The existence ofelectronspin angular momentum isinferred from experiments, such as theStern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum.[3] The relativisticspin–statistics theorem connects electron spin quantization to thePauli 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 aspinor orbispinor for other particles such as electrons. Spinors and bispinors behave similarly tovectors: 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 aspin quantum number.[2]: 183–184
TheSI units of spin are the same as classical angular momentum (i.e.,N·m·s,J·s, orkg·m2·s−1). In quantum mechanics, angular momentum and spin angular momentum take discrete values proportional to thePlanck constant. In practice, spin is usually given as adimensionless spin quantum number by dividing the spin angular momentum by thereduced Planck constantħ. Often, the "spin quantum number" is simply called "spin".
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 theelectron radius: the required rotation speed exceeds the speed of light.[4] In theStandard Model, the fundamental particles are all considered "point-like": 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.
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 fundamentalPauli exclusion principle, a proof now called thespin-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]
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 applyingFrederik 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 inquantum numbers below.
As the name suggests, spin was originally conceived as the rotation of a particle around some axis. Historicallyorbital angular momentum related to particle orbits.[12]: 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 for rotation of angleθ around the axis parallel to the spinS. This is equivalent to the quantum-mechanical interpretation ofmomentum as phase dependence in the position, and oforbital angular momentum as phase dependence in the angular position.
For fermions, the picture is less clear: From theEhrenfest theorem, theangular velocity is equal to the derivative of theHamiltonian to itsconjugate momentum, which is the totalangular momentum operatorJ =L +S . Therefore, if the HamiltonianH has any dependence on the spinS, then ∂ H/ ∂ S must be non-zero; consequently, forclassical 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 / 2ℏ , and one might decide that since it cannot change, nopartial (∂) can exist. Therefore it is a matter of interpretation whether the Hamiltonian must include such a term, and whether this aspect ofclassical mechanics extends intoquantum mechanics (any particle's intrinsic spin angular momentum,S, is aquantum number arising from a "spinor" in the mathematical solution to theDirac equation, rather than being a more nearly physical quantity, likeorbital angular momentumL). Nevertheless, spin appears in theDirac equation, and thus the relativistic Hamiltonian of the electron, treated as aDirac field, can be interpreted as including a dependence in the spinS.[9]
The conventional definition of thespin quantum number iss =n/2, wheren can be anynon-negative integer. Hence the allowed values ofs are 0,1/2, 1,3/2, 2, etc. The value ofs for anelementary particle depends only on the type of particle and cannot be altered in any known way (in contrast to thespin direction described below). The spin angular momentumS of any physical system isquantized. The allowed values ofS arewhereh is thePlanck constant, and is the reduced Planck constant. In contrast,orbital angular momentum can only take on integer values ofs; i.e., even-numbered values ofn.
Those particles with half-integer spins, such as1/2,3/2,5/2, are known asfermions, while those particles with integer spins, such as 0, 1, 2, are known asbosons. The two families of particles obey different rules andbroadly have different roles in the world around us. A key distinction between the two families is that fermions obey thePauli 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 ofFermi–Dirac statistics. In contrast, bosons obey the rules ofBose–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, ahelium-4 atom in the ground state has spin 0 and behaves like a boson, even though thequarks and electrons which make it up are all fermions.
This has some profound consequences:
Quarks andleptons (includingelectrons andneutrinos), which make up what is classically known asmatter, are all fermions withspin 1/2. The common idea that "matter takes up space" actually comes from thePauli 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 ofpressure (sometimes known asdegeneracy 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 ascarrying forces are all bosons with spin 1. They include thephoton, which carries theelectromagnetic force, thegluon (strong force), and theW and Z bosons (weak force). The ability of bosons to occupy the same quantum state is used in thelaser, which aligns many photons having the same quantum number (the same direction and frequency),superfluidliquid helium resulting from helium-4 atoms being bosons, andsuperconductivity, wherepairs 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 thegraviton (predicted to exist by somequantum gravity theories) with spin 2, and theHiggs boson (explainingelectroweak symmetry breaking) with spin 0. Since 2013, the Higgs boson with spin 0 has been considered proven to exist.[13] It is the firstscalar elementary particle (spin 0) known to exist in nature.
