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Spin (physics)

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**spin**is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.BOOK, Merzbacher, Eugen, Quantum Mechanics, 3rd, 1998, 372–3, BOOK, Griffiths, David, Introduction to Quantum Mechanics, 2nd, 2005, 183–4, Spin is one of two types of angular momentum in quantum mechanics, the other being

*orbital angular momentum*. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies."Angular Momentum Operator Algebra", class notes by Michael Fowler

*A modern approach to quantum mechanics*, by Townsend, p. 31 and p. 80 The existence of spin angular momentum is inferred from experiments, such as the Sternâ€“Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum.BOOK, Eisberg, Robert, Resnick, Robert, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd, 1985, 272–3, In some ways, spin is like a 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.The SI unit of spin is the (NÂ·mÂ·s) or (kgÂ·m2Â·sâˆ’1), just as with classical angular momentum. In practice, spin is given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant {{mvar|Ä§}}, which has the same units of angular momentum, although this is not the full computation of this value. Very often, the "spin quantum number" is simply called "spin", leaving its meaning as the unitless "spin quantum number" to be inferred from context.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 elements.Wolfgang Pauli in 1924 was the first to propose a doubling of electron states due to a two-valued non-classical "hidden rotation".BOOK, Pais, Abraham, 1991, Niels Bohr's Times, Oxford, Clarendon Press, 201, 978-0-19-852049-8, Abraham Pais, In 1925, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the old quantum theory of Bohr and Sommerfeld.BOOK, Pais, Abraham, 1991, Niels Bohr's Times, Oxford, Clarendon Press, 241â€“244, 978-0-19-852049-8, Abraham Pais, Ralph Kronig anticipated the Uhlenbeck-Goudsmit model in discussion with Hendrik Kramers several months earlier in Copenhagen, but did not publish. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it.

## Quantum number

As the name suggests, spin was originally conceived as the rotation of a particle around some axis. This picture is correct so far as spin obeys the same mathematical laws as quantized angular momenta do. On the other hand, spin has some peculiar properties that distinguish it from orbital angular momenta:- Spin quantum numbers may take half-integer values.
- Although the direction of its spin can be changed, an elementary particle cannot be made to spin faster or slower.
- The spin of a charged particle is associated with a magnetic dipole moment with a {{mvar|g}}-factor differing from 1. This could only occur classically if the internal charge of the particle were distributed differently from its mass.

**spin quantum number**, {{mvar|s}}, is {{math|

*s*{{=}} {{sfrac|

*n*|2}}}}, where {{mvar|n}} can be any non-negative integer. Hence the allowed values of {{mvar|s}} are 0, {{sfrac|1|2}}, 1, {{sfrac|3|2}}, 2, etc. The value of {{mvar|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, {{mvar|S}}, of any physical system is quantized. The allowed values of {{mvar|S}} are

S = hbar , sqrt{s (s+1)}=frac{h}{4pi} , sqrt{n(n+2)},

where {{mvar|h}} is the Planck constant and {{math| hbar {{=}} {{sfrac|*h*|2Ï€}}}} is the reduced Planck constant. In contrast, orbital angular momentum can only take on integer values of {{mvar|s}}; i.e., even-numbered values of {{mvar|n}}.

### Fermions and bosons

Those particles with half-integer spins, such as {{sfrac|1|2}}, {{sfrac|3|2}}, {{sfrac|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). In contrast, bosons obey the rules of Boseâ€“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 their component particles. For example, a helium 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 {{sfrac|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 degeneracy pressure of electrons) acts to resist the fermions being overly close.

Elementary fermions with other spins ({{sfrac|3|2}}, {{sfrac|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.Information about Higgs Boson in CERN's official website. It is the first scalar elementary particle (spin 0) known to exist in nature.

