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| {{about|spin in quantum mechanics|rotation in classical mechanics|angular momentum}}
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| {{Standard model of particle physics}}
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| In [[quantum mechanics]] and [[particle physics]], '''spin''' is an intrinsic form of [[angular momentum]] carried by [[elementary particle]]s, composite particles ([[hadron]]s), and [[atomic nucleus|atomic nuclei]].<ref name="merzbacher372">{{cite book|last=Merzbacher|first=Eugen|title=Quantum Mechanics|edition=3rd|year=1998|pages=372–3}}</ref><ref name="griffiths183">{{cite book|last=Griffiths|first=David|title=Introduction to Quantum Mechanics|edition=2nd|year=2005|pages=183–4}}</ref> Spin is a solely quantum-mechanical phenomenon; it does not have a counterpart in [[classical mechanics]] (despite the term ''spin'' being reminiscent of classical phenomena such as a planet spinning on its axis).<ref name="griffiths183" />
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| Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. Orbital angular momentum is the quantum-mechanical counterpart to the classical notion of angular momentum: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus).<ref>[http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/AngularMomentum.htm "Angular Momentum Operator Algebra", class notes by Michael Fowler]</ref><ref>[http://books.google.com/books?id=3_7uriPX028C&pg=PA31 ''A modern approach to quantum mechanics'', by Townsend, p31 and p80]</ref> The existence of spin angular momentum is inferred from experiments, such as the [[Stern–Gerlach experiment]], in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.<ref name="eisberg272">{{cite book|last=Eisberg|first=Robert|last2=Resnick|first2=Robert|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles|edition=2nd|year=1985|pages=272–3}}</ref>
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| In some ways, spin is like a [[Euclidean vector|vector]] quantity; it has a definite magnitude; and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a ''spin [[quantum number]]''.<ref name="griffiths183" /> However, in a technical sense, spins are not strictly vectors, and they are instead described as a related quantity: a [[spinor]]. In particular, unlike a Euclidean vector, a spin when rotated by 360 degrees can have its sign reversed.<ref name="tomonaga">{{cite book|last=Tomonaga|first=Sin-Itiro|coauthors=Oka, Takeshi (trans.)|authorlink=Shin-Ichiro Tomonaga|title=The Story of Spin|year=1997|chapter=Chapter 7: The Quantity Which is Neither Vector Nor Tensor|publisher=University of Chicago Press|isbn=0-226-80794-0}}</ref>
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| The [[SI unit]] of spin is the [[joule-second]], just as with classical angular momentum. In practice, however, SI units are never used to describe spin: instead, it is written as a multiple of the [[reduced Planck constant]] ''ħ''. In [[natural units]], the ''ħ'' is omitted, so spin is written as a [[unitless]] number. The spin quantum numbers are always unitless numbers by definition.
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| When combined with the [[spin-statistics theorem]], the spin of electrons results in the [[Pauli exclusion principle]], which in turn underlies the [[periodic table]] of [[chemical element]]s.
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| [[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.
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| ==Spin quantum number==
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| {{main|Spin quantum number}}
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| As the name suggests, spin was originally conceived as the rotation of a particle around some axis. This picture is correct so far as spin obeys the same mathematical laws as [[angular momentum quantization|quantized]] [[angular momentum|angular momenta]] do. On the other hand, spin has some peculiar properties that distinguish it from orbital angular momenta:
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| *Spin quantum numbers may take half-integer values.
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| *Although the direction of its spin can be changed, an elementary particle cannot be made to spin faster or slower.
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| *The spin of a charged particle is associated with a [[magnetic dipole moment]] with a [[g-factor (physics)|g-factor]] differing from 1. This could only occur classically [[Gyromagnetic ratio#Gyromagnetic ratio for a classical rotating body|if the internal charge of the particle were distributed differently from its mass]].
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| The conventional definition of the '''spin quantum number''' ''s'' is ''s'' = ''n''/2, where ''n'' can be any [[non-negative]] [[integer]]. Hence the allowed values of ''s'' are 0, 1/2, 1, 3/2, 2, etc. The value of ''s'' for an [[elementary particle]] depends only on the type of particle, and cannot be altered in any known way (in contrast to the ''spin direction'' described below). The spin angular momentum ''S'' of any physical system is [[angular momentum quantization|quantized]]. The allowed values of ''S'' are:
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| :<math>S = \frac{h}{2\pi} \, \sqrt{s (s+1)}=\frac{h}{4\pi} \, \sqrt{n(n+2)},</math>
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| where ''h'' is the [[Planck constant]]. In contrast, [[angular momentum operator|orbital angular momentum]] can only take on integer values of ''s'', even values of ''n''.