Atomic nuclei havenuclear spin which may be either half-integer or integer, so that the nuclei may be either fermions or bosons.
Thespin–statistics theorem splits particles into two groups:bosons andfermions, where bosons obeyBose–Einstein statistics, and fermions obeyFermi–Dirac statistics (and therefore thePauli exclusion principle). Specifically, the theorem requires that particles with half-integer spins obey thePauli 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 byWolfgang Pauli in 1940; it relies on both quantum mechanics and the theory ofspecial relativity. Pauli described this connection between spin and statistics as "one of the most important applications of the special relativity theory".[14]
Schematic diagram depicting the spin of the neutron as the black arrow and magnetic field lines associated with theneutron 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.
The intrinsic magnetic momentμ of aspin-1/2 particle with chargeq, massm, and spin angular momentumS is[15]
where thedimensionless quantitygs is called the sping-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 anonzero magnetic moment. One of the triumphs of the theory ofquantum electrodynamics is its accurate prediction of the electrong-factor, which has been experimentally determined to have the value−2.00231930436092(36), with the digits in parentheses denotingmeasurement uncertainty in the last two digits at onestandard deviation.[16] The value of 2 arises from theDirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties; and thedeviation from−2 arises from the electron's interaction with the surrounding quantum fields, including its own electromagnetic field andvirtual particles.[17]
Composite particles also possess magnetic moments associated with their spin. In particular, theneutron 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 ofquarks, which are electrically charged particles. Themagnetic 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 extendedStandard Model that takes into account non-zero neutrino masses predicts neutrino magnetic moments of:[18][19][20]
where theμν are the neutrino magnetic moments,mν are the neutrino masses, andμB is theBohr 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 10−14μB 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 about1 eV/c2,fine-tuning would be necessary in order to prevent large contributions to the neutrino mass via radiative corrections.[21] The measurement of neutrino magnetic moments is an active area of research. Experimental results have put the neutrino magnetic moment at less than1.2×10−10 times the electron's magnetic moment.
On the other hand, elementary particles with spin but without electric charge, such as thephoton andZ boson, do not have a magnetic moment.
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 theirCurie temperature, however, exhibitmagnetic 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.
Inparamagnetic materials, the magnetic dipole moments of individual atoms will partially align with an externally applied magnetic field. Indiamagnetic 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 incondensed matter physics. For instance, theIsing model describes spins (dipoles) that have only two possible states, up and down, whereas in theHeisenberg 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 ofphase transitions.
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 theaxis of rotation of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that thecomponent of angular momentum for a spin-s particle measured along any direction can only take on the values[22]
whereSi is the spin component along thei-th axis (eitherx,y, orz),si is the spin projection quantum number along thei-th axis, ands is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is thez axis:
whereSz is the spin component along thez axis,sz is the spin projection quantum number along thez axis.
One can see that there are2s + 1 possible values ofsz. The number "2s + 1" is themultiplicity of the spin system. For example, there are only two possible values for aspin-1/2 particle:sz = +1/2 andsz = −1/2. These correspond toquantum states 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-3/2 particle, like adelta baryon, the possible values are +3/2, +1/2, −1/2, −3/2.
For a givenquantum state, one could think of a spin vector whose components are theexpectation values of the spin components along each axis, i.e.,. This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of theaxis of rotation. It turns out that the spin vector is not very useful in actual quantum-mechanical calculations, because it cannot be measured directly:sx,sy andsz cannot possess simultaneous definite values, because of a quantumuncertainty 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 aStern–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 probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180°—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%.
As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum-mechanical spin can exhibit phenomena analogous to classicalgyroscopic effects. For example, one can exert a kind of "torque" on an electron by putting it in amagnetic field (the field acts upon the electron's intrinsicmagnetic dipole moment—see the following section). The result is that the spin vector undergoesprecession, just like a classical gyroscope. This phenomenon is known aselectron spin resonance (ESR). The equivalent behaviour of protons in atomic nuclei is used innuclear magnetic resonance (NMR) spectroscopy and imaging.
