Spin (physics)

Encyclopedia

In quantum mechanics

and particle physics

,

s, composite particles (hadron

s), and atomic nuclei

.It is worth noting that the intrinsic property of subatomic particles called

All elementary particles of a given kind have the same

, the spin of electrons results in the Pauli exclusion principle

, which in turn underlies the periodic table

of chemical element

s. The

.

Wolfgang Pauli

was the first to propose the concept of spin, but he did not name it. In 1925, Ralph Kronig

, George Uhlenbeck, and Samuel Goudsmit suggested a physical interpretation of particles spinning around their own axis. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac

derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it.

Spin is a type of angular momentum

, where angular momentum is defined in the modern way as the "generator of rotations" (see Noether's theorem

). This modern definition of angular momentum is

definition,

".)

Since spin is a type of angular momentum, it has the same dimensions

: J

·s

in SI units. In practice, however, SI units are almost never used to describe spin: instead, it is written as a multiple of the reduced Planck constant

, the

angular momenta

do. On the other hand, spins have some peculiar properties that distinguish them from orbital angular momenta:

(see classical electron radius

); as far as can be determined, these elementary particles have no inner structure. The spin of an elementary particle is a truly intrinsic physical property, akin to the particle's electric charge

and rest mass.

It turns out that a convenient definition of the

. Hence the allowed values of

depends only on the type of particle, and cannot be altered in any known way (in contrast to the

. The allowed values of

where

All known matter

is ultimately composed of elementary particles called fermion

s, and all elementary fermions have

and positron

, the quark

s making up proton

s and neutron

s, and the neutrino

s. Elementary particles emit and receive one or more particles called boson

s. This boson exchange gives rise to the three fundamental interaction

s ("forces") of the Standard model of particle physics; hence bosons are also called

. Electromagnetism

is the force that results when charged particles exchange photons.

Theory predicts the existence of two bosons whose

; theory suggests that it has

predicts that elementary particles acquire nonzero rest mass by exchanging hypothetical Higgs boson

s with an all-pervasive Higgs field. Theory predicts that the Higgs boson has

s, neutron

s, and atomic nuclei

is usually understood to mean the total angular momentum. This is the sum of the spins and orbital angular momenta of the constituent particles. Such a composite spin is subject to the same quantization condition as any other angular momentum.

Composite particles are often referred to as having a definite spin, just like elementary particles; for example, the proton is a spin-1/2 particle. This is understood to refer to the spin of the lowest-energy internal state of the composite particle (i.e., a given spin and orbital configuration of the constituents).

It is not always easy to deduce the spin of a composite particle from first principles. For example, even though we know that the proton is a spin-1/2 particle, the question of how this spin is distributed

among the three internal valence quarks and the surrounding sea quarks and gluon

s is an active area of research.

Delta baryon

s, which decay into protons and neutrons, have spin 3/2. All the three quarks inside a Delta baryon

(Δ) have their spin axis pointing in the same direction, unlike the nearly identical proton and neutron (called "nucleons") in which the intrinsic spin of one of the three constituent quarks is always opposite the spin of the other two. This difference in spin alignment is the only quantum number distinction between the Δ

.

. Particles with half-integer spin obey Fermi-Dirac statistics

, and are known as fermion

s. They are required to occupy antisymmetric quantum states (see the article on identical particles

.) This property forbids fermions from sharing quantum states – a restriction known as the Pauli exclusion principle

. Particles with integer spin, on the other hand, obey Bose-Einstein statistics, and are known as boson

s. These particles occupy "symmetric states", and can therefore share quantum states. The proof of this is known as the spin-statistics theorem

, which relies on both quantum mechanics and the theory of special relativity

. In fact, "the connection between spin and statistics is one of the most important applications of the special relativity theory".

body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic field

s in a Stern–Gerlach experiment

, or by measuring the magnetic fields generated by the particles themselves.

The intrinsic magnetic moment

where the dimensionless quantity

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

, which has been experimentally determined to have the value , with the digits in parentheses denoting measurement uncertainty

in the last two digits at one standard deviation

. The value of 2 arises from the Dirac equation

, 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

, 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, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.

Neutrino

s are both elementary and electrically neutral. The minimally extended Standard Model

that takes into account non-zero neutrino masses predicts neutrino magnetic moments of:

where the

The measurement of neutrino magnetic moments is an active area of research. , the latest experimental results have put the neutrino magnetic moment at less than times the electron's magnetic moment.

In ordinary material

s, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, 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 spontaneously align with an externally applied magnetic field. In diamagnetic materials, on the other hand, the magnetic dipole moments of individual atoms spontaneously align oppositely to any externally applied magnetic field, even if it requires energy to do so.