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| ===Fermions and Bosons===
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| Those particles with half-integer spins, such as 1/2, 3/2, 5/2, are known as [[fermion]]s, while those particles with integer spins, such as 0, 1, 2, are known as [[bosons]]. The two families of particles obey different rules and ''broadly'' have different roles in the world around us. A key distinction between the two families is that fermions obey the [[Pauli exclusion principle]]; that is, there cannot be two identical fermions simultaneously having the same quantum numbers (meaning roughly, being in the same place with the same velocity). In contrast, Bosons obey the rules of [[Bose Einstein statistics]] and have no such restriction, so they may "bunch" together even if in identical states. Also, composite particles can have spins different from the particles which comprise them. For example, a [[helium atom]] can have spin 0 and therefore can behave like a boson even though the [[quarks]] and electrons which make it up are all fermions.
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| This has profound practical applications:
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| :* '''Fermions:''' [[Quarks]] and [[leptons]] (including [[electrons]] and [[neutrinos]]), which make up what is classically known as [[matter]], are all fermions with [[spin-1/2]]. The common idea that "matter takes up space" actually comes from the Pauli exclusion principle acting on these particles to prevent the fermions that make up matter from being in the same quantum state. Further compaction would require electrons to occupy the same energy states, and therefore a kind of [[pressure]] (sometimes known as [[degenerate matter|degeneracy pressure of electrons]]) acts to resist the fermions being overly close. It is also this pressure which prevents [[star]]s collapsing inwardly, and which, when it finally gives way under immense gravitational pressure in a dying massive star, triggers inward collapse and the dramatic explosion into a [[supernova]].
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| ::: ''Elementary fermions with other spins (3/2, 5/2 etc.)'' are not known to exist, as of 2013.
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| :* '''Bosons:''' Elementary particles which are thought of as [[force carrier|carrying forces]] are all bosons with spin-1. They include the [[photon]] which carries the [[electromagnetic force]], the [[gluon]] ([[strong force]]), and the [[W and Z bosons]] ([[weak force]]). The ability of bosons to occupy the same quantum state is used in the [[laser]], which aligns many photons having the same quantum number (the same direction and frequency), [[superfluid]] [[liquid helium]] resulting from helium-4 atoms being bosons, and [[superconductivity]] where [[Cooper pair|pairs of electrons]] (which individually are fermions) act as single composite bosons.
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| ::: ''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.
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| ::: As of 2013 the Higgs boson with spin-0 is considered proven to exist. It is the first [[scalar boson|scalar particle]] (spin-0) known to exist in nature.
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| Theoretical and experimental studies have shown that the spin possessed by [[elementary particle]]s cannot be explained by postulating that they are made up of even smaller particles rotating about a common [[center of mass]] (see [[classical electron radius]]); as far as can be presently determined, these elementary particles have no inner structure. The spin of an elementary particle is therefore seen as a truly intrinsic physical property, akin to the particle's [[electric charge]] and [[rest mass]].
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| ===The spin-statistics theorem===
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| The spin of a particle has crucial consequences for its properties in [[statistical mechanics]]. 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 state]]s – 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".<ref>{{cite journal | last1 = Pauli | first1 = Wolfgang | authorlink = Wolfgang Pauli | year = 1940 | title = The Connection Between Spin and Statistics | url = http://web.ihep.su/dbserv/compas/src/pauli40b/eng.pdf | format = PDF | journal = Phys. Rev | volume = 58 | issue = 8| pages = 716–722 | doi = 10.1103/PhysRev.58.716 |bibcode = 1940PhRv...58..716P }}</ref>
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| ==Magnetic moments==
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| {{main|Spin magnetic moment}}
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| [[Image:Magnetic ring dipole field lines.svg|thumbnail|Magnetic field lines around a ''magnetostatic dipole''; the [[magnetic dipole]] itself is in the center and is seen from the side.]]
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| Particles with spin can possess a [[magnetic dipole moment]], just like a rotating [[electric charge|electrically charged]] body in [[classical electrodynamics]]. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous [[magnetic field]]s in a [[Stern–Gerlach experiment]], or by measuring the magnetic fields generated by the particles themselves.
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| The intrinsic magnetic moment '''μ''' of a [[spin-1/2]] particle with charge ''q'', mass ''m'', and spin angular momentum '''S''', is<ref>Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2</ref>
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| :<math>\boldsymbol{\mu} = \frac{g_s q}{2m} \mathbf{S} </math>
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| where the [[dimensionless quantity]] ''g<sub>s</sub>'' is called the spin [[g-factor (physics)#Electron spin g-factor|g-factor]]. For exclusively orbital rotations it would be 1 (assuming that the mass and the charge occupy spheres of equal radius).
