only a few steps are allowed. For many qualitative purposes, the complexity of spin quantum wavefields can be ignored, and systems are discussed in terms of "integer" or "half-integer" spin models.

**In Bohmian mechanics** Spin can be understood differently depending on the interpretation of quantum mechanics. In the de Broglie–Bohm interpretation, particles have definite trajectories, but their motion is driven by the wave function or pilot wave. Here, spin is a property of the pilot wave, not of the particle itself.

**Dirac's relativistic electron** Relativistic calculations of spin properties for electrons require the Dirac equation.

**Relation to orbital angular momentum** The name "spin" originally came from the idea of a particle rotating around an axis, while orbital angular momentum referred to particle orbits. Although these mechanical-model names persist, the physical explanation has changed. Quantization fundamentally alters the nature of both spin and orbital angular momentum. Since elementary particles are point-like, self-rotation is not well-defined. However, spin implies that a particle's phase depends on the rotation angle θ around the axis parallel to its spin S, as e^(iSθ). This mirrors the quantum-mechanical interpretation of momentum as phase dependence in position, and orbital angular momentum as phase dependence in angular position. For fermions, the picture is less clear: from the Ehrenfest theorem, angular velocity equals the derivative of the Hamiltonian with respect to its conjugate momentum, which is the total angular momentum operator J = L + S. If the Hamiltonian H depends on spin S, then ∂H/∂S must be nonzero; consequently, in classical mechanics, the presence of spin in the Hamiltonian produces an actual angular velocity and thus a real physical rotation.