The study of the behavior of such "spin model

s" 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

s.

where

, the possible values are +3/2, +1/2, −1/2, −3/2.

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-1/2 particles, this maximum probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180 degrees—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

" 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

, just like a classical gyroscope. This phenomenon is used in nuclear magnetic resonance

sensing.

Mathematically, quantum mechanical spin is not described by vectors as in classical angular momentum, but by objects known as spinor

s. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin-1/2 particle by 360 degrees 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 experiments. To return the particle to its exact original state, one needs a 720 degree rotation. A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180 degrees can bring it back to the same quantum state and a spin-4 particle should be rotated 90 degrees 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 degrees and a spin 0 particle can be imagined as sphere which looks the same after whatever angle it is turned through.

where is the Levi-Civita symbol

. It follows (as with angular momentum

) that the eigenvectors of

s in the total

) are:

The spin raising and lowering operators

acting on these eigenvectors give:

, where

But unlike orbital angular momentum the eigenvectors are not spherical harmonics

. They are not functions of

In addition to their other properties, all quantum mechanical particles possess an intrinsic spin (though it may have the intrinsic spin 0, too). The spin is quantized in units of the reduced Planck constant, such that the state function of the particle is, e.g., not , but where is out of the following discrete set of values:

.

One distinguishes boson

s (integer spin) and fermion

s (half-integer spin). The total angular momentum conserved in interaction processes is then the

operators associated with spin observables are:

In the special case of spin-1/2 , and are the three Pauli matrices

, given by:

, which states that by interchanges of any two of the N particles one must have

Thus, for bosons the prefactor (−1)

" particles exist, where linear combinations of bosonic and fermionic components appear. In two dimensions, the prefactor (−1)

).

Electrons are fermions with

s") are bosons with

The above permutation postulate for

of the chemists or biologists.

For a generic particle with spin s, we would need 2s+1 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's 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:

Mathematically speaking, these matrices furnish a unitary projective

representation

of the rotation group

SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2). There is one n-dimensional irreducible representation of SU(2) for each dimension, though this representation is n-dimensional real for odd n and n-dimensional complex for even n (hence of real dimension 2

where 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

:

An irreducible representation of this group of operators is furnished by the Wigner D-matrix

:

where

is Wigner's small d-matrix. Note that for γ = 2π and α = β = 0, i.e. a full rotation about the z-axis, the Wigner D-matrix elements become

Recalling that a generic spin state can be written as a superposition of states with definite

.

s, but we'd immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact

and therefore does not have any faithful unitary finite-dimensional representations.

In case of spin 1/2 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 with each particle. These spinors transform under Lorentz transformations according to the law

where are gamma matrices and is an antisymmetric 4x4 matrix parametrizing the transformation. It can be shown that the scalar product

is preserved. (It is not, however, positive definite, so the representation is not unitary.)

By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the x, y or z axis can only yield an eigenvalue of the corresponding spin operator ( or ) on that axis, i.e. or The quantum state of a particle (with respect to spin), can be represented by a two component spinor

:

When the spin of this particle is measured with respect to a given axis (in this example, the x-axis), the probability that its spin will be measured as is just . Correspondingly, the probability that its spin will be measured as is just . Following the measurement, the spin state of the particle will collapse

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 , etc), provided that no measurements of the spin are made along other axes (see compatibility section below).

A normalized spinor for spin-1/2 in the direction (which works for all spin states except spin down where it will give 0/0), is:

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.

, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the x-axis, and we then measure the spin along the y-axis, we have invalidated our previous knowledge of the x-axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that:

So when we measure the spin of a particle along the x-axis as, for example, , the particle's spin state collapses

into the eigenstate . When we then subsequently measure the particle's spin along the y-axis, the spin state will now collapse into either or , each with probability . Let us say, in our example, that we measure . When we now return to measure the particle's spin along the x-axis again, the probabilities that we will measure or are each (i.e. they are and ). This implies that our original measurement of the spin along the x-axis is no longer valid, since the spin along the x-axis will now be measured to have either eigenvalue with equal probability.

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

.

Electron spin plays an important role in magnetism

, with applications for instance in computer memories. The manipulation of

) is important in chemical spectroscopy and medical imaging.

Spin-orbit coupling leads to the fine structure

of atomic spectra, which is used in atomic clock

s and in the modern definition of the second

. Precise measurements of the g-factor of the electron have played an important role in the development and verification of quantum electrodynamics

.

A possible future direct application of spin is as a binary information carrier in spin transistor

s. Original concept proposed in 1990 is known as Datta-Das spin transistor. Electronics based on spin transistors is called spintronics

, which includes the manipulation of spins in semiconductor devices.