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| The electron, being a charged elementary particle, possesses a [[Electron magnetic dipole moment|nonzero magnetic moment]]. One of the triumphs of the theory of [[quantum electrodynamics]] is its accurate prediction of the electron [[Landé g-factor|''g''-factor]], which has been experimentally determined to have the value {{val|-2.0023193043622|(15)}}, with the digits in parentheses denoting [[measurement uncertainty]] in the last two digits at one [[standard deviation]].<ref>
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| {{cite web
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| |title=CODATA Value: electron g factor
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| |url=http://physics.nist.gov/cgi-bin/cuu/Value?gem{{!}}search_for=all!
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| |year=2006
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| |work=The NIST Reference on Constants, Units, and Uncertainty
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| |publisher=[[National Institute of Standards and Technology|NIST]]
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| |accessdate=2013-11-15
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| }}</ref> 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 {{val|0.002319304}}... arises from the electron's interaction with the surrounding [[electromagnetic field]], including its own field.<ref>
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| {{cite book
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| |author=[[Richard Feynman|R.P. Feynman]]
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| |title=[[QED: The Strange Theory of Light and Matter]]
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| |chapter=Electrons and Their Interactions
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| |page=115
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| |publisher=[[Princeton University Press]] |location=[[Princeton, New Jersey]]
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| |year=1985
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| |isbn=0-691-08388-6
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| }}<br>
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| :"After some years, it was discovered that this value [−''g''/2] was not exactly 1, but slightly more—something like 1.00116. This correction was worked out for the first time in 1948 by Schwinger as ''j''*''j'' divided by 2 pi {{sic}} [where ''j'' is the square root of the [[fine-structure constant]]], and was due to an alternative way the electron can go from place to place: instead of going directly from one point to another, the electron goes along for a while and suddenly emits a photon; then (horrors!) it absorbs its own photon."</ref> 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.
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| [[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:<ref>
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| {{cite journal
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| |author=W.J. Marciano, A.I. Sanda
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| |title=Exotic decays of the muon and heavy leptons in gauge theories
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| |journal=[[Physics Letters]]
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| |volume=B67 |issue=3 |pages=303–305
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| |year=1977
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| |doi=10.1016/0370-2693(77)90377-X
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| |bibcode = 1977PhLB...67..303M }}</ref><ref>
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| {{cite journal
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| |author=B.W. Lee, R.E. Shrock
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| |title=Natural suppression of symmetry violation in gauge theories: Muon- and electron-lepton-number nonconservation
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| |journal=[[Physical Review]]
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| |volume=D16 |issue=5 |pages=1444–1473
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| |year=1977
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| |doi=10.1103/PhysRevD.16.1444
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| |bibcode = 1977PhRvD..16.1444L }}</ref><ref>
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| {{cite journal
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| |author=K. Fujikawa, R. E. Shrock
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| |title=Magnetic Moment of a Massive Neutrino and Neutrino-Spin Rotation
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| |journal=[[Physical Review Letters]]
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| |volume=45 |issue=12 |pages=963–966
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| |year=1980
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| |doi=10.1103/PhysRevLett.45.963
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| |bibcode=1980PhRvL..45..963F
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| }}</ref>
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| :<math>\mu_{\nu}\approx 3\times 10^{-19}\mu_\mathrm{B}\frac{m_{\nu}}{\text{eV}}</math>
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| where the ''μ''<sub>ν</sub> are the neutrino magnetic moments, ''m''<sub>ν</sub> are the neutrino masses, and ''μ''<sub>B</sub> 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<sup>−14</sup> μ<sub>B</sub> 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.<ref>
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| {{cite journal
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| |author=N.F. Bell ''et al''.
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| |title=How Magnetic is the Dirac Neutrino?
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| |journal=[[Physical Review Letters]]
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| |volume=95 |issue=15 |page=151802
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| |year=2005
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| |arxiv=hep-ph/0504134
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| |doi=10.1103/PhysRevLett.95.151802
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| |bibcode=2005PhRvL..95o1802B
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| |pmid=16241715
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| |last2=Cirigliano
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| |first2=V.
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| |last3=Ramsey-Musolf
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| |first3=M.
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| |last4=Vogel
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| |first4=P.
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| |last5=Wise
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| |first5=Mark
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| }}</ref>
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| The measurement of neutrino magnetic moments is an active area of research. {{As of|2001}}, the latest experimental results have put the neutrino magnetic moment at less than {{val|1.2|e=-10}} times the electron's magnetic moment.
