spin (physics)

TheInfoList

OR:

Spin is a conserved quantity carried by elementary particles, and thus by composite particles ( hadrons) and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. For photons, spin is the quantum-mechanical counterpart of the
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds * Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion b ...
of light; for electrons, the spin has no classical counterpart. The existence of electron spin angular momentum is
inferred Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
from experiments, such as the
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throu ...
, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The existence of the electron spin can also be inferred theoretically from the
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
and from the Pauli exclusion principle—and vice versa, given the particular spin of the electron, one may derive the Pauli exclusion principle. Spin is described mathematically as a vector for some particles such as photons, and as
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
s and
bispinor In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, specif ...
s 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 In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
. The
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
of spin is the same as classical angular momentum (i.e., N· m· s, J·s, or kg·m2·s−1). In practice, spin is given as a
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
spin quantum number by dividing the spin angular momentum by the reduced Planck constant , which has the same
dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
as angular momentum, although this is not the full computation of this value. Very often, the "spin quantum number" is simply called "spin". The fact that it is a quantum number is implicit.

# History

Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...
in 1924 was the first to propose a doubling of the number of available electron states due to a two-valued non-classical "hidden rotation". In 1925, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the old quantum theory of Bohr and Sommerfeld.
Ralph Kronig Ralph Kronig (10 March 1904 – 16 November 1995) was a German physicist. He is noted for the discovery of particle spin and for his theory of X-ray absorption spectroscopy. His theories include the Kronig–Penney model, the Coster–Kronig t ...
anticipated the Uhlenbeck–Goudsmit model in discussion with Hendrik Kramers several months earlier in Copenhagen, but did not publish. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his
relativistic quantum mechanics In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light '' ...
in 1928, electron spin was an essential part of it.

# Quantum number

As the name suggests, spin was originally conceived as the rotation of a particle around some axis. While the question of whether elementary particles actually rotate is ambiguous (as they appear point-like), this picture is correct insofar as spin obeys the same mathematical laws as quantized angular momenta do; in particular, spin implies that the particle's phase changes with angle. On the other hand, spin has some peculiar properties that distinguish it from orbital angular momenta: * Spin quantum numbers may take half-integer values. * Although the direction of its spin can be changed, an elementary particle cannot be made to spin faster or slower. * The spin of a charged particle is associated with a magnetic dipole moment with a -factor differing from 1. This could occur classically only if the internal charge of the particle were distributed differently from its mass. The conventional definition of the spin quantum number is , where can be any non-negative integer. Hence the allowed values of are 0, , 1, , 2, etc. The value of for an elementary particle depends only on the type of particle and cannot be altered in any known way (in contrast to the ''spin direction'' described below). The spin angular momentum of any physical system is quantized. The allowed values of are $S = \hbar \, \sqrt = \frac \, \sqrt = \frac \, \sqrt,$ where is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, and $\hbar = \frac$ 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 In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
. 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 (meaning, roughly, having the same position, velocity and spin direction). Fermions obey the rules of
Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac ...
. 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 Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consis ...
atom in the ground state has spin 0 and behaves like a boson, even though the
quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All comm ...
and electrons which make it up are all fermions. This has some profound consequences: *
Quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All comm ...
and
leptons In particle physics, a lepton is an elementary particle of half-integer spin (spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutr ...
(including
electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
and
neutrinos A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
), 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 (sometimes known as degeneracy pressure of electrons) acts to resist the fermions being overly close. Elementary fermions with other spins (, , etc.) 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 (
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called th ...
), and the
W and Z bosons In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , ...
(
weak force Weak may refer to: Songs * "Weak" (AJR song), 2016 * "Weak" (Melanie C song), 2011 * "Weak" (SWV song), 1993 * "Weak" (Skunk Anansie song), 1995 * "Weak", a song by Seether from '' Seether: 2002-2013'' Television episodes * "Weak" (''Fear t ...
). The ability of bosons to occupy the same quantum state is used in the laser, which aligns many photons having the same quantum number (the same direction and frequency),
superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two i ...
liquid helium Liquid helium is a physical state of helium at very low temperatures at standard atmospheric pressures. Liquid helium may show superfluidity. At standard pressure, the chemical element helium exists in a liquid form only at the extremely low te ...
resulting from helium-4 atoms being bosons, and
superconductivity Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
, where pairs of electrons (which individually are fermions) act as single composite bosons. Elementary bosons with other spins (0, 2, 3, etc.) 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 (predicted to exist by some quantum gravity theories) with spin 2, and the
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Stand ...
(explaining electroweak symmetry breaking) with spin 0. Since 2013, the Higgs boson with spin 0 has been considered proven to exist. It is the first scalar elementary particle (spin 0) 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