Mathematically, quantum-mechanical spin states are described by vector-like objects known asspinors. There are subtle differences between the behavior of spinors and vectors undercoordinate rotations. For example, rotating a spin-1/2 particle by 360° does not bring it back to the same quantum state, but to the state with the opposite quantumphase; this is detectable, in principle, withinterference experiments. To return the particle to its exact original state, one needs a 720° rotation. (Theplate trick andMöbius strip give non-quantum analogies.) A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180° can bring it back to the same quantum state, and a spin-4 particle should be rotated 90° to bring it back to the same quantum state. The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated 180°, and a spin-0 particle can be imagined as sphere, which looks the same after whatever angle it is turned through.
But unlike orbital angular momentum, the eigenvectors are notspherical harmonics. They are not functions ofθ andφ. There is also no reason to exclude half-integer values ofs andms.
All quantum-mechanical particles possess an intrinsic spin (though this value may be equal to zero). The projection of the spin on any axis is quantized in units of thereduced Planck constant, such that the state function of the particle is, say, not, but, where can take only the values of the following discrete set:
One distinguishesbosons (integer spin) andfermions (half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.
ThePauli exclusion principle states that thewavefunction for a system ofN identical particles having spins must change upon interchanges of any two of theN particles as
Thus, forbosons the prefactor(−1)2s will reduce to +1, forfermions to −1. This permutation postulate forN-particle state functions has most important consequences in daily life, e.g. theperiodic table of the chemical elements.
As described above, quantum mechanics states thatcomponents 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 numbersa±1/2, giving amplitudes of finding it with projection of angular momentum equal to+ħ/2 and−ħ/2, satisfying the requirement
For a generic particle with spins, we would need2s + 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 is clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to phase) to the matrix representing rotation AB. Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices:
Mathematically speaking, these matrices furnish a unitaryprojective representation of therotation group SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which isSU(2).[24] There is onen-dimensional irreducible representation of SU(2) for each dimension, though this representation isn-dimensional real for oddn andn-dimensional complex for evenn (hence of real dimension2n). For a rotation by angleθ in the plane with normal vector,where, andS is the vector ofspin operators.
Proof
Working in the coordinate system where, we would like to show thatSx andSy are rotated into each other by the angleθ. Starting withSx. Using units whereħ = 1:
Using thespin operator commutation relations, we see that the commutators evaluate toi Sy for the odd terms in the series, and toSx for all of the even terms. Thus: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 numbers)[25]: 164
A generic rotation in 3-dimensional space can be built by compounding operators of this type usingEuler angles:
An irreducible representation of this group of operators is furnished by theWigner D-matrix:whereisWigner's small d-matrix. Note that forγ = 2π andα =β = 0; i.e., a full rotation about thez axis, the Wigner D-matrix elements become
Recalling that a generic spin state can be written as a superposition of states with definitem, we see that ifs is an integer, the values ofm are all integers, and this matrix corresponds to the identity operator. However, ifs is a half-integer, the values ofm are also all half-integers, giving(−1)2m = −1 for allm, and hence upon rotation by 2π the state picks up a minus sign. This fact is a crucial element of the proof of thespin–statistics theorem.
We could try the same approach to determine the behavior of spin under generalLorentz transformations, but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformationsSO(3,1) isnon-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-componentDirac spinorψ with each particle. These spinors transform under Lorentz transformations according to the lawwhereγν aregamma matrices, andωμν is an antisymmetric 4 × 4 matrix parametrizing the transformation. It can be shown that the scalar productis preserved. It is not, however, positive-definite, so the representation is not unitary.
(Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign. In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such asSymPy; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.)
By thepostulates of quantum mechanics, an experiment designed to measure the electron spin on thex,y, orz axis can only yield an eigenvalue of the corresponding spin operator (Sx,Sy orSz) on that axis, i.e.ħ/2 or–ħ/2. Thequantum state of a particle (with respect to spin), can be represented by a two-componentspinor:
When the spin of this particle is measured with respect to a given axis (in this example, thex axis), the probability that its spin will be measured asħ/2 is just. Correspondingly, the probability that its spin will be measured as–ħ/2 is just. Following the measurement, the spin state of the particlecollapses into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to have a given eigenvalue, all measurements will yield the same eigenvalue (since, etc.), provided that no measurements of the spin are made along other axes.