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| 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. [[Ferromagnet]]ic materials below their [[Curie temperature]], however, exhibit [[magnetic domain]]s 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.
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| 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.
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| 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.
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| ==Spin "direction"==
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| {{further|Angular momentum operator}}
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| ===Spin projection quantum number and spin multiplicity===
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| 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 [[Spatial vector|component]] of angular momentum measured along any direction can only take on the values <ref>Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1</ref>
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| :<math> S_i = \hbar s_i, \quad s_i \in \{ - s, -(s-1), \dots, s-1, s \} \,\!</math>
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| where ''S<sub>i</sub>'' is the spin component along the ''i''-axis (either ''x'', ''y'', or ''z''), ''s<sub>i</sub>'' is the spin projection quantum number along the ''i''-axis, and ''s'' is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the ''z''-axis:
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| :<math> S_z = \hbar s_z, \quad s_z \in \{ - s, -(s-1), \dots, s - 1, s \} \,\!</math>
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| where ''S<sub>z</sub>'' is the spin component along the ''z''-axis, ''s<sub>z</sub>'' is the spin projection quantum number along the ''z''-axis.
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| One can see that there are 2''s''+1 possible values of ''s''<sub>z</sub>. 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''<sub>z</sub> = +1/2 and ''s''<sub>z</sub> = −1/2. These correspond to [[quantum state]]s 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 baryon]], the possible values are +3/2, +1/2, −1/2, −3/2.
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| ===Spin vector===
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| For a given [[quantum state]], one could think of a spin vector <math>\lang S \rang </math> whose components are the [[expectation value (quantum physics)|expectation value]]s of the spin components along each axis, i.e., <math>\lang S \rang = [\lang S_x \rang, \lang S_y \rang, \lang S_z \rang]</math>. 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<sub>x</sub>'', ''s<sub>y</sub>'' and ''s<sub>z</sub>'' cannot possess simultaneous definite values, because of a quantum [[uncertainty principle|uncertainty relation]] between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a [[Stern–Gerlach apparatus]], the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin-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%.
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| 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 [[gyroscope|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.
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| Mathematically, quantum mechanical spin states are described by vector-like objects known as [[spinor]]s. There are subtle differences between the behavior of spinors and vectors under [[coordinate rotation]]s. 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 (waves)|phase]]; this is detectable, in principle, with [[Interference (wave propagation)|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.
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| ==Mathematical formulation of spin==
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| ===Spin operator===
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| Spin obeys [[commutation relations]] analogous to those of the [[angular momentum operator|orbital angular momentum]]:
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| :<math>[S_i, S_j ] = i \hbar \epsilon_{ijk} S_k</math>
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| where <math>\epsilon_{ijk}</math> is the [[Levi-Civita symbol]]. It follows (as with [[angular momentum]]) that the [[eigenvectors]] of ''S''<sup>2</sup> and ''S''<sub>z</sub> (expressed as [[Bra-ket notation|ket]]s in the total ''S'' [[Basis (linear algebra)|basis]]) are:
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| :<math>S^2 |s,m\rangle = \hbar^2 s(s + 1) |s,m\rangle</math>
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| :<math>S_z |s,m\rangle = \hbar m |s,m\rangle.</math>
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| The spin [[Creation and annihilation operators|raising and lowering operators]] acting on these eigenvectors give:
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| :<math>S_\pm |s,m\rangle = \hbar\sqrt{s(s+1)-m(m\pm 1)} |s,m\pm 1 \rangle</math>, where <math>S_\pm = S_x \pm i S_y. </math>
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| But unlike orbital angular momentum the eigenvectors are not [[spherical harmonics]]. They are not functions of ''θ'' and ''φ''. There is also no reason to exclude half-integer values of ''s'' and ''m''.
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| 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's constant|Planck constant]], such that the state function of the particle is, say, not <math>\psi = \psi(\mathbf r)</math>, but <math>\psi =\psi(\mathbf r,\sigma)\,,</math> where <math>\sigma </math> is out of the following discrete set of values:
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| :<math>\sigma \in \{-s\hbar , -(s-1)\hbar , \cdots ,+(s-1)\hbar ,+s\hbar\}.\,\!</math>
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| One distinguishes [[boson]]s (integer spin) and [[fermion]]s (half-integer spin). The total angular momentum conserved in interaction processes is then the ''sum'' of the orbital angular momentum and the spin.