The
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
splits particles into two groups:
bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
and
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
, where bosons obey Bose–Einstein statistics, and fermions obey
Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac ...
(and therefore the Pauli exclusion principle). Specifically, the theory states that particles with an integer spin are bosons, while all other particles have half-integer spins and are fermions. As an example, electrons have half-integer spin and are fermions that obey the Pauli exclusion principle, while photons have integer spin and do not. The theorem relies on both quantum mechanics and the theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The la ...
, and this connection between spin and statistics has been called "one of the most important applications of the special relativity theory".

## Relation to classical rotation

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 $e^$, for rotation of angle ''θ'' around the axis parallel to the spin ''S''. This is equivalent to the quantum-mechanical interpretation of
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
as phase dependence in the position, and of orbital angular momentum as phase dependence in the angular position. Photon spin is the quantum-mechanical description of light
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds * Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion b ...
, where spin +1 and spin −1 represent two opposite directions of
circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
. Thus, light of a defined circular polarization consists of photons with the same spin, either all +1 or all −1. Spin represents polarization for other vector bosons as well. For fermions, the picture is less clear.
Angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
is equal by Ehrenfest theorem to the derivative of the Hamiltonian to its conjugate momentum, which is the total angular momentum operator . Therefore, if the Hamiltonian ''H'' is dependent upon the spin ''S'', ''dH''/''dS'' is non-zero, and the spin causes angular velocity, and hence actual rotation, i.e. a change in the phase-angle relation over time. However, whether this holds for free electron is ambiguous, since for an electron, ''S''2 is constant, and therefore it is a matter of interpretation whether the Hamiltonian includes such a term. Nevertheless, spin appears in the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac pa ...
, and thus the relativistic Hamiltonian of the electron, treated as a
Dirac field In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bo ...
, can be interpreted as including a dependence in the spin ''S''. Under this interpretation, free electrons also self-rotate, with the
Zitterbewegung In physics, the zitterbewegung ("jittery motion" in German, ) is the predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first discussed by Gregory Breit in 1928 and ...
effect understood as this rotation.

# Magnetic moments

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throu ...
, or by measuring the magnetic fields generated by the particles themselves. The intrinsic magnetic moment of a spin- particle with charge , mass , and spin angular momentum , is : $\boldsymbol = \frac \mathbf,$ where the dimensionless quantity is called the spin -factor. For exclusively orbital rotations it would be 1 (assuming that the mass and the charge occupy spheres of equal radius). The electron, being a charged elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron -factor, which has been experimentally determined to have the value , with the digits in parentheses denoting
measurement uncertainty In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a measured quantity. All measurements are subject to uncertainty and a measurement result is complete only when it is accompanied by ...
in the last two digits at one
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
. The value of 2 arises from the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac pa ...
, a fundamental equation connecting the electron's spin with its electromagnetic properties, and the correction of ... arises from the electron's interaction with the surrounding
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
, including its own field. Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of
quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All comm ...
, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.
Neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s are both elementary and electrically neutral. The minimally extended
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
that takes into account non-zero neutrino masses predicts neutrino magnetic moments of: : $\mu_\nu \approx 3 \times 10^ \mu_\text \frac,$ where the are the neutrino magnetic moments, are the neutrino masses, and is the
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\mathr ...
. New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments. It can be shown in a model-independent way that neutrino magnetic moments larger than about 10−14  are "unnatural" because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses are known to be at most about 1 eV, the large radiative corrections would then have to be "fine-tuned" to cancel each other, to a large degree, and leave the neutrino mass small. The measurement of neutrino magnetic moments is an active area of research. Experimental results have put the neutrino magnetic moment at less than  times the electron's magnetic moment. On the other hand elementary particles with spin but without electric charge, such as a photon or a
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , ...
, do not have a magnetic moment.