The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Letu = (ux,uy,uz) be an arbitrary unit vector. Then the operator for spin in this direction is simply
The operatorSu 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 thedot product of the direction with a vector of the three operators for the threex-,y-,z-axis directions.
A normalized spinor for spin-1/2 in the(ux,uy,uz) direction (which works for all spin states except spin down, where it will give0/0) is
The above spinor is obtained in the usual way by diagonalizing theσu 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.
Since the Pauli matrices do notcommute, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along thex axis, and we then measure the spin along they axis, we have invalidated our previous knowledge of thex axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that
So whenphysicists measure the spin of a particle along thex axis as, for example,ħ/2, the particle's spin statecollapses into the eigenstate. When we then subsequently measure the particle's spin along they axis, the spin state will now collapse into either or, each with probability1/2. Let us say, in our example, that we measure−ħ/2. When we now return to measure the particle's spin along thex axis again, the probabilities that we will measureħ/2 or−ħ/2 are each1/2 (i.e. they are and respectively). This implies that the original measurement of the spin along thex axis is no longer valid, since the spin along thex axis will now be measured to have either eigenvalue with equal probability.
The spin-1/2 operatorS =ħ/2σ forms thefundamental representation ofSU(2). By takingKronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resultingspin operators for higher-spin systems in three spatial dimensions can be calculated for arbitrarily larges using thisspin operator andladder operators. For example, taking the Kronecker product of two spin-1/2 yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 (triplet states) and a 1-dimensional spin-0 representation (singlet state).
The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis:
For spin 1 they are
For spin3/2 they are
For spin5/2 they are
The generalization of these matrices for arbitrary spins iswhere indices are integer numbers such that
Also useful in thequantum mechanics of multiparticle systems, the generalPauli groupGn is defined to consist of alln-foldtensor products of Pauli matrices.
In tables of thespin quantum numbers for nuclei or particles, the spin is often followed by a "+" or "−".[citation needed] This refers to theparity with "+" for even parity (wave function unchanged by spatial inversion) and "−" for odd parity (wave function negated by spatial inversion). For example, see theisotopes of bismuth, in which the list of isotopes includes the columnnuclear spin and parity. For Bi-209, the longest-lived isotope, the entry 9/2– means that the nuclear spin is 9/2 and the parity is odd.
The nuclear spin of atoms can be determined by sophisticated improvements to the originalStern-Gerlach experiment.[27] A single-energy (monochromatic)molecular beam of atoms in an inhomogeneous magnetic field will split into beams representing each possible spin quantum state. For an atom with electronic spinS and nuclear spinI, there are(2S + 1)(2I + 1) spin states. For example, neutralNa atoms, which haveS = 1/2, were passed through a series of inhomogeneous magnetic fields that selected one of the two electronic spin states and separated the nuclear spin states, from which four beams were observed. Thus, the nuclear spin for23Na atoms was found to beI = 3/2.[28][29]
The spin ofpions, a type of elementary particle, was determined by the principle ofdetailed balance applied to those collisions of protons that produced charged pions anddeuterium.The known spin values for protons and deuterium allows analysis of the collision cross-section to show that has spin. A different approach is needed for neutral pions. In that case the decay produced twogamma ray photons with spin one:This result supplemented with additional analysis leads to the conclusion that the neutral pion also has spin zero.[30]: 66
Electron spin plays an important role inmagnetism, with applications for instance in computer memories. The manipulation ofnuclear spin by radio-frequency waves (nuclear magnetic resonance) is important in chemical spectroscopy and medical imaging.