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| === Pauli matrices and spin operators ===
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| The [[quantum mechanics|quantum mechanical]] [[Operator (mathematics)|operators]] associated with spin [[observables]] are:
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| :<math> S_x = {\hbar \over 2} \sigma_x,\quad S_y = {\hbar \over 2} \sigma_y,\quad S_z = {\hbar \over 2} \sigma_z.</math>
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| In the special case of spin-1/2 particles, ''σ<sub>x</sub>'', ''σ<sub>y</sub>'' and ''σ<sub>z</sub>'' are the three [[Pauli matrices]], given by:
| |
| | |
| :<math>
| |
| \sigma_x =
| |
| \begin{pmatrix}
| |
| 0 & 1\\
| |
| 1 & 0
| |
| \end{pmatrix}
| |
| ,\quad
| |
| \sigma_y =
| |
| \begin{pmatrix}
| |
| 0 & -i\\
| |
| i & 0
| |
| \end{pmatrix}
| |
| ,\quad
| |
| \sigma_z =
| |
| \begin{pmatrix}
| |
| 1 & 0\\
| |
| 0 & -1
| |
| \end{pmatrix}.
| |
| </math>
| |
| | |
| ===Spin and the Pauli exclusion principle===
| |
| For systems of ''N'' identical particles this is related to the [[Pauli exclusion principle]], which states that by interchanges of any two of the ''N'' particles one must have
| |
| | |
| :<math>\psi ( \cdots \mathbf r_i,\sigma_i\cdots \mathbf r_j,\sigma_j\cdots ) = (-1)^{2s}\psi ( \cdots \mathbf r_j,\sigma_j\cdots \mathbf r_i,\sigma_i\cdots ) .</math>
| |
| | |
| Thus, for bosons the prefactor (−1)<sup>2''s''</sup> will reduce to +1, for fermions to −1. In quantum mechanics all particles are either bosons or fermions. In some speculative relativistic quantum field theories "[[Supersymmetry|supersymmetric]]" particles also exist, where linear combinations of bosonic and fermionic components appear. In two dimensions, the prefactor (−1)<sup>2''s''</sup> can be replaced by any complex number of magnitude 1 (see [[Anyon]]).
| |
|
| |
| The above permutation postulate for ''N''-particle state functions has most-important consequences in daily life, e.g. the [[periodic table]] of the chemists or biologists.
| |
| | |
| ===Spin and rotations===
| |
| | |
| {{see also|symmetries in quantum mechanics}}
| |
| | |
| As described above, quantum mechanics states that [[Spatial vector|components]] 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''<sub>±1/2</sub>, giving amplitudes of finding it with projection of angular momentum equal to ''ħ''/2 and −''ħ''/2, satisfying the requirement
| |
| | |
| :<math>|a_{1/2}|^2 + |a_{-1/2}|^2 \, = 1.</math>
| |
| | |
| 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:
| |
| | |
| :<math> \sum_{m=-j}^{j} a_m^* b_m = \sum_{m=-j}^{j} \left(\sum_{n=-j}^j U_{nm} a_n\right)^* \left(\sum_{k=-j}^j U_{km} b_k\right)</math>
| |
| :<math> \sum_{n=-j}^j \sum_{k=-j}^j U_{np}^* U_{kq} = \delta_{pq}.</math>
| |
| | |
| Mathematically speaking, these matrices furnish a unitary [[projective representation|projective]] [[group representation|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)]].<ref>
| |
| {{cite book
| |
| |author=B.C. Hall
| |
| |title=Quantum Theory for Mathematicians
| |
| |publisher=Springer
| |
| |pages=354–358
| |
| |year=2013 }}</ref> 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 <math>\hat{\theta}</math>, ''U'' can be written
| |
| | |
| :<math> U = e^{\frac{-i}{\hbar} \vec{\theta} \cdot \vec{S}},</math>
| |
| | |
| where <math>\vec{\theta} = \theta \hat{\theta}</math> and <math>\vec{S}</math> is the vector of [[#Spin operator|spin operators]].
| |
| | |
| <blockquote>
| |
| {{hidden
| |
| |(Click "show" at right to see a proof or "hide" to hide it.)