There are manyThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

, starting with the periodic table

of chemistry.

of alkali metal

s. In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell

. This allowed him to formulate the Pauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

, stating that no two electrons can share the same quantum state at the same time.

The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig

, 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

in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity

. Largely due to Pauli's criticism, Kronig decided not to publish his idea.

In the autumn of 1925, the same thought came to two Dutch physicists, George Uhlenbeck and Samuel Goudsmit. Under the advice of Paul Ehrenfest

, 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 ones). 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

effect is additive and relativistic; that is, it vanishes if

).

Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics

discovered by Schrödinger

and Heisenberg

. He pioneered the use of Pauli matrices

as a representation

of the spin operators, and introduced a two-component spinor

wave-function.

Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac

published the Dirac equation

, which described the relativistic electron

. In the Dirac equation, a four-component spinor (known as a "Dirac spinor") was used for the electron wave-function. In 1940, Pauli proved theIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s have half-integer spin and bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s integer spin.

In retrospect, the first direct experimental evidence of the electron spin was the Stern-Gerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1927.

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

and particle physics

Particle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

,

**spin**is a fundamental characteristic property of elementary particleElementary particle

In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

s, composite particles (hadron

Hadron

In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

s), and atomic nuclei

Atomic nucleus

The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

.It is worth noting that the intrinsic property of subatomic particles called

*spin*and discussed in this article, is related in some small ways, but is very different from the everyday concept of spin (for example, as used when describing a spinning ball).*Spin*, as used by particle physicists in the quantum world, is a property of subatomic particles, which has certain qualities and obeys certain rules.All elementary particles of a given kind have the same

**spin quantum number**

, an important part of the quantum state of a particle. When combined with the spin-statistics theoremQuantum number

Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

Spin-statistics theorem

In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...

, the spin of electrons results in the Pauli exclusion principle

Pauli exclusion principle

The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

, which in turn underlies the periodic table

Periodic table

The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

of chemical element

Chemical element

A chemical element is a pure chemical substance consisting of one type of atom distinguished by its atomic number, which is the number of protons in its nucleus. Familiar examples of elements include carbon, oxygen, aluminum, iron, copper, gold, mercury, and lead.As of November 2011, 118 elements...

s. The

**spin direction**(also called*spin*for short) of a particle is an important intrinsic degree of freedomDegrees of freedom (physics and chemistry)

A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

.

Wolfgang Pauli

Wolfgang Pauli

Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...

was the first to propose the concept of spin, but he did not name it. In 1925, Ralph Kronig

Ralph Kronig

Ralph Kronig was a German-American physicist . He is noted for the discovery of particle spin and for his theory of x-ray absorption spectroscopy...

, George Uhlenbeck, and Samuel Goudsmit suggested a physical interpretation of particles spinning around their own axis. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac

Paul Dirac

Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it.

Spin is a type of angular momentum

Angular momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

, where angular momentum is defined in the modern way as the "generator of rotations" (see Noether's theorem

Noether's theorem

Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...

). This modern definition of angular momentum is

*not*the same as the historical classical mechanicsClassical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

definition,

**L**=**r**×**p**. (The historical definition, which does not include spin, is more specifically called "orbital angular momentumAngular momentum operator

In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry...

".)

Since spin is a type of angular momentum, it has the same dimensions

Dimensions

Dimensions is a French project that makes educational movies about mathematics, focusing on spatial geometry. It uses POV-Ray to render some of the animations, and the films are release under a Creative Commons licence....

: J

Joule

The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

·s

Second

The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

in SI units. In practice, however, SI units are almost never used to describe spin: instead, it is written as a multiple of the reduced Planck constant

*ħ*. In natural unitsNatural units

In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

, the

*ħ*is omitted, so the units of spin are implied. However, by definition the "spin quantum number" is always dimensionless.## Spin quantum number

As the name suggests, spin was originally conceived as the rotation of a particle around some axis. This picture is correct so far as spins obey the same mathematical laws as quantizedQuantization (physics)

In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...

angular momenta

Angular momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

do. On the other hand, spins have some peculiar properties that distinguish them from orbital angular momenta:

- Spin quantum numbers may take on half-odd-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 g-factor differing from 1. This could only occur classically if the internal charge of the particle were distributed differently from its mass.

### Elementary particles

Elementary particles are particles for which there is no known method of division into smaller units. Theoretical and experimental studies have shown that the spin possessed by such particles cannot be explained by postulating that they are made up of even smaller particles rotating about a common center of massCenter of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

(see classical electron radius

Classical electron radius

The classical electron radius, also known as the Lorentz radius or the Thomson scattering length, is based on a classical relativistic model of the electron...