# Curie temperature and loss of alignment

In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction, with the overall average being very near zero.
Ferromagnet Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic material ...
ic materials below their Curie temperature, however, exhibit
magnetic domain A magnetic domain is a region within a magnetic material in which the magnetization is in a uniform direction. This means that the individual magnetic moments of the atoms are aligned with one another and they point in the same direction. When c ...
s in which the atomic dipole moments spontaneously align locally, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar. In paramagnetic materials, the magnetic dipole moments of individual atoms will partially align with an externally applied magnetic field. In
diamagnetic Diamagnetic materials are repelled by a magnetic field; an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast, paramagnetic and ferromagnetic materials are attracte ...
materials, on the other hand, the magnetic dipole moments of individual atoms align oppositely to any externally applied magnetic field, even if it requires energy to do so. The study of the behavior of such " spin models" is a thriving area of research in condensed matter physics. For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to interesting results in the theory of
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
s.

# Direction

## Spin projection quantum number and multiplicity

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the
axis of rotation Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rot ...
of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum for a spin-''s'' particle measured along any direction can only take on the values : $S_i = \hbar s_i, \quad s_i \in \,$ where is the spin component along the -th axis (either , , or ), is the spin projection quantum number along the -th axis, and is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the  axis: : $S_z = \hbar s_z, \quad s_z \in \,$ where is the spin component along the  axis, is the spin projection quantum number along the  axis. One can see that there are possible values of . The number "" is the multiplicity of the spin system. For example, there are only two possible values for a spin- particle: and . These correspond to
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s in which the spin component is pointing in the +''z'' or −''z'' directions respectively, and are often referred to as "spin up" and "spin down". For a spin- particle, like a delta baryon, the possible values are +, +, −, −.

## Vector

For a given
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
, one could think of a spin vector $\lang S \rang$ whose components are the expectation values of the spin components along each axis, i.e.,
axis of rotation Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rot ...
. It turns out that the spin vector is not very useful in actual quantum-mechanical calculations, because it cannot be measured directly: , and cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a Stern–Gerlach apparatus, the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin- particles, this probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180°—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%. As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum-mechanical spin can exhibit phenomena analogous to classical gyroscopic effects. For example, one can exert a kind of "
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of the ...
" on an electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole moment—see the following section). The result is that the spin vector undergoes
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...
, just like a classical gyroscope. This phenomenon is known as
electron spin resonance Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the sp ...
(ESR). The equivalent behaviour of protons in atomic nuclei is used in
nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near and far field, near field) and respond by producing an electromagn ...
(NMR) spectroscopy and imaging. Mathematically, quantum-mechanical spin states are described by vector-like objects known as
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
s. There are subtle differences between the behavior of spinors and vectors under
coordinate rotation Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have sign ( ...
s. For example, rotating a spin- particle by 360° does not bring it back to the same quantum state, but to the state with the opposite quantum phase; this is detectable, in principle, with
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extr ...
experiments. To return the particle to its exact original state, one needs a 720° rotation. (The Plate trick and
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Aug ...
give non-quantum analogies.) A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180° can bring it back to the same quantum state, and a spin-4 particle should be rotated 90° to bring it back to the same quantum state. The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated 180°, and a spin-0 particle can be imagined as sphere, which looks the same after whatever angle it is turned through.