| |
| |2=
| |
| ----
| |
| Working in the coordinate system where <math>\hat{\theta} = \hat{z}</math>, we would like to show that ''S''<sub>x</sub> and ''S''<sub>y</sub> are rotated into each other by the angle ''θ''. Starting with ''S''<sub>x</sub>. Using units where ''ħ'' = 1:
| |
| : <math>
| |
| \begin{align}
| |
| S_x \rightarrow U^\dagger S_x U &{}= e^{i \theta S_z} S_x e^{-i \theta S_z} \\
| |
| &{} = S_x + (i \theta) [S_z, S_x] + \left(\frac{1}{2!}\right) (i \theta)^2 [S_z, [S_z, S_x]] + \left(\frac{1}{3!}\right) (i \theta)^3 [S_z, [S_z, [S_z, S_x]]] + \cdots\\
| |
| \end{align}
| |
| </math>
| |
| Using the [[#Spin operator|spin operator commutation relations]], we see that the commutators evaluate to i''S''<sub>y</sub> for the odd terms in the series, and to ''S''<sub>x</sub> for all of the even terms. Thus:
| |
| : <math>
| |
| \begin{align}
| |
| U^\dagger S_x U &{}= S_x \left[ 1 - \frac{\theta^2}{2!} + ... \right] - S_y \left[ \theta - \frac{\theta^3}{3!} \cdots \right] \\
| |
| &{} = S_x \cos\theta - S_y \sin\theta\\
| |
| \end{align}
| |
| </math>
| |
| as expected. Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension (i.e. for any principal spin quantum number ''s'').<ref>[http://books.google.com/books?id=w2a8QgAACAAJ ''Modern Quantum Mechanics'', by J. J. Sakurai, p159]</ref>
| |
| ----
| |
| }}
| |
| </blockquote>
| |
| | |
| A generic rotation in 3-dimensional space can be built by compounding operators of this type using [[Euler angles]]:
| |
| :<math> \mathcal{R}(\alpha,\beta,\gamma) = e^{-i\alpha S_x}e^{-i\beta S_y}e^{-i\gamma S_z}</math>
| |
| An irreducible representation of this group of operators is furnished by the [[Wigner D-matrix]]:
| |
| :<math> D^s_{m'm}(\alpha,\beta,\gamma) \equiv
| |
| \langle sm' | \mathcal{R}(\alpha,\beta,\gamma)| sm \rangle =
| |
| e^{-im'\alpha } d^s_{m'm}(\beta)e^{-i m\gamma},
| |
| </math>
| |
| where
| |
| :<math>d^s_{m'm}(\beta)= \langle sm' |e^{-i\beta s_y} | sm \rangle</math>
| |
| is [[Wigner D-matrix#Wigner (small) d-matrix|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
| |
| :<math> D^s_{m'm}(0,0,2\pi) = d^s_{m'm}(0) e^{-i m 2 \pi} = \delta_{m'm} (-1)^{2m}.</math>
| |
| 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)<sup>2''m''</sup> = −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 theorem]].
| |
| | |
| ===Spin and Lorentz transformations===
| |
| We could try the same approach to determine the behavior of spin under general [[Lorentz transformation]]s, but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations [[SO(3,1)]] is [[Compact group|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]] <math>\psi</math> with each particle. These spinors transform under Lorentz transformations according to the law
| |
| | |
| :<math>\psi' = \exp{\left(\frac{1}{8} \omega_{\mu\nu} [\gamma_{\mu}, \gamma_{\nu}]\right)} \psi</math>
| |
| | |
| where <math>\gamma_{\mu}</math> are [[gamma matrices]] and <math>\omega_{\mu\nu}</math> is an antisymmetric 4x4 matrix parametrizing the transformation. It can be shown that the scalar product
| |
| | |
| :<math>\langle\psi|\phi\rangle = \bar{\psi}\phi = \psi^{\dagger}\gamma_0\phi</math>
| |
| | |
| 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 matrix|Hermitian]]) Pauli matrices has two [[eigenvalues]], +1 and −1. The corresponding [[Normalisable wavefunction|normalized]] [[eigenvectors]] are:
| |
| | |
| :<math>
| |
| \begin{array}{lclc}
| |
| \psi_{x+}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{1}\end{pmatrix}, & \psi_{x-}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{-1}\end{pmatrix}, \\
| |
| \psi_{y+}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{i}\end{pmatrix}, & \psi_{y-}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{-i}\end{pmatrix}, \\
| |
| \psi_{z+}= & \begin{pmatrix}{1}\\{0}\end{pmatrix}, & \psi_{z-}= & \begin{pmatrix}{0}\\{1}\end{pmatrix}.
| |
| \end{array}
| |
| </math>
| |
| | |
| By the [[Postulates of quantum mechanics#Postulates of quantum mechanics|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 (''S<sub>x</sub>'', ''S<sub>y</sub>'' or ''S<sub>z</sub>'') on that axis, i.e. ''ħ''/2 or –''ħ''/2. The [[quantum state]] of a particle (with respect to spin), can be represented by a two component [[spinor]]:
| |
| | |
| :<math> \psi = \begin{pmatrix} {a+bi}\\{c+di}\end{pmatrix}.</math>
| |
| | |
| 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 ''ħ''/2 is just <math>\left\vert \langle \psi_{x+} \vert \psi \rangle \right\vert ^2</math>. Correspondingly, the probability that its spin will be measured as –''ħ''/2 is just <math>\left\vert \langle \psi_{x-} \vert \psi \rangle \right\vert ^2</math>. Following the measurement, the spin state of the particle will [[wavefunction collapse|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 <math>\left\vert \langle \psi_{x+} \vert \psi_{x+} \rangle \right\vert ^2 = 1 </math>, 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 ''u'' = (''u<sub>x</sub>'', ''u<sub>y</sub>'', ''u<sub>z</sub>'') be an arbitrary unit vector. Then the operator for spin in this direction is simply
| |
| :<math> S_u = \frac{\hbar}{2}(u_x\sigma_x + u_y\sigma_y + u_z\sigma_z)</math>.