); as far as can be determined, these elementary particles have no inner structure. The spin of an elementary particle is a truly intrinsic physical property, akin to the particle's electric charge

Electric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

and rest mass.

It turns out that a convenient definition of the

**spin quantum number***s*is*s*=*n*/2, where*n*can be any non-negative integerInteger

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

. Hence the allowed values of

*s*are 0, 1/2, 1, 3/2, 2, etc. The value of*s*for an elementary particleElementary particle

In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

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*S*of any physical system is quantisedQuantization (physics)

In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...

. The allowed values of

*S*are:where

*ħ*is the reduced Planck's constant. In contrast, orbital angular momentum can only take on integer quantum numbers.All known matter

Matter

Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

is ultimately composed of elementary particles called fermion

Fermion

In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s, and all elementary fermions have

*s*= 1/2. Examples of fermions are the electronElectron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

and positron

Positron

The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...

, the quark

Quark

A quark is an 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. Due to a phenomenon known as color confinement, quarks are never directly...

s making up proton

Proton

The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

s and neutron

Neutron

The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

s, and the neutrino

Neutrino

A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...

s. Elementary particles emit and receive one or more particles called boson

Boson

In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s. This boson exchange gives rise to the three fundamental interaction

Fundamental interaction

In particle physics, fundamental interactions are the ways that elementary particles interact with one another...

s ("forces") of the Standard model of particle physics; hence bosons are also called

*force carriers*. These bosons have*s*=1. A basic example for a boson is the photonPhoton

In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

. Electromagnetism

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

is the force that results when charged particles exchange photons.

Theory predicts the existence of two bosons whose

*s*differs from 1. The force carrier for gravity is the hypothetical gravitonGraviton

In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...

; theory suggests that it has

*s*= 2. The Higgs mechanismHiggs mechanism

In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....

predicts that elementary particles acquire nonzero rest mass by exchanging hypothetical Higgs boson

Higgs boson

The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

s with an all-pervasive Higgs field. Theory predicts that the Higgs boson has

*s*= 0. If so, it would be the only elementary particle for which this is the case.### Composite particles

The spin of composite particles, such as protonProton

The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

s, neutron

Neutron

The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

s, and atomic nuclei

Atomic nucleus

The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

is usually understood to mean the total angular momentum. This is the sum of the spins and orbital angular momenta of the constituent particles. Such a composite spin is subject to the same quantization condition as any other angular momentum.

Composite particles are often referred to as having a definite spin, just like elementary particles; for example, the proton is a spin-1/2 particle. This is understood to refer to the spin of the lowest-energy internal state of the composite particle (i.e., a given spin and orbital configuration of the constituents).

It is not always easy to deduce the spin of a composite particle from first principles. For example, even though we know that the proton is a spin-1/2 particle, the question of how this spin is distributed

Nucleon spin structure

Nucleon spin structure describes the partonic structure of nucleon intrinsic angular momentum . The key question is how nucleon spin 1/2 is carried by its constituents, say partons . It was originally expected before the 1980s that quarks carry all of the nucleon spin, but later experiments...

among the three internal valence quarks and the surrounding sea quarks and gluon

Gluon

Gluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....

s is an active area of research.

Delta baryon

Delta baryon

The Delta baryons are a family of subatomic hadron particles which have the symbols , , , and and electric charges +2, +1, 0 and -1 elementary charge respectively...

s, which decay into protons and neutrons, have spin 3/2. All the three quarks inside a Delta baryon

Delta baryon

The Delta baryons are a family of subatomic hadron particles which have the symbols , , , and and electric charges +2, +1, 0 and -1 elementary charge respectively...

(Δ) have their spin axis pointing in the same direction, unlike the nearly identical proton and neutron (called "nucleons") in which the intrinsic spin of one of the three constituent quarks is always opposite the spin of the other two. This difference in spin alignment is the only quantum number distinction between the Δ

^{+}and Δ^{0}and ordinary nucleons.### Atoms and molecules

The spin of atoms and molecules is the sum of the spins of unpaired electrons, which may be parallel or antiparallel. It is responsible for paramagnetismParamagnetism

Paramagnetism is a form of magnetism whereby the paramagnetic material is only attracted when in the presence of an externally applied magnetic field. In contrast with this, diamagnetic materials are repulsive when placed in a magnetic field...

.

### The spin-statistics theorem

The spin of a particle has crucial consequences for its properties in statistical mechanicsStatistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

. Particles with half-integer spin obey Fermi-Dirac statistics

Fermi-Dirac statistics

Fermi–Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...

