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rtened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikiquote wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia intrinsic quantum property of particles this article is about the concept in quantum mechanics for the concept in classical mechanics see spin geometry spin is an intrinsic form of angular momentum carried by elementary particles and thus by composite particles such as hadrons atomic nuclei and atoms 1 2 183 184 spin is quantized and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory the existence of electron spin angular momentum is inferred from experiments such as the stern gerlach experiment in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum 3 the relativistic spin statistics theorem connects electron spin quantization to the pauli exclusion principle observations of exclusion imply half integer spin and observations of half integer spin imply exclusion spin is described mathematically as a vector for some particles such as photons and as a spinor for other particles such as electrons spinors 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 2 183 184 the si units of spin are the same as classical angular momentum i e n m s j s or kg m 2 s 1 in quantum mechanics angular momentum and spin angular momentum take discrete values proportional to the planck constant in practice spin is usually given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced planck constant ħ often the spin quantum number is simply called spin models edit rotating charged mass edit the earliest models for electron spin imagined a rotating charged mass but this model fails when examined in detail the required space distribution does not match limits on the electron radius and the required rotation speed exceeds the speed of light 4 in the standard model the fundamental particles are all considered point like and they have their effects through the field that surrounds them 5 any model for spin based on mass rotation would need to be consistent with that model pauli s classically non describable two valuedness edit wolfgang pauli a central figure in the history of quantum spin initially rejected any idea that the degree of freedom he introduced to explain experimental observations was related to rotation he called it classically non describable two valuedness later he allowed that it is related to angular momentum but insisted on considering spin an abstract property 6 this approach allowed pauli to develop a proof of his fundamental pauli exclusion principle a proof now called the spin statistics theorem 7 in retrospect this insistence and the style of his proof initiated the modern particle physics era where abstract quantum properties derived from symmetry properties dominate concrete interpretation became secondary and optional 6 circulation of classical fields edit the first classical model for spin proposed a small rigid particle rotating about an axis as ordinary use of the word may suggest angular momentum can be computed from a classical field as well 8 9 63 by applying frederik belinfante s approach to calculating the angular momentum of a field hans c ohanian showed that spin is essentially a wave property generated by a circulating flow of charge in the wave field of the electron 10 this same concept of spin can be applied to gravity waves in water spin is generated by subwavelength circular motion of water particles 11 unlike classical wavefield circulation which allows continuous values of angular momentum quantum wavefields allow only discrete values 10 consequently energy transfer to or from spin states always occurs in fixed quantum steps only a few steps are allowed for many qualitative purposes the complexity of the spin quantum wavefields can be ignored and the system properties can be discussed in terms of integer or half integer spin models as discussed in quantum numbers below in bohmian mechanics edit spin can be understood differently depending on the interpretations of quantum mechanics in the de broglie bohm interpretation particles have definitive trajectories but their motion is driven by the wave function or pilot wave in this interpretation the spin is a property of the pilot wave and not of the particle itself 12 dirac s relativistic electron edit relativistic calculations of spin properties for electrons requires the dirac equation 7 relation to orbital angular momentum edit as the name suggests spin was originally conceived as the rotation of a particle around some axis historically orbital angular momentum related to particle orbits 13 131 while the names based on mechanical models have survived the physical explanation has not quantization fundamentally alters the character of both spin and orbital angular momentum since elementary particles are point like self rotation is not well defined for them however spin implies that the phase of the particle depends on the angle as e i s θ displaystyle e is theta 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 for fermions the picture is less clear from the ehrenfest theorem the angular velocity is equal to the derivative of the hamiltonian to its conjugate momentum which is the total angular momentum operator j l s therefore if the hamiltonian h has any dependence on the spin s then h s must be non zero consequently for classical mechanics the existence of spin in the hamiltonian will produce an actual angular velocity and hence an actual physical rotation that is a change in the phase angle θ over time however whether this holds true for free electrons is ambiguous since for an electron s ² is a constant 1 2 ℏ and one might decide that since it cannot change no partial can exist therefore it is a matter of interpretation whether the hamiltonian must include such a term and whether this aspect of classical mechanics extends into quantum mechanics any particle s intrinsic spin angular momentum s is a quantum number arising