An emerging application of spin is as a binary information carrier inspin transistors. The original concept, proposed in 1990, is known as Datta–Dasspin transistor.[31] Electronics based on spin transistors are referred to asspintronics. The manipulation of spin indilute magnetic semiconductor materials, such as metal-dopedZnO orTiO2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.[32]
Spin was first discovered in the context of theemission spectrum ofalkali metals. Starting around 1910, many experiments on different atoms produced a collection of relationships involvingquantum numbers for atomic energy levels partially summarized inBohr's model for the atom[33]: 106 Transitions between levels obeyedselection rules and the rules were known to be correlated with even or oddatomic number. Additional information was known from changes to atomic spectra observed in strong magnetic fields, known as theZeeman effect. In 1924,Wolfgang Pauli used this large collection of empirical observations to propose a new degree of freedom,[7] introducing what he called a "two-valuedness not describable classically"[34] associated with the electron in the outermostshell.
The physical interpretation of Pauli's "degree of freedom" was initially unknown.Ralph Kronig, one ofAlfred 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 thespeed of light in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate thetheory of relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea.[35]
Pauli was especially unconvinced and continued to pursue his two-valued degree of freedom. This allowed him to formulate thePauli exclusion principle, stating that no two electrons can have the samequantum state in the same quantum system.
Fortunately, by February 1926,Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results for thefine structure in the hydrogen spectrum and calculations based on Uhlenbeck and Goudsmit's (and Kronig's unpublished) model.[2]: 385 This discrepancy was due to a relativistic effect, the difference between the electron's rotating rest frame and the nuclear rest frame; the effect is now known asThomas precession.[7] Thomas' result convinced Pauli that electron spin was the correct interpretation of his two-valued degree of freedom, while he continued to insist that the classical rotating charge model is invalid.[34][6]
Pauli's theory of spin was non-relativistic. In 1928,Paul Dirac published his relativistic electron equation, using a four-component spinor (known as a "Dirac spinor") for the electron wave-function. In 1940, Pauli proved thespin–statistics theorem, which states thatfermions have half-integer spin, andbosons have integer spin.[7]
In retrospect, the first direct experimental evidence of the electron spin was theStern–Gerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1927.[38]The original interpretation assumed the two spots observed in the experiment were due to quantizedorbital angular momentum. However, in 1927 Ronald Fraser showed that Sodium atoms are isotropic with no orbital angular momentum and suggested that the observed magnetic properties were due to electron spin.[39] In the same year, Phipps and Taylor applied the Stern-Gerlach technique to hydrogen atoms; the ground state of hydrogen has zero angular momentum but the measurements again showed two peaks.[40]Once the quantum theory became established, it became clear that the original interpretation could not have been correct: the possible values of orbital angular momentum along one axis is always an odd number, unlike the observations. Hydrogen atoms have a single electron with two spin states giving the two spots observed; silver atoms have closed shells which do not contribute to the magnetic moment and only the unmatched outer electron's spin responds to the field.
^abcdeFröhlich, Jürg (2009). "Spin, or actually: Spin and Quantum Statistics". In Duplantier, Bertrand; Raimond, Jean-Michel; Rivasseau, Vincent (eds.).The Spin. Progress in Mathematical Physics, vol 55. Basel: Birkhäuser Basel. pp. 1–60.arXiv:0801.2724.doi:10.1007/978-3-7643-8799-0_1.ISBN978-3-7643-8798-3.
^Whittaker, Edmund, Sir (1989).A History of the Theories of Aether and Electricity. Vol. 2. Courier Dover Publications. p. 87, 131.ISBN0-486-26126-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
^Feynman, R. P. (1985). "Electrons and their interactions".QED: The Strange Theory of Light and Matter.Princeton, New Jersey:Princeton University Press. p. 115.ISBN978-0-691-08388-9.After some years, it was discovered that this value [−1/2g] was not exactly 1, but slightly more – something like 1.00116. This correction was worked out for the first time in 1948 by Schwinger asj ×j divided by2π [sic] [wherej is the square root of thefine-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.
^Whittaker, Edmund T. (1989).A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 (Repr ed.). New York: Dover Publ.ISBN978-0-486-26126-3.
^Uhlenbeck, G., G.; Goudsmit, S. (November 1925). "Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons".Die Naturwissenschaften (in German).13 (47):953–954.doi:10.1007/bf01558878.ISSN0028-1042.S2CID32211960.
Thompson, William J. (1994).Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. Wiley.ISBN978-0-471-55264-2.
Tipler, Paul (2004).Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman.ISBN978-0-7167-0809-4.