, and are known as fermion

Fermion

In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s. They are required to occupy antisymmetric quantum states (see the article on identical particles

Identical particles

Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

.) This property forbids fermions from sharing quantum states – a restriction known as the Pauli exclusion principle

Pauli exclusion principle

The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

. Particles with integer spin, on the other hand, obey Bose-Einstein statistics, and are known as boson

Boson

In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s. These particles occupy "symmetric states", and can therefore share quantum states. The proof of this is known as the spin-statistics theorem

Spin-statistics theorem

In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...

, which relies on both quantum mechanics and the theory of special relativity

Special relativity

Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

. In fact, "the connection between spin and statistics is one of the most important applications of the special relativity theory".

## Magnetic moments

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically chargedElectric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

s in a Stern–Gerlach experiment

Stern–Gerlach experiment

Important in the field of quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is a 1922 experiment on the deflection of particles, often used to illustrate basic principles of quantum mechanics...

, or by measuring the magnetic fields generated by the particles themselves.

The intrinsic magnetic moment

*μ*of an elementary particle with charge*q*, mass*m*, and spin angular momentum*S*, iswhere the dimensionless quantity

Dimensionless quantity

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

*g*is called the g-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

Electron magnetic dipole moment

In atomic physics, the electron magnetic dipole moment is the magnetic moment of an electron caused by its intrinsic property of spin.-Magnetic moment of an electron:...

. One of the triumphs of the theory of quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

is its accurate prediction of the electron

*g*-factorLandé g-factor

In physics, the Landé g-factor is a particular example of a g-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921....

, which has been experimentally determined to have the value , with the digits in parentheses denoting measurement uncertainty

Measurement uncertainty

In metrology, measurement uncertainty is a non-negative parameter characterizing the dispersion of the values attributed to a measured quantity. The uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity. All measurements are subject to uncertainty and a measured...

in the last two digits at one standard deviation

Standard deviation

Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

. The value of 2 arises from the Dirac equation

Dirac equation

The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...

, 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

Electromagnetic field

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

, 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, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.

Neutrino

Neutrino

A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...

s are both elementary and electrically neutral. The minimally extended Standard Model

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

that takes into account non-zero neutrino masses predicts neutrino magnetic moments of:

where the

*μ*_{ν}are the neutrino magnetic moments,*m*_{ν}are the neutrino masses, and*μ*_{B}is the Bohr magneton. 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}μ_{B}are unnatural, because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses cannot exceed about 1 eV, these radiative corrections must then be assumed to be fine tuned to cancel out to a large degree.The measurement of neutrino magnetic moments is an active area of research. , the latest experimental results have put the neutrino magnetic moment at less than times the electron's magnetic moment.

In ordinary material

Material

Material is anything made of matter, constituted of one or more substances. Wood, cement, hydrogen, air and water are all examples of materials. Sometimes the term "material" is used more narrowly to refer to substances or components with certain physical properties that are used as inputs to...

s, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, 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 spontaneously align with an externally applied magnetic field. In diamagnetic materials, on the other hand, the magnetic dipole moments of individual atoms spontaneously align oppositely to any externally applied magnetic field, even if it requires energy to do so.

The study of the behavior of such "spin model

Spin model

A spin model is a mathematical model used in physics primarily to explain magnetism. Spin models may either be classical or quantum mechanical in nature. Spin models have been studied in quantum field theory as examples of integrable models. Spin models are also used in quantum information theory...

s" is a thriving area of research in condensed matter physics

Condensed matter physics

Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

. For instance, the Ising model

Ising model

The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...

describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model

Heisenberg model

The Heisenberg model can refer to two models in statistical mechanics:*Heisenberg model , a classical nearest neighbour spin model*Heisenberg model , a model where the spins are treated quantum mechanically using Pauli matrices....

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

Phase transition

A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....

s.

### Spin projection quantum number and spin 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 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 measured along any direction (say along the*z*-axis) can only take on the valueswhere

*s*is the principal spin quantum number discussed in the previous section. One can see that there are 2*s*+1 possible values of*s*_{z}. The number "2*s*+ 1" is the multiplicity of the spin system. For example, there are only two possible values for a spin-1/2 particle:*s*_{z}= +1/2 and*s*_{z}= −1/2. These correspond to quantum states in which the spin is pointing in the +z or −z directions respectively, and are often referred to as "spin up" and "spin down". For a spin-3/2 particle, like a delta baryonDelta baryon

The Delta baryons are a family of subatomic hadron particles which have the symbols , , , and and electric charges +2, +1, 0 and -1 elementary charge respectively...