# Mathematical formulation

## Operator

Spin obeys commutation relations analogous to those of the orbital angular momentum: : where is the Levi-Civita symbol. It follows (as with angular momentum) that the
eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of $\hat S^2$ and $\hat S_z$ (expressed as
kets Kets (russian: Кеты; Ket: Ostygan) are a tribe of Yeniseian speaking people in Siberia. During the Russian Empire, they were known as Ostyaks, without differentiating them from several other Siberian people. Later, they became known as ''Y ...
in the total
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items * Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
) are : $\begin \hat S^2 , s, m_s\rangle &= \hbar^2 s\left(s + 1\right) , s, m_s\rangle, \\ \hat S_z , s, m_s\rangle &= \hbar m_s , s, m_s\rangle. \end$ The spin raising and lowering operators acting on these eigenvectors give : $\hat S_\pm , s, m_s\rangle = \hbar \sqrt , s, m_s \pm 1\rangle,$ where $\hat S_\pm = \hat S_x \pm i \hat S_y$. But unlike orbital angular momentum, the eigenvectors are not
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics for ...
. They are not functions of and . There is also no reason to exclude half-integer values of and . All quantum-mechanical particles possess an intrinsic spin $s$ (though this value may be equal to zero). The projection of the spin $s$ on any axis is quantized in units of the reduced
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, such that the state function of the particle is, say, not $\psi=\psi\left(\vec r\right)$, but $\psi=\psi\left(\vec r,s_z\right)$, where $s_z$ can take only the values of the following discrete set: : $s_z \in \.$ One distinguishes bosons (integer spin) and fermions (half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.

## Pauli matrices

The quantum-mechanical operators associated with spin-
observables In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum p ...
are : $\hat = \frac \boldsymbol,$ where in Cartesian components : $S_x = \frac \sigma_x, \quad S_y = \frac \sigma_y, \quad S_z = \frac \sigma_z.$ For the special case of spin- particles, , and are the three
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when ...
: :$\sigma_x = \begin 0 & 1\\ 1 & 0 \end, \quad \sigma_y = \begin 0 & -i\\ i & 0 \end, \quad \sigma_z = \begin 1 & 0\\ 0 & -1 \end.$

## Pauli exclusion principle

For systems of identical particles this is related to the Pauli exclusion principle, which states that its
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
$\psi\left(\mathbf r_1, \sigma_1, \dots, \mathbf r_N, \sigma_N\right)$ must change upon interchanges of any two of the particles as : $\psi\left(\dots, \mathbf r_i, \sigma_i, \dots, \mathbf r_j, \sigma_j, \dots \right) = \left(-1\right)^ \psi\left(\dots, \mathbf r_j, \sigma_j, \dots, \mathbf r_i, \sigma_i, \dots\right).$ Thus, for bosons the prefactor will reduce to +1, for fermions to −1. In quantum mechanics all particles are either bosons or fermions. In some speculative relativistic quantum field theories " supersymmetric" particles also exist, where linear combinations of bosonic and fermionic components appear. In two dimensions, the prefactor can be replaced by any complex number of magnitude 1 such as in the
anyon In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of excha ...
. The above permutation postulate for -particle state functions has most important consequences in daily life, e.g. the periodic table of the chemical elements.

## Rotations

As described above, quantum mechanics states that
components Circuit Component may refer to: •Are devices that perform functions when they are connected in a circuit.   In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assemb ...
of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum-mechanical description of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis. For instance, for a spin- particle, we would need two numbers , giving amplitudes of finding it with projection of angular momentum equal to and , satisfying the requirement : $, a_, ^2 + , a_, ^2 = 1.$ For a generic particle with spin , we would need such parameters. Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It is clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to phase) to the matrix representing rotation AB. Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices: : $\sum_^j a_m^* b_m = \sum_^j \left\left(\sum_^j U_ a_n\right\right)^* \left\left(\sum_^j U_ b_k\right\right),$ : $\sum_^j \sum_^j U_^* U_ = \delta_.$ Mathematically speaking, these matrices furnish a unitary
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...
of the rotation group SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
. There is one -dimensional irreducible representation of SU(2) for each dimension, though this representation is -dimensional real for odd and -dimensional complex for even (hence of real dimension ). For a rotation by angle in the plane with normal vector $\hat$, : $U = e^,$ where $\boldsymbol = \theta \hat$, and is the vector of spin operators. A generic rotation in 3-dimensional space can be built by compounding operators of this type using
Euler angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> The ...
: : $\mathcal\left(\alpha, \beta, \gamma\right) = e^ e^ e^.$ An irreducible representation of this group of operators is furnished by the
Wigner D-matrix The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex con ...
: :$D^s_\left(\alpha, \beta, \gamma\right) \equiv \langle sm\text{'} , \mathcal\left(\alpha, \beta, \gamma\right) , sm \rangle = e^ d^s_\left(\beta\right)e^,$ where : $d^s_\left(\beta\right) = \langle sm\text{'} , e^ , sm \rangle$ is Wigner's small d-matrix. Note that for and ; i.e., a full rotation about the  axis, the Wigner D-matrix elements become : $D^s_\left(0, 0, 2\pi\right) = d^s_\left(0\right) e^ = \delta_ \left(-1\right)^.$ Recalling that a generic spin state can be written as a superposition of states with definite , we see that if is an integer, the values of are all integers, and this matrix corresponds to the identity operator. However, if is a half-integer, the values of are also all half-integers, giving for all , and hence upon rotation by 2 the state picks up a minus sign. This fact is a crucial element of the proof of the
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
.

