Spin (physics)
Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory.
The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The relativistic spin–statistics theorem connects electron spin quantization to the Pauli exclusion principle: observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion.
Spin is described mathematically as a vector for some particles such as photons, and as a spinor or bispinor for other particles such as electrons. Spinors and bispinors behave similarly to vectors: they have definite magnitudes and change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change. These are indicated by assigning the particle a spin quantum number.
The SI units of spin are the same as classical angular momentum. In quantum mechanics, angular momentum and spin angular momentum take discrete values proportional to the Planck constant. In practice, spin is usually given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant. Often, the "spin quantum number" is simply called "spin".
Models
Rotating charged mass
The earliest models for electron spin imagined a rotating charged mass, but this model fails when examined in detail. The required space distribution does not match limits on the electron radius, and the required rotation speed exceeds the speed of light. In the Standard Model, the fundamental particles are all considered "point-like", and they have their effects through the field that surrounds them. Any model for spin based on mass rotation would need to be consistent with that model.Pauli's "classically non-describable two-valuedness"
, a central figure in the history of quantum spin, initially rejected any idea that the "degree of freedom" he introduced to explain experimental observations was related to rotation. He called it "classically non-describable two-valuedness". Later, he allowed that it is related to angular momentum, but insisted on considering spin an abstract property. This approach allowed Pauli to develop a proof of his fundamental Pauli exclusion principle, a proof now called the spin-statistics theorem. In retrospect, this insistence and the style of his proof initiated the modern particle-physics era, where abstract quantum properties derived from symmetry properties dominate. Concrete interpretation became secondary and optional.Circulation of classical fields
The first classical model for spin proposed a small rigid particle rotating about an axis, as ordinary use of the word may suggest. Angular momentum can be computed from a classical field as well. By applying Frederik Belinfante's approach to calculating the angular momentum of a field, 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 of spin can be applied to gravity waves in water: "spin is generated by subwavelength circular motion of water particles".Unlike classical wavefield circulation, which allows continuous values of angular momentum, quantum wavefields allow only discrete values. Consequently, energy transfer to or from spin states always occurs in fixed quantum steps. Only a few steps are allowed: for many qualitative purposes, the complexity of the spin quantum wavefields can be ignored and the system properties can be discussed in terms of "integer" or "half-integer" spin models as discussed in [|quantum numbers] below.
In Bohmian mechanics
Spin can be understood differently depending on the interpretations of quantum mechanics. In the de Broglie–Bohm interpretation, particles have definitive trajectories but their motion is driven by the wave function or pilot wave. In this interpretation, the spin is a property of the pilot wave and not of the particle itself.Dirac's relativistic electron
Relativistic calculations of spin properties for electrons requires the Dirac equation.Relation to orbital angular momentum
As the name suggests, spin was originally conceived as the rotation of a particle around some axis. Historically orbital angular momentum related to particle orbits. While the names based on mechanical models have survived, the physical explanation has not. Quantization fundamentally alters the character of both spin and orbital angular momentum.Since elementary particles are point-like, self-rotation is not well-defined for them. However, spin implies that the phase of the particle depends on the angle as for rotation of angle around the axis parallel to the spin. This is equivalent to the quantum-mechanical interpretation of momentum as phase dependence in the position, and of orbital angular momentum as phase dependence in the angular position.
For fermions, the picture is less clear: From the Ehrenfest theorem, the angular velocity is equal to the derivative of the Hamiltonian to its conjugate momentum, which is the total angular momentum operator Therefore, if the Hamiltonian has any dependence on the spin, then must be non-zero; consequently, for classical mechanics, the existence of spin in the Hamiltonian will produce an actual angular velocity, and hence an actual physical rotation – that is, a change in the phase-angle,, over time. However, whether this holds true for free electrons is ambiguous, since for an electron, ² is a constant and one might decide that since it cannot change, no partial can exist. Therefore it is a matter of interpretation whether the Hamiltonian must include such a term, and whether this aspect of classical mechanics extends into quantum mechanics. Nevertheless, spin appears in the Dirac equation, and thus the relativistic Hamiltonian of the electron, treated as a Dirac field, can be interpreted as including a dependence in the spin.
Quantum number
Spin obeys the mathematical laws of angular momentum quantization. The specific properties of spin angular momenta include:- Spin quantum numbers may take either half-integer or integer values.
- Although the direction of its spin can be changed, the magnitude of the spin of an elementary particle cannot be changed.
- The spin of a charged particle is associated with a magnetic dipole moment with a -factor that differs from 1.
where is the Planck constant, and is the reduced Planck constant. In contrast, orbital angular momentum can only take on integer values of ; i.e., even-numbered values of.
Fermions and bosons
Those particles with half-integer spins, such as,,, are known as fermions, while those particles with integer spins, such as 0, 1, 2, are known as bosons. The two families of particles obey different rules and broadly have different roles in the world around us. A key distinction between the two families is that fermions obey the Pauli exclusion principle: that is, there cannot be two identical fermions simultaneously having the same quantum numbers. Fermions obey the rules of Fermi–Dirac statistics. In contrast, bosons obey the rules of Bose–Einstein statistics and have no such restriction, so they may "bunch together" in identical states. Also, composite particles can have spins different from their component particles. For example, a helium-4 atom in the ground state has spin 0 and behaves like a boson, even though the quarks and electrons which make it up are all fermions.This has some profound consequences:
- Quarks and leptons, which make up what is classically known as matter, are all fermions with spin . The common idea that "matter takes up space" actually comes from the Pauli exclusion principle acting on these particles to prevent the fermions from being in the same quantum state. Further compaction would require electrons to occupy the same energy states, and therefore a kind of pressure acts to resist the fermions being overly close. Elementary fermions with other spins are not known to exist.
- Elementary particles which are thought of as carrying forces are all bosons with spin 1. They include the photon, which carries the electromagnetic force, the gluon, and the W and Z bosons. The ability of bosons to occupy the same quantum state is used in the laser, which aligns many photons having the same quantum number, superfluid liquid helium resulting from helium-4 atoms being bosons, and superconductivity, where pairs of electrons act as single composite bosons. Elementary bosons with other spins were not historically known to exist, although they have received considerable theoretical treatment and are well established within their respective mainstream theories. In particular, theoreticians have proposed the graviton with spin 2, and the Higgs boson with spin 0. Since 2013, the Higgs boson with spin 0 has been considered proven to exist. It is the first scalar elementary particle known to exist in nature.
- Atomic nuclei have nuclear spin which may be either half-integer or integer, so that the nuclei may be either fermions or bosons.
Spin–statistics theorem