, the possible values are +3/2, +1/2, −1/2, −3/2.

### Spin vector

For a given quantum state, one could think of a spin vector whose components are the expectation values of the spin components along each axis, i.e., . This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation. It turns out that the spin vector is not very useful in actual quantum mechanical calculations, because it cannot be measured directly —*s*_{x},*s*_{y}and*s*_{z}cannot possess simultaneous definite values, because of a quantum uncertainty relationUncertainty principle

In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

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-1/2 particles, this maximum probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180 degrees—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

Gyroscope

A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...

. For example, one can exert a kind of "torque

Torque

Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

" on an electron by putting it in a magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

(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

Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...

, just like a classical gyroscope. This phenomenon is used in nuclear magnetic resonance

Nuclear magnetic resonance

Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...

sensing.

Mathematically, quantum mechanical spin is not described by vectors as in classical angular momentum, but by objects known as spinor

Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin-1/2 particle by 360 degrees does not bring it back to the same quantum state, but to the state with the opposite quantum phase

Phase (waves)

Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

; this is detectable, in principle, with interference experiments. To return the particle to its exact original state, one needs a 720 degree rotation. A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180 degrees can bring it back to the same quantum state and a spin-4 particle should be rotated 90 degrees 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 degrees and a spin 0 particle can be imagined as sphere which looks the same after whatever angle it is turned through.

### Spin operator

Spin obeys commutation relations analogous to those of the orbital angular momentum:where is the Levi-Civita symbol

Levi-Civita symbol

The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...

. It follows (as with angular momentum

Angular momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

) that the eigenvectors of

*S*^{2}and*S*_{z}(expressed as ketBra-ket notation

Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

s in the total

*S*basisBasis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

) are:

The spin raising and lowering operators

Creation and annihilation operators

Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one...

acting on these eigenvectors give:

, where

But unlike orbital angular momentum the eigenvectors are not spherical harmonics

Spherical harmonics

In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

. They are not functions of

*θ*and*φ*. There is also no reason to exclude half-integer values of*s*and*m*.In addition to their other properties, all quantum mechanical particles possess an intrinsic spin (though it may have the intrinsic spin 0, too). The spin is quantized in units of the reduced Planck constant, such that the state function of the particle is, e.g., not , but where is out of the following discrete set of values:

.

One distinguishes boson

Boson

In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s (integer spin) and fermion

Fermion

In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s (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 and spin operators

The quantum mechanicalQuantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

operators associated with spin observables are:

In the special case of spin-1/2 , and are the three Pauli matrices

Pauli matrices

The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

, given by:

### Spin and the Pauli exclusion principle

For systems of*N*identical particles this is related to the Pauli exclusion principlePauli exclusion principle

The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

, which states that by interchanges of any two of the N particles one must have

Thus, for bosons the prefactor (−1)

^{2s}will reduce to +1, for fermions to −1. In quantum mechanics all particles are either bosons or fermions. In relativistic quantum field theories also "supersymmetricSupersymmetry

In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

" particles exist, where linear combinations of bosonic and fermionic components appear. In two dimensions, the prefactor (−1)

^{2s}can be replaced by any complex number of magnitude 1 (see AnyonAnyon

In physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...

).

Electrons are fermions with

*s*= 1/2; quanta of light ("photonPhoton

In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s") are bosons with

*s*= 1. This shows also explicitly that the property*spin*cannot be fully explained as a classical intrinsic orbital angular momentum, e.g., similar to that of a "spinning top", since orbital angular rotations would lead to integer values of*s*. Instead one is dealing with an essential legacy of relativity. The photon, in contrast, is always relativistic (velocity*v*≈*c*

), and the corresponding classical theory, that of Maxwell, is also relativistic.Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

The above permutation postulate for

*N*-particle state functions has most-important consequences in daily life, e.g. the periodic tablePeriodic table

The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

of the chemists or biologists.

### Spin and rotations

As described above, quantum mechanics states that component 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 1/2 particle, we would need two numbers*a*_{±1/2}, giving amplitudes of finding it with projection of angular momentum equal to*ħ*/2 and −*ħ*/2, satisfying the requirementFor a generic particle with spin s, we would need 2s+1 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's 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:

Mathematically speaking, these matrices furnish a unitary projective

Projective representation

In the mathematical field of representation theory, 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 groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...

representation

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

of the rotation group

Rotation group

In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...

SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2). There is one n-dimensional irreducible representation of SU(2) for each dimension, though this representation is n-dimensional real for odd n and n-dimensional complex for even n (hence of real dimension 2

*n*). For a rotation by angle*θ*in the plane with normal vector ,*U*can be writtenwhere 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

Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...