from a spinor in the mathematical solution to the dirac equation rather than being a more nearly physical quantity like orbital angular momentum l 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 9 quantum number edit main article spin quantum number spin obeys the mathematical laws of angular momentum quantization the specific properties of spin angular momenta include spin quantum numbers may take either half integer or integer values although the direction of its spin can be changed the magnitude of the spin of an elementary particle cannot be changed the spin of a charged particle is associated with a magnetic dipole moment with a g factor that differs from 1 in the classical context this would imply the internal charge and mass distributions differing for a rotating nonconductive object 4 the conventional definition of the spin quantum number 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 quantized the allowed values of s are s ℏ s s 1 h 2 π n 2 n 2 2 h 4 π n n 2 displaystyle s hbar sqrt s s 1 frac h 2 pi sqrt frac n 2 frac n 2 2 frac h 4 pi sqrt n n 2 where h is the planck constant and ℏ h 2 π textstyle hbar frac h 2 pi is the reduced planck constant in contrast orbital angular momentum can only take on integer values of s i e even numbered values of n fermions and bosons edit those particles with half integer spins such as 1 2 3 2 5 2 are known as fermions while those particles with integer spins such as 0 1 2 are known as bosons the two families of particles obey different rules and broadly have different roles in the world around us a key distinction between the two families is that fermions obey the pauli exclusion principle that is there cannot be two identical fermions simultaneously having the same quantum numbers 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 atom in the ground state has spin 0 and behaves like a boson even though the quarks and electrons which make it up are all fermions this has some profound consequences quarks and leptons 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 from being in the same quantum state further compaction would require electrons to occupy the same energy states and therefore a kind of pressure sometimes known as degeneracy pressure of electrons acts to resist the fermions being overly close elementary fermions with other spins 3 2 5 2 etc are not known to exist elementary particles which are thought of as carrying forces are all bosons with spin 1 they include the photon which carries the electromagnetic force the gluon strong force 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 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 14 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 edit main article spin statistics theorem the spin statistics theorem splits particles into two groups bosons and fermions where bosons obey bose einstein statistics and fermions obey fermi dirac statistics and therefore the pauli exclusion principle specifically the theorem requires that particles with half integer spins obey the pauli exclusion principle while particles with integer spin do not as an example electrons have half integer spin and are fermions that obey the pauli exclusion principle while photons have integer spin and do not the theorem was derived by wolfgang pauli in 1940 it relies on both quantum mechanics and the theory of special relativity pauli described this connection between spin and statistics as one of the most important applications of the special relativity theory 15 magnetic moments edit schematic diagram depicting the spin of the neutron as the black arrow and magnetic field lines associated with the neutron magnetic moment the neutron has a negative magnetic moment while the spin of the neutron is upward in this diagram the magnetic field lines at the center of the dipole are downward particles with spin can possess a magnetic dipole moment just like a rotating electrically charged body in classical electrodynamics these magnetic moments can be experimentally observed in several ways e g by the deflection of particles by inhomogeneous magnetic fields in a stern gerlach experiment or by measuring the magnetic fields generated by the particles themselves the intrinsic magnetic moment μ of a spin 1 2 particle with charge q mass m and spin angular momentum s is 16 μ g s q 2 m s displaystyle boldsymbol mu frac g_ text s q 2m mathbf s where the dimensionless quantity g s is called the spin g factor for exclusively orbital rotations it would be 1 assuming that the mass and the charge occupy spheres of equal radius the electron being a charged elementary particle possesses a nonzero magnetic moment one of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g factor which has been experimentally determined to have the value 2 002 319 304 360 92 36 with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation 17 the value of 2 arises from the dirac equation a fundamental equation connecting the electron s spin with its electromagnetic properties and the deviation from 2 arises from the electron s interaction with the surrounding quantum fields including its own electromagnetic field and virtual particles 18 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 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 19 20 21 μ ν 3 10 19 μ b m ν c 2 ev displaystyle mu _ nu approx 3 times 10 19 mu _ text b frac m_ nu c 2 text ev where the μ ν are the neutrino magnetic moments m ν are the neutrino masses and μ b is the bohr magneton new physics above the electroweak scale could however lead to significantly higher neutrino magnetic moments it can be shown in a model independent way that neutrino magnetic moments larger than about 10 14 μ b are unnatural because they would also lead to large radiative contributions to the neutrino mass since the neutrino masses are known to be at most about 1 ev c 2 fine tuning would be necessary ...
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