| |
| The operator ''S<sub>u</sub>'' has eigenvalues of ±''ħ''/2, 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 (''u<sub>x</sub>'', ''u<sub>y</sub>'', ''u<sub>z</sub>'') direction (which works for all spin states except spin down where it will give 0/0), is:
| |
| | |
| :<math> \frac{1}{\sqrt{2+2u_z}}\begin{pmatrix} 1+u_z \\ u_x+iu_y \end{pmatrix}.</math>
| |
| | |
| The above spinor is obtained in the usual way by diagonalizing the <math>\sigma_u</math> 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 [[commutativity|commute]], 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:
| |
| | |
| :<math> \mid \langle \psi_{x\pm} \mid \psi_{y\pm} \rangle \mid ^ 2 = \mid \langle \psi_{x\pm} \mid \psi_{z\pm} \rangle \mid ^ 2 = \mid \langle \psi_{y\pm} \mid \psi_{z\pm} \rangle \mid ^ 2 = \frac{1}{2}. </math>
| |
| | |
| So when [[physicist]]s measure the spin of a particle along the ''x''-axis as, for example, ''ħ''/2, the particle's spin state [[wavefunction collapse|collapses]] into the eigenstate <math>\mid \psi_{x+} \rangle</math>. When we then subsequently measure the particle's spin along the y-axis, the spin state will now collapse into either <math>\mid \psi_{y+} \rangle</math> or <math>\mid \psi_{y-} \rangle</math>, each with probability 1/2. Let us say, in our example, that we measure –''ħ''/2. When we now return to measure the particle's spin along the x-axis again, the probabilities that we will measure ''ħ''/2 or –''ħ''/2 are each 1/2 (i.e. they are <math> \mid \langle \psi_{x+} \mid \psi_{y-} \rangle \mid ^ 2</math> and <math> \mid \langle \psi_{x-} \mid \psi_{y-} \rangle \mid ^ 2 </math> respectively). This implies that the 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 [[Parity (physics)|parity]] with "+" for even parity (wave function unchanged by spatial inversion) and "−" for odd parity (wave function negated by spatial inversion). For example, see the [[isotopes of bismuth]].
| |
| | |
| == Applications ==
| |
| Spin has important theoretical implications and practical applications. Well-established ''direct'' applications of spin include:
| |
| * [[Nuclear magnetic resonance]] spectroscopy in chemistry;
| |
| * [[Electron spin resonance]] spectroscopy in chemistry and physics;
| |
| * [[Magnetic resonance imaging]] (MRI) in medicine, which relies on proton spin density;
| |
| * [[Giant magnetoresistive effect|Giant magnetoresistive]] (GMR) drive head technology in modern [[hard disk]]s.
| |
| | |
| Electron spin plays an important role in [[magnetism]], with applications for instance in computer memories. The manipulation of ''nuclear spin'' by radiofrequency 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]]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]]. ''Photon spin'' is associated with the [[Polarization (waves)|polarization]] of light.
| |
| | |
| 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.<ref>{{cite journal| doi = 10.1063/1.102730| author = Datta. S and B. Das |title = Electronic analog of the electrooptic modulator|journal = Applied Physics Letters| volume = 56| pages = 665–667|year = 1990| issue = 7|bibcode = 1990ApPhL..56..665D }}</ref> Electronics based on spin transistors is called [[spintronics]], which includes the manipulation of spins in semiconductor devices.
| |
| | |
| There are many ''indirect'' applications and manifestations of spin and the associated [[Pauli exclusion principle]], starting with the [[periodic table]] of chemistry.
| |
| | |
| ==History==
| |
| Spin was first discovered in the context of the [[emission spectrum]] of [[alkali metal]]s. In 1924 [[Wolfgang Ernst Pauli|Wolfgang Pauli]] introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost [[electron shell|shell]]. This allowed him to formulate the [[Pauli exclusion principle]], stating that no two electrons can share the same [[quantum state]] at the same time.