:

An irreducible representation of this group of operators is furnished by the Wigner D-matrix

Wigner D-matrix

The Wigner D-matrix is a matrix in an irreducible representation of the groups SU and SO. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced in 1927 by Eugene Wigner.- Definition Wigner D-matrix :Let j_x,...

:

where

is Wigner's small d-matrix. Note that for γ = 2π and α = β = 0, i.e. a full rotation about the z-axis, the Wigner D-matrix elements become

Recalling that a generic spin state can be written as a superposition of states with definite

*m*, we see that if*s*is an integer, the values of*m*are all integers, and this matrix corresponds to the identity operator. However, if*s*is a half-integer, the values of*m*are also all half-integers, giving (-1)^{2m}= -1 for all*m*, and hence upon rotation by 2π the state picks up a minus sign. This fact is a crucial element of the proof of the spin-statistics theoremSpin-statistics theorem

In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...

.

### Spin and Lorentz transformations

We could try the same approach to determine the behavior of spin under general Lorentz transformationLorentz transformation

In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

s, but we'd immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact

Compact group

In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

and therefore does not have any faithful unitary finite-dimensional representations.

In case of spin 1/2 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 with each particle. These spinors transform under Lorentz transformations according to the law

where are gamma matrices and is an antisymmetric 4x4 matrix parametrizing the transformation. It can be shown that the scalar product

is preserved. (It is not, however, positive definite, so the representation is not unitary.)

### Measuring spin along the x, y, and z axes

Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the x, y or z axis can only yield an eigenvalue of the corresponding spin operator ( or ) on that axis, i.e. or The quantum state of a particle (with respect to spin), can be represented by a two component spinor

Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

:

When the spin of this particle is measured with respect to a given axis (in this example, the x-axis), the probability that its spin will be measured as is just . Correspondingly, the probability that its spin will be measured as is just . Following the measurement, the spin state of the particle will collapse

Wavefunction collapse

In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...

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 , etc), provided that no measurements of the spin are made along other axes (see compatibility section below).

### Measuring 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 . 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 x,y,z axis directions.A normalized spinor for spin-1/2 in the direction (which works for all spin states except spin down where it will give 0/0), is:

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 commuteCommutativity

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the x-axis, and we then measure the spin along the y-axis, we have invalidated our previous knowledge of the x-axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that:

So when we measure the spin of a particle along the x-axis as, for example, , the particle's spin state collapses

Wavefunction collapse

In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...

into the eigenstate . When we then subsequently measure the particle's spin along the y-axis, the spin state will now collapse into either or , each with probability . Let us say, in our example, that we measure . When we now return to measure the particle's spin along the x-axis again, the probabilities that we will measure or are each (i.e. they are and ). This implies that our original measurement of the spin along the x-axis is no longer valid, since the spin along the x-axis will now be measured to have either eigenvalue with equal probability.

## Spin and parity

In tables of the spin quantum number*s*for nuclei or particles, the spin is often followed by a "+" or "-". This refers to the parityParity (physics)

In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

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

Isotopes of bismuth

Bismuth has no stable isotopes, but does have one very long-lived isotope, thus the standard atomic mass can be given. Although bismuth-209 is now known to be unstable, it has classically been considered to be a "stable" isotope because it has a half-life of over 1.9×1019 years, which is...

.

## Applications

Spin has important theoretical implications and practical applications. Well-established*direct*applications of spin include:- Nuclear magnetic resonanceNuclear magnetic resonanceNuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...

spectroscopy in chemistry; - Electron spin resonance spectroscopy in chemistry and physics;
- Magnetic resonance imagingMagnetic resonance imagingMagnetic resonance imaging , nuclear magnetic resonance imaging , or magnetic resonance tomography is a medical imaging technique used in radiology to visualize detailed internal structures...

(MRI) in medicine, which relies on proton spin density; - Giant magnetoresistiveGiant magnetoresistive effectGiant magnetoresistance is a quantum mechanical magnetoresistance effect observed in thin-film structures composed of alternating ferromagnetic and non-magnetic layers...

(GMR) drive head technology in modern hard diskHard diskA hard disk drive is a non-volatile, random access digital magnetic data storage device. It features rotating rigid platters on a motor-driven spindle within a protective enclosure. Data is magnetically read from and written to the platter by read/write heads that float on a film of air above the...

s.

Electron spin plays an important role in magnetism

Magnetism

Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

, with applications for instance in computer memories. The manipulation of

*nuclear spin*by radiofrequency waves (nuclear magnetic resonanceNuclear magnetic resonance

Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...