### Spin-statistics theorem

The spin-statistics theorem states (1) that particles with half-integer spin (fermions) obey Fermiâ€“Dirac statistics and the Pauli Exclusion Principle, and (2) that particles with integer spin (bosons) obey Boseâ€“Einstein statistics, occupy "symmetric states", and thus can share quantum states. The theorem relies on both quantum mechanics and the theory of special relativity, and this connection between spin and statistics has been called "one of the most important applications of the special relativity theory".JOURNAL, Pauli, Wolfgang, Wolfgang Pauli, 1940, The Connection Between Spin and Statistics,weblink Phys. Rev., 58, 8, 716â€“722, 10.1103/PhysRev.58.716, 1940PhRv...58..716P,## Magnetic moments

File:Neutron spin dipole field.jpg|thumbnail|right|Schematic diagram depicting the spin of the neutron as the black arrow and magnetic field lines associated with the neutron magnetic momentneutron magnetic momentParticles 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 Sternâ€“Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves.The intrinsic magnetic moment {{math|**Î¼**}} of a spin {{sfrac|1|2}} particle with charge {{mvar|q}}, mass {{mvar|m}}, and spin angular momentum {{math|

**S**}}, isPhysics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, {{ISBN|0-582-44401-2}}

boldsymbol{mu} = frac{g_s q}{2m} mathbf{S}

where the dimensionless quantity {{mvar|gs}} is called the spin {{mvar|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 {{mvar|g}}-factor, which has been experimentally determined to have the value {{val|-2.00231930436256|(35)}}, with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation.WEB, CODATA Value: electron *g*factor,weblink 2018, The NIST Reference on Constants, Units, and Uncertainty, National Institute of Standards and Technology, NIST, 2019-06-04, 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.BOOK, Richard Feynman, Feynman, R.P., QED: The Strange Theory of Light and Matter, Electrons and their interactions, 115, Princeton University Press, Princeton, New Jersey, 1985, 978-0-691-08388-9, After some years, it was discovered that this value [{{math, âˆ’Â½

*g*, ] was not exactly 1, but slightly more â€“ something like 1.00116 . This correction was worked out for the first time in 1948 by Schwinger as {{math|

*j*Ã—

*j*}} divided by 2 {{mvar|Ï€}} {{sic}} [where {{mvar|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.|title-link=QED: The Strange Theory of Light and Matter }} 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:JOURNAL, Marciano, W.J., Sanda, A.I., Exotic decays of the muon and heavy leptons in gauge theories, Physics Letters, B67, 3, 303â€“305, 1977, 10.1016/0370-2693(77)90377-X, 1977PhLB...67..303M, JOURNAL, Lee, B.W., Shrock, R.E., Natural suppression of symmetry violation in gauge theories: Muon- and electron-lepton-number nonconservation, Physical Review, D16, 5, 1444â€“1473, 1977, 10.1103/PhysRevD.16.1444, 1977PhRvD..16.1444L, JOURNAL, K. Fujikawa, R. E. Shrock, Magnetic Moment of a Massive Neutrino and Neutrino-Spin Rotation, Physical Review Letters, 45, 12, 963â€“966, 1980, 10.1103/PhysRevLett.45.963, 1980PhRvL..45..963F,

mu_{nu}approx 3times 10^{-19}mu_mathrm{B}frac{m_{nu}}{text{eV}}

where the {{math|*Î¼*Î½}} are the neutrino magnetic moments, {{math|

*m*Î½}} are the neutrino masses, and {{math|

*Î¼*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 10âˆ’14 {{math|

*Î¼*B}} are â€œunnaturalâ€ because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses are known to be at most about 1 eV, the large radiative corrections would then have to be â€œfine-tunedâ€ to cancel each other, to a large degree, and leave the neutrino mass small.JOURNAL, Bell, N.F., How Magnetic is the Dirac neutrino?, Physical Review Letters, 95, 15, 151802, 2005, hep-ph/0504134, 10.1103/PhysRevLett.95.151802, 2005PhRvL..95o1802B, 16241715, Cirigliano, V., Ramsey-Musolf, M., Vogel, P., Wise, Mark, etal, The measurement of neutrino magnetic moments is an active area of research. Experimental results have put the neutrino magnetic moment at less than {{val|1.2|e=-10}} times the electron's magnetic moment.

