Spin is a
conserved quantity carried by
elementary particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
s, and thus by composite particles (
hadron
In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the ...
s) and
atomic nuclei
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...
.
Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbital
angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum p ...
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 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 i ...
from experiments, such as the
Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum.
The existence of the electron spin can also be inferred theoretically from the
spin–statistics theorem and from the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
—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
spinors 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, spe ...
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.
The
SI unit of spin is the same as classical angular momentum (i.e.,
N·
m·
s,
J·s, or
kg·m
2·s
−1). In practice, spin is given as a
dimensionless spin quantum number by dividing the spin angular momentum by the
reduced Planck constant , which has the same
dimensions 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 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
Leiden University (abbreviated as ''LEI''; nl, Universiteit Leiden) is a public research university in Leiden, Netherlands. The university was founded as a Protestant university in 1575 by William, Prince of Orange, as a reward to the city o ...
suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the
old quantum theory of
Bohr
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. B ...
and
Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
.
Ralph Kronig 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
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
derived his relativistic quantum mechanics 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
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
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
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 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 spi ...
. 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
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
: 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. 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 cons ...
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 common ...
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 common ...
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 neutra ...
(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 n ...
and neutrinos), which make up what is classically known as matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
, are all fermions with spin . The common idea that "matter takes up space" actually comes from the Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
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
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
(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
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
, which carries the electromagnetic force, the gluon ( strong force), 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 , , an ...
( weak force). The ability of bosons to occupy the same quantum state is used in the laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The ...
, which aligns many photons having the same quantum number (the same direction and frequency), superfluid liquid helium 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 (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 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 spi ...
and fermions, where bosons obey Bose–Einstein statistics, and fermions obey Fermi–Dirac statistics (and therefore the Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
). 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, electron
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 n ...
s 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 law ...
, 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 , for rotation of angle ''θ'' around the axis parallel to the spin ''S''. This is equivalent to the quantum-mechanical interpretation of momentum as phase dependence in the position, and of orbital angular momentum as phase dependence in the angular position.
Photon spin is the quantum-mechanical description of light polarization, where spin +1 and spin −1 represent two opposite directions of circular polarization. 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 is equal by Ehrenfest theorem
The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of the ...
to the derivative of the Hamiltonian to its conjugate momentum
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
, which is the total angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum p ...
. 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, and thus the relativistic Hamiltonian of the electron, treated as a Dirac field, 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 la ...
effect understood as this rotation.
Magnetic moments
Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectiv ...
body in classical electrodynamics
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fi ...
. 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, 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
:
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 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
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
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 common ...
, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.
Neutrinos 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 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_\m ...
. 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
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
or a Z boson, 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. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains 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 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 of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component
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 assem ...
of angular momentum for a spin-''s'' particle measured along any direction can only take on the values
:
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:
:
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 whose components are the expectation values of the spin components along each axis, i.e.,
, 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, ...
of multiparticle systems, the general Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\begin
...
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 tensor ...
products of Pauli matrices.
The analog formula of Euler's formula in terms of the Pauli matrices
:
\hat(\theta, \hat) = e^ =
I \cos \frac + i \left(\hat \cdot \boldsymbol\right) \sin \frac
for higher spins is tractable, but less simple.
Parity
In tables of the spin quantum number for nuclei or particles, the spin is often followed by a "+" or "−". This refers to the parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
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 (NMR) spectroscopy in chemistry;
* Electron spin resonance (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
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
, with applications for instance in computer memories. The manipulation of ''nuclear spin'' by radio-frequency waves ( nuclear magnetic resonance) 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
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 betwe ...
s and in the modern definition of the second. 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 of light ( photon polarization).
An emerging application of spin is as a binary information carrier in spin transistor
The magnetically sensitive transistor (also known as the spin transistor or spintronic transistor—named for spintronics, the technology which this development spawned), originally proposed in 1990 by Supriyo Datta and Biswajit Das, currently stil ...
s. The original concept, proposed in 1990, is known as Datta–Das spin transistor
The magnetically sensitive transistor (also known as the spin transistor or spintronic transistor—named for spintronics, the technology which this development spawned), originally proposed in 1990 by Supriyo Datta and Biswajit Das, currently stil ...
. 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- ...
. 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
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
, 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 names ...
s. In 1924, Wolfgang Pauli introduced what he called a "two-valuedness not describable classically" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
, 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, 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 fo ...
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 Dutch physicists George Uhlenbeck and Samuel Goudsmit at Leiden University
Leiden University (abbreviated as ''LEI''; nl, Universiteit Leiden) is a public research university in Leiden, Netherlands. The university was founded as a Protestant university in 1575 by William, Prince of Orange, as a reward to the city o ...
. 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 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, 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 ...
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, ...
invented 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. 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
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
published the Dirac equation, which described the relativistic electron
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 n ...
. 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 co ...
") was used for the electron wave-function. Relativistic spin explained gyromagnetic anomaly, which was (in retrospect) first observed by Samuel Jackson Barnett Samuel Jackson Barnett (December 14, 1873 – May 22, 1956) was an American physicist. He was a professor at the University of California, Los Angeles.
Barnett was born in Woodson County, Kansas, the son of a minister. In 1894, he received a B.A. ...
in 1914 (see Einstein–de Haas effect). In 1940, Pauli proved the '' spin–statistics theorem'', 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 of 1922. However, the correct explanation of this experiment was only given in 1927.
See also
* 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 impo ...
* Kramers theorem
In quantum mechanics, the Kramers' degeneracy theorem states that for every energy eigenstate of a time-reversal symmetric system with half-integer total spin, there is another eigenstate with the same energy related by time-reversal. In other wor ...
* 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 f ...
* Pauli–Lubanski pseudovector
* Rarita–Schwinger equation
In theoretical physics, the Rarita–Schwinger equation is the
relativistic field equation of spin-3/2 fermions. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwing ...
* Representation theory of SU(2)
* Spin angular momentum of light
* Spin engineering
Spin engineering describes the control and manipulation of quantum spin systems to develop devices and materials. This includes the use of the spin degrees of freedom as a probe for spin based phenomena.
Because of the basic importance of quant ...
* Spin-flip
A black hole spin-flip occurs when the spin axis of a rotating black hole undergoes a sudden change in orientation due to absorption of a second (smaller) black hole. Spin-flips are believed to be a consequence of galaxy mergers, when two supe ...
* 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–orb ...
* Spin tensor
* Spin wave
* Yrast
''Yrast'' ( , ) is a technical term in nuclear physics that refers to a state of a nucleus with a minimum of energy (when it is least excited) for a given angular momentum. ''Yr'' is a Swedish adjective sharing the same root as the English ''whirl ...
References
Further reading
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*Sin-Itiro Tomonaga, The Story of Spin, 1997
External links
*
Goudsmit on the discovery of electron spin.
*''Nature
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
'':
Milestones in 'spin' since 1896.
ECE 495N Lecture 36: Spin
Online lecture by S. Datta
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Rotational symmetry
Quantum field theory
Physical quantities