) is important in chemical spectroscopy and medical imaging.

Spin-orbit coupling leads to the fine structure

Fine structure

In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy...

of atomic spectra, which is used in atomic clock

Atomic clock

An atomic clock is a clock that uses an electronic transition frequency in the microwave, optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element...

s and in the modern definition of the second

Second

The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

. Precise measurements of the g-factor of the electron have played an important role in the development and verification of quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

.

*Photon spin*is associated with the polarization of light.A possible future direct application of spin is as a binary information carrier in spin transistor

Spin transistor

The magnetically-sensitive transistor , originally proposed in 1990 and currently still being developed, is an improved design on the common transistor invented in the 1940s...

s. Original concept proposed in 1990 is known as Datta-Das spin transistor. Electronics based on spin transistors is called spintronics

Spintronics

Spintronics , also known as magnetoelectronics, is an emerging technology that exploits both the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental electronic charge, in solid-state devices.An additional effect occurs when a spin-polarized current is...

, which includes the manipulation of spins in semiconductor devices.

There are many

*indirect*applications and manifestations of spin and the associated Pauli exclusion principlePauli exclusion principle

, starting with the periodic table

Periodic table

The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

of chemistry.

## History

Spin was first discovered in the context of the emission spectrumEmission spectrum

The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted by the element's atoms or the compound's molecules when they are returned to a lower energy state....

of alkali metal

Alkali metal

The alkali metals are a series of chemical elements in the periodic table. In the modern IUPAC nomenclature, the alkali metals comprise the group 1 elements, along with hydrogen. The alkali metals are lithium , sodium , potassium , rubidium , caesium , and francium...

s. In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell

Electron shell

An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....

. This allowed him to formulate the Pauli exclusion principle

Pauli exclusion principle

, stating that no two electrons can share the same quantum state at the same time.

The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig

Ralph Kronig

Ralph Kronig was a German-American physicist . He is noted for the discovery of particle spin and for his theory of x-ray absorption spectroscopy...

, one of Landé

Alfred Landé

Alfred Landé was a German-American physicist known for his contributions to quantum theory. He is responsible for the Landé g-factor an explanation of the Zeeman Effect.-Life and Achievements:...

'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

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity

Theory of relativity

The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

. Largely due to Pauli's criticism, Kronig decided not to publish his idea.

In the autumn of 1925, the same thought came to two Dutch physicists, George Uhlenbeck and Samuel Goudsmit. Under the advice of Paul Ehrenfest

Paul Ehrenfest

Paul Ehrenfest was an Austrian and Dutch physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition and the Ehrenfest theorem.- Biography :Paul Ehrenfest was born and grew up in Vienna in a Jewish...

, they published their results. It met a favorable response, especially after Llewellyn Thomas

Llewellyn Thomas

Llewellyn Hilleth Thomas was a British physicist and applied mathematician. He is best known for his contributions to atomic physics, in particular:...

managed to resolve a factor-of-two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished ones). This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.

Mathematically speaking, a fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

description is needed. The tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

effect is additive and relativistic; that is, it vanishes if

*c*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 precessionThomas precession

In physics the Thomas precession, named after Llewellyn Thomas, is a special 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 particle following a curvilinear orbit to the angular...

).

Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

discovered by Schrödinger

Erwin Schrödinger

Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

and Heisenberg

Werner Heisenberg

Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

. He pioneered the use of Pauli matrices

Pauli matrices

The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

as a representation

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

of the spin operators, and introduced a two-component spinor

Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

wave-function.

Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac

Paul Dirac

Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

published the Dirac equation

Dirac equation

The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...

, which described the relativistic electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

. In the Dirac equation, a four-component spinor (known as a "Dirac spinor") was used for the electron wave-function. In 1940, Pauli proved the

*spin-statistics theoremIn quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...*

, which states that fermionSpin-statistics theorem

Fermion

s have half-integer spin and boson

Boson

s integer spin.

In retrospect, the first direct experimental evidence of the electron spin was the Stern-Gerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1927.

## External links

- "Spintronics. Feature Article" in
*Scientific American*, June 2002.Scientific AmericanScientific American is a popular science magazine. It is notable for its long history of presenting science monthly to an educated but not necessarily scientific public, through its careful attention to the clarity of its text as well as the quality of its specially commissioned color graphics... - Goudsmit on the discovery of electron spin.
*Nature*: " Milestones in 'spin' since 1896."NatureNature, in the broadest sense, is equivalent to the natural world, physical world, or material world. "Nature" refers to the phenomena of the physical world, and also to life in general...- ECE 495N Lecture 36: Spin Online lecture by S. Datta