## Lorentz transformations

We could try the same approach to determine the behavior of spin under general Lorentz transformations, but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations
SO(3,1) In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
is non-compact and therefore does not have any faithful, unitary, finite-dimensional representations. In case of spin- particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product that is preserved by this representation. We associate a 4-component
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain comb ...
with each particle. These spinors transform under Lorentz transformations according to the law : $\psi\text{'} = \exp \psi,$ where are
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\m ...
, and is an antisymmetric 4 × 4 matrix parametrizing the transformation. It can be shown that the scalar product : $\langle\psi, \phi\rangle = \bar\phi = \psi^\dagger \gamma_0 \phi$ is preserved. It is not, however, positive-definite, so the representation is not unitary.

## Measurement of spin along the , , or axes

Each of the (
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature ...
) Pauli matrices of spin- particles has two
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
, +1 and −1. The corresponding normalized
eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
are : $\begin \psi_ = \left, \frac, \frac\right\rangle_x = \displaystyle\frac \!\!\!\!\! & \begin\\\end, & \psi_ = \left, \frac, \frac\right\rangle_x = \displaystyle\frac \!\!\!\!\! & \begin\\\end, \\ \psi_ = \left, \frac, \frac\right\rangle_y = \displaystyle\frac \!\!\!\!\! & \begin\\\end, & \psi_ = \left, \frac, \frac\right\rangle_y = \displaystyle\frac \!\!\!\!\! & \begin\\\end, \\ \psi_ = \left, \frac, \frac\right\rangle_z = & \begin\\\end, & \psi_ = \left, \frac, \frac\right\rangle_z = & \begin\\\end. \end$ (Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign. In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as SymPy; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.) By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the , , or  axis can only yield an eigenvalue of the corresponding spin operator (, or ) on that axis, i.e. or . The
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of a particle (with respect to spin), can be represented by a two-component
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
: : $\psi = \begin a + bi \\ c + di \end.$ When the spin of this particle is measured with respect to a given axis (in this example, the  axis), the probability that its spin will be measured as is just $\big, \langle \psi_, \psi\rangle\big, ^2$. Correspondingly, the probability that its spin will be measured as is just $\big, \langle\psi_, \psi\rangle\big, ^2$. Following the measurement, the spin state of the particle collapses into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to have a given eigenvalue, all measurements will yield the same eigenvalue (since $\big, \langle\psi_, \psi_\rangle\big, ^2 = 1$, etc.), provided that no measurements of the spin are made along other axes.

## Measurement of spin along an arbitrary axis

The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let be an arbitrary unit vector. Then the operator for spin in this direction is simply : $S_u = \frac\left(u_x \sigma_x + u_y \sigma_y + u_z \sigma_z\right).$ The operator has eigenvalues of , just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three -, -, -axis directions. A normalized spinor for spin- in the direction (which works for all spin states except spin down, where it will give ) is : $\frac \begin 1 + u_z \\ u_x + iu_y \end.$ The above spinor is obtained in the usual way by diagonalizing the matrix and finding the eigenstates corresponding to the eigenvalues. In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity.