| |
| [[File:Wolfgang Pauli young.jpg|thumb|160px|Wolfgang Pauli]]
| |
| The physical interpretation of Pauli's "degree of freedom" was initially unknown. [[Ralph Kronig]], one of [[Alfred Landé|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 ''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 precession]]).
| |
| | |
| Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of [[quantum mechanics]] invented by [[Erwin Schrödinger|Schrödinger]] and [[Werner Heisenberg|Heisenberg]]. He pioneered the use of [[Pauli matrices]] as a [[group representation|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 the ''[[spin-statistics theorem]]'', which states that [[fermion]]s have half-integer spin and [[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.<ref>{{cite journal
| |
| |author=B. Friedrich, D. Herschbach
| |
| |title=Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics
| |
| |journal=[[Physics Today]]
| |
| |volume=56 |issue=12 |page=53
| |
| |year=2003
| |
| |doi=10.1063/1.1650229
| |
| |bibcode = 2003PhT....56l..53F }}</ref>
| |
| | |
| == See also ==
| |
| <div style="-moz-column-count:3; column-count:3;">
| |
| * [[Spinor]]
| |
| * [[Stern–Gerlach experiment]]
| |
| * [[Einstein–de Haas effect]]
| |
| * [[Spin-orbital]]
| |
| * [[Angular momentum]]
| |
| * [[Chirality (physics)]]
| |
| * [[Dynamic nuclear polarisation]]
| |
| * [[Helicity (particle physics)]]
| |
| * [[Pauli equation]]
| |
| * [[Pauli–Lubanski pseudovector]]
| |
| * [[Rarita–Schwinger equation]]
| |
| * [[Representation theory of SU(2)]]
| |
| * [[Spin-½]]
| |
| * [[Spin-flip]]
| |
| * [[Spin isomers of hydrogen]]
| |
| * [[Spin magnetic moment]]
| |
| * [[Spin quantum number]]
| |
| * [[Spin-statistics theorem]]
| |
| * [[Spin tensor]]
| |
| * [[Spin wave]]
| |
| * [[Spin Engineering]]
| |
| * [[Spintronics]]
| |
| * [[Yrast]]
| |
| * [[Zitterbewegung]]
| |
| </div>
| |
| | |
| ==Notes==
| |
| {{reflist|group=note}}
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{cite book|last1=Cohen-Tannoudji|first1= Claude|last2= Diu|first2= Bernard|last3= Laloë|first3=Franck|year=2006|title=Quantum Mechanics|publisher= John Wiley & Sons|isbn=978-0-471-56952-7|edition=2 volume set}}
| |
| * {{cite book|first1=E. U.|last1= Condon |first2= G. H. |last2=Shortley |year=1935|title=The Theory of Atomic Spectra|publisher= Cambridge University Press| isbn= 0-521-09209-4|chapter=Especially Chapter 3}}
| |
| * {{cite book|last=Edmonds|first= A. R. |year=1957|title=Angular Momentum in Quantum Mechanics|publisher= Princeton University Press|isbn= 0-691-07912-9}}
| |
| * {{cite book|last=Jackson|first= John David |year=1998|title=Classical Electrodynamics|edition=3rd |publisher=John Wiley & Sons| isbn =978-0-471-30932-1}}
| |
| * {{cite book | last1=Serway|first1= Raymond A.|last2= Jewett|first2= John W. | title=Physics for Scientists and Engineers|edition=6th | publisher=Brooks/Cole | year=2004 | isbn=0-534-40842-7}}
| |
| * {{cite book|last=Thompson|first= William J. |year=1994|title=Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems|publisher=Wiley| isbn =0-471-55264-X}}
| |
| * {{cite book | last=Tipler|first= Paul | title=Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics|edition= 5th | publisher=W. H. Freeman | year=2004 | isbn=0-7167-0809-4}}
| |
| | |
| ==External links==
| |
| *"[http://www.sciam.com/article.cfm?articleID=0007A735-759A-1CDD-B4A8809EC588EEDF Spintronics. Feature Article]" in ''[[Scientific American]]'', June 2002.
| |
| *[http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html Goudsmit on the discovery of electron spin.]
| |
| *''[[Nature]]'': "[http://www.nature.com/milestones/milespin/index.html Milestones in 'spin' since 1896.]"
| |
| * [http://nanohub.org/resources/6025 ECE 495N Lecture 36: Spin] Online lecture by S. Datta
| |
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| {{Physics operator}}
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| [[Category:Concepts in physics]]
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| [[Category:Rotational symmetry]]
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| [[Category:Quantum field theory]]
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| [[Category:Spintronics]]
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| [[ro:Spin (fizică)]]
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