## Compatibility of spin measurements

Since the Pauli matrices do not
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the  axis, and we then measure the spin along the  axis, we have invalidated our previous knowledge of the  axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that : $\big, \langle \psi_ , \psi_ \rangle \big, ^2 = \big, \langle \psi_ , \psi_ \rangle \big, ^2 = \big, \langle \psi_ , \psi_ \rangle \big, ^2 = \tfrac.$ So when physicists measure the spin of a particle along the  axis as, for example, , the particle's spin state collapses into the eigenstate $, \psi_\rangle$. When we then subsequently measure the particle's spin along the  axis, the spin state will now collapse into either $, \psi_\rangle$ or $, \psi_\rangle$, each with probability . Let us say, in our example, that we measure . When we now return to measure the particle's spin along the  axis again, the probabilities that we will measure or are each (i.e. they are $\big, \langle \psi_ , \psi_ \rangle \big, ^2$ and $\big, \langle \psi_ , \psi_ \rangle \big, ^2$ respectively). This implies that the original measurement of the spin along the  axis is no longer valid, since the spin along the  axis will now be measured to have either eigenvalue with equal probability.

## Higher spins

The spin- operator forms the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the def ...
of
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
. By taking
Kronecker product In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
s of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher-spin systems in three spatial dimensions can be calculated for arbitrarily large using this spin operator and ladder operators. For example, taking the Kronecker product of two spin- yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 (
triplet state In quantum mechanics, a triplet is a quantum state of a system with a spin of quantum number =1, such that there are three allowed values of the spin component, = −1, 0, and +1. Spin, in the context of quantum mechanics, is not a mechanical r ...
s) and a 1-dimensional spin-0 representation ( singlet state). The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis: \begin 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end, & \left, 1, +1\right\rangle_x &= \frac \begin 1 \\\\ 1 \end, & \left, 1, 0\right\rangle_x &= \frac \begin -1 \\ 0 \\ 1 \end, & \left, 1, -1\right\rangle_x &= \frac \begin 1 \\\\ 1 \end \\ S_y &= \frac \begin 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end, & \left, 1, +1\right\rangle_y &= \frac \begin -1 \\ -i\sqrt \\ 1 \end, & \left, 1, 0\right\rangle_y &= \frac \begin 1 \\ 0 \\ 1 \end, & \left, 1, -1\right\rangle_y &= \frac \begin -1 \\ i\sqrt \\ 1 \end \\ S_z &= \hbar \begin 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end, & \left, 1, +1\right\rangle_z &= \begin 1 \\ 0 \\ 0 \end, & \left, 1, 0\right\rangle_z &= \begin 0 \\ 1 \\ 0 \end, & \left, 1, -1\right\rangle_z &= \begin 0 \\ 0 \\ 1 \end \\ \end , For spin they are $\begin S_x = \frac\hbar2 \begin 0 &\sqrt &0 &0\\ \sqrt &0 &2 &0\\ 0 &2 &0 &\sqrt\\ 0 &0 &\sqrt &0 \end, \!\!\! & \left, \frac, \frac\right\rangle_x =\!\!\! & \frac \begin 1 \\\\\\ 1 \end, \!\!\! & \left, \frac, \frac\right\rangle_x =\!\!\! & \frac \begin\\ -1 \\ 1 \\\end, \!\!\! & \left, \frac, \frac\right\rangle_x =\!\!\! & \frac \begin\\ -1 \\ -1 \\\end, \!\!\! & \left, \frac, \frac\right\rangle_x =\!\!\! & \frac \begin -1 \\\\\\ 1 \end \\ S_y = \frac\hbar2 \begin 0 &-i\sqrt &0 &0\\ i\sqrt &0 &-2i &0\\ 0 &2i &0 &-i\sqrt\\ 0 &0 &i\sqrt &0 \end, \!\!\! & \left, \frac, \frac\right\rangle_y =\!\!\! & \frac \begin\\\\\\ 1 \end, \!\!\! & \left, \frac, \frac\right\rangle_y =\!\!\! & \frac \begin\\ 1 \\\\\end, \!\!\! & \left, \frac, \frac\right\rangle_y =\!\!\! & \frac \begin\\ 1 \\\\\end, \!\!\! & \left, \frac, \frac\right\rangle_y =\!\!\! & \frac \begin\\\\\\ 1 \end \\ S_z = \frac\hbar2 \begin 3 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &-1 &0\\ 0 &0 &0 &-3 \end, \!\!\! & \left, \frac, \frac\right\rangle_z =\!\!\! & \begin 1 \\ 0 \\ 0 \\ 0 \end, \!\!\! & \left, \frac, \frac\right\rangle_z =\!\!\! & \begin 0 \\ 1 \\ 0 \\ 0 \end, \!\!\! & \left, \frac, \frac\right\rangle_z =\!\!\! & \begin 0 \\ 0 \\ 1 \\ 0 \end, \!\!\! & \left, \frac, \frac\right\rangle_z =\!\!\! & \begin 0 \\ 0 \\ 0 \\ 1 \end \\ \end$ , For spin they are $\begin \boldsymbol_x &= \frac \begin 0 &\sqrt &0 &0 &0 &0 \\ \sqrt &0 &2\sqrt &0 &0 &0 \\ 0 &2\sqrt &0 &3 &0 &0 \\ 0 &0 &3 &0 &2\sqrt &0 \\ 0 &0 &0 &2\sqrt &0 &\sqrt \\ 0 &0 &0 &0 &\sqrt &0 \end, \\ \boldsymbol_y &= \frac \begin 0 &-i\sqrt &0 &0 &0 &0 \\ i\sqrt &0 &-2i\sqrt &0 &0 &0 \\ 0 &2i\sqrt &0 &-3i &0 &0 \\ 0 &0 &3i &0 &-2i\sqrt &0 \\ 0 &0 &0 &2i\sqrt &0 &-i\sqrt \\ 0 &0 &0 &0 &i\sqrt &0 \end, \\ \boldsymbol_z &= \frac \begin 5 &0 &0 &0 &0 &0 \\ 0 &3 &0 &0 &0 &0 \\ 0 &0 &1 &0 &0 &0 \\ 0 &0 &0 &-1 &0 &0 \\ 0 &0 &0 &0 &-3 &0 \\ 0 &0 &0 &0 &0 &-5 \end. \end$ , The generalization of these matrices for arbitrary spin is $\begin \left(S_x\right)_ & = \frac \left(\delta_ + \delta_\right) \sqrt, \\ \left(S_y\right)_ & = \frac \left(\delta_ - \delta_\right) \sqrt, \\ \left(S_z\right)_ & = \hbar (s + 1 - a) \delta_ = \hbar (s + 1 - b) \delta_, \end$ where indices $a, b$ are integer numbers such that $1 \le a \le 2s + 1, \quad 1 \le b \le 2s + 1.$ Also useful in the
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
of multiparticle systems, the general Pauli group is defined to consist of all -fold
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
products of Pauli matrices. The analog formula of Euler's formula in terms of the Pauli matrices : $\hat\left(\theta, \hat\right) = e^ = I \cos \frac + i \left\left(\hat \cdot \boldsymbol\right\right) \sin \frac$ for higher spins is tractable, but less simple.

# Parity

In tables of the
spin quantum number In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
for nuclei or particles, the spin is often followed by a "+" or "−". This refers to the parity with "+" for even parity (wave function unchanged by spatial inversion) and "−" for odd parity (wave function negated by spatial inversion). For example, see the isotopes of bismuth, in which the list of isotopes includes the column nuclear spin and parity. For Bi-209, the only stable isotope, the entry 9/2– means that the nuclear spin is 9/2 and the parity is odd.

# Applications

Spin has important theoretical implications and practical applications. Well-established ''direct'' applications of spin include: *
Nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near and far field, near field) and respond by producing an electromagn ...
(NMR) spectroscopy in chemistry; *
Electron spin resonance Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the sp ...
(ESR or EPR) spectroscopy in chemistry and physics; * Magnetic resonance imaging (MRI) in medicine, a type of applied NMR, which relies on proton spin density; * Giant magnetoresistive (GMR) drive-head technology in modern hard disks. Electron spin plays an important role in magnetism, with applications for instance in computer memories. The manipulation of ''nuclear spin'' by radio-frequency waves (
nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near and far field, near field) and respond by producing an electromagn ...
) is important in chemical spectroscopy and medical imaging. Spin–orbit coupling leads to the
fine structure In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom b ...
of atomic spectra, which is used in
atomic clock An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
s and in the modern definition of the
second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ea ...
. Precise measurements of the -factor of the electron have played an important role in the development and verification of quantum electrodynamics. ''Photon spin'' is associated with the
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds * Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion b ...
of light ( photon polarization). An emerging application of spin is as a binary information carrier in spin transistors. The original concept, proposed in 1990, is known as Datta–Das spin transistor. Electronics based on spin transistors are referred to as
spintronics Spintronics (a portmanteau meaning spin transport electronics), also known as spin electronics, is the study of the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental electronic charge, in solid-sta ...
. The manipulation of spin in dilute magnetic semiconductor materials, such as metal-doped ZnO or TiO2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics. There are many ''indirect'' applications and manifestations of spin and the associated Pauli exclusion principle, starting with the periodic table of chemistry.

# History

Spin was first discovered in the context of the emission spectrum of
alkali metal The alkali metals consist of the chemical elements lithium (Li), sodium (Na), potassium (K),The symbols Na and K for sodium and potassium are derived from their Latin names, ''natrium'' and ''kalium''; these are still the origins of the nam ...
s. In 1924,
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...
introduced what he called a "two-valuedness not describable classically" associated with the electron in the outermost
shell Shell may refer to: Architecture and design * Shell (structure), a thin structure ** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses ** Thin-shell structure Science Biology * Seashell, a hard ...
. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can have the same
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
in the same quantum system. The physical interpretation of Pauli's "degree of freedom" was initially unknown.
Ralph Kronig Ralph Kronig (10 March 1904 – 16 November 1995) was a German physicist. He is noted for the discovery of particle spin and for his theory of X-ray absorption spectroscopy. His theories include the Kronig–Penney model, the Coster–Kronig t ...
, one of Landé's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...
in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena ...
. Largely due to Pauli's criticism, Kronig decided not to publish his idea. In the autumn of 1925, the same thought came to Dutch physicists George Uhlenbeck and Samuel Goudsmit at Leiden University. Under the advice of
Paul Ehrenfest Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition ...
, they published their results. It met a favorable response, especially after Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished results). This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position. Mathematically speaking, a fiber bundle description is needed. The
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...
effect is additive and relativistic; that is, it vanishes if goes to infinity. It is one half of the value obtained without regard for the tangent-space orientation, but with opposite sign. Thus the combined effect differs from the latter by a factor two (
Thomas precession In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a par ...
, known to
Ludwik Silberstein Ludwik Silberstein (1872 – 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmillan in 1914 with a se ...
in 1914). Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
invented by Schrödinger and
Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
. He pioneered the use of
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when ...
as a
representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
of the spin operators and introduced a two-component
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
wave-function. Uhlenbeck and Goudsmit treated spin as arising from classical rotation, while Pauli emphasized, that spin is non-classical and intrinsic property. Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac published the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac pa ...
, which described the relativistic electron. In the Dirac equation, a four-component spinor (known as a "
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain comb ...
") was used for the electron wave-function. Relativistic spin explained gyromagnetic anomaly, which was (in retrospect) first observed by Samuel Jackson Barnett in 1914 (see Einstein–de Haas effect). In 1940, Pauli proved the ''
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
'', which states that fermions have half-integer spin, and bosons have integer spin. In retrospect, the first direct experimental evidence of the electron spin was the
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throu ...
of 1922. However, the correct explanation of this experiment was only given in 1927.

* Chirality (physics) * Dynamic nuclear polarisation * Helicity (particle physics) *
Holstein–Primakoff transformation The Holstein–Primakoff transformation in quantum mechanics is a mapping to the spin operators from boson creation and annihilation operators, effectively truncating their infinite-dimensional Fock space to finite-dimensional subspaces. One ...
* Kramers theorem *
Pauli equation In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic fiel ...
* Pauli–Lubanski pseudovector * Rarita–Schwinger equation *
Representation theory of SU(2) In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-ab ...
*
Spin angular momentum of light The spin angular momentum of light (SAM) is the component of angular momentum of light that is associated with the quantum spin and the rotation between the polarization degrees of freedom of the photon. Introduction Spin is the fundamental ...
* Spin engineering * Spin-flip * Spin isomers of hydrogen *
Spin–orbit interaction In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–or ...
*
Spin tensor In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The tensor has application in general relativity and special relativity, as well as qu ...
*
Spin wave A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known a ...
* Yrast

# References

* * * * * * * * *Sin-Itiro Tomonaga, The Story of Spin, 1997