## Curie temperature and loss of alignment

In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction, with the overall average being very near zero. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.In paramagnetic materials, the magnetic dipole moments of individual atoms spontaneously align with an externally applied magnetic field. In diamagnetic materials, on the other hand, the magnetic dipole moments of individual atoms spontaneously align oppositely to any externally applied magnetic field, even if it requires energy to do so.The study of the behavior of such "spin models" is a thriving area of research in condensed matter physics. For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to interesting results in the theory of phase transitions.## Direction

{{further|Angular momentum operator}}### Spin projection quantum number and multiplicity

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Quantum mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum measured along any direction can only take on the values Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, {{ISBN|0-19-855493-1}}
S_i = hbar s_i, quad s_i in { - s, -(s-1), dots, s-1, s } ,!

where {{mvar|Si}} is the spin component along the {{mvar|i}}-axis (either {{mvar|x}}, {{mvar|y}}, or {{mvar|z}}), {{mvar|si}} is the spin projection quantum number along the {{mvar|i}}-axis, and {{mvar|s}} is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the {{mvar|z}}-axis:
S_z = hbar s_z, quad s_z in { - s, -(s-1), dots, s - 1, s } ,!

where {{mvar|Sz}} is the spin component along the {{mvar|z}}-axis, {{mvar|sz}} is the spin projection quantum number along the {{mvar|z}}-axis.One can see that there are {{math|2*s*+ 1}} possible values of {{mvar|sz}}. The number "{{math|2

*s*+ 1}}" is the multiplicity of the spin system. For example, there are only two possible values for a spin-{{sfrac|1|2}} particle: {{math|

*sz*{{=}} +{{sfrac|1|2}}}} and {{math|

*sz*{{=}} âˆ’{{sfrac|1|2}}}}. These correspond to quantum states in which the spin component is pointing in the +z or âˆ’z directions respectively, and are often referred to as "spin up" and "spin down". For a spin-{{sfrac|3|2}} particle, like a delta baryon, the possible values are +{{sfrac|3|2}}, +{{sfrac|1|2}}, âˆ’{{sfrac|1|2}}, âˆ’{{sfrac|3|2}}.

### Vector

File:Spin One-Half (Slow).gif|thumb|A single point in space can spin continuously without becoming tangled. Notice that after a 360-degree rotation, the spiral flips between clockwise and counterclockwise orientations. It returns to its original configuration after spinning a full 720 degrees. ]]For a given quantum state, one could think of a spin vector lang S rang whose components are the expectation values of the spin components along each axis, i.e., lang S rang = [lang S_x rang, lang S_y rang, lang S_z rang]. 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: {{mvar|sx}}, {{mvar|sy}} and {{mvar|sz}} cannot possess simultaneous definite values, because of a quantum 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 Sternâ€“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-{{sfrac|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 degreesâ€”that is, for detectors oriented in the opposite direction to the spin vectorâ€”the expectation of detecting particles from the collection reaches a minimum of 0%.As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum mechanical spin can exhibit phenomena analogous to classical gyroscopic effects. For example, one can exert a kind of "torque" on an electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole momentâ€”see the following section). The result is that the spin vector undergoes precession, just like a classical gyroscope. This phenomenon is known as electron spin resonance (ESR). The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance (NMR) spectroscopy and imaging.Mathematically, quantum-mechanical spin states are described by vector-like objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin-{{sfrac|1|2}} particle by 360 degrees does not bring it back to the same quantum state, but to the state with the opposite quantum phase; this is detectable, in principle, with interference experiments. To return the particle to its exact original state, one needs a 720-degree rotation. (The Plate trick and MÃ¶bius 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 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.## Mathematical formulation

### Operator

Spin obeys commutation relations analogous to those of the orbital angular momentum:
[S_j, S_k ] = i hbar varepsilon_{jkl} S_l

where {{mvar|Îµjkl}} is the Levi-Civita symbol. It follows (as with angular momentum) that the eigenvectors of {{math|*S*2}} and {{math|

*S*z}} (expressed as kets in the total {{mvar|S}} basis) are:

begin{align}

S^2 |s,m_{s}rangle &= hbar^2 s(s + 1) |s,m_{s}rangle

S_z |s,m_{s}rangle &= hbar m_{s} |s,m_{s}rangle.

end{align}The spin raising and lowering operators acting on these eigenvectors give:
S_z |s,m_{s}rangle &= hbar m_{s} |s,m_{s}rangle.

S_pm |s,m_{s}rangle = hbarsqrt{s(s+1)-m_{s}(m_{s}pm 1)} |s,m_{s}pm 1 rangle

where {{math|*S*Â± {{=}}

*Sx*Â±

*i Sy*}}.But unlike orbital angular momentum the eigenvectors are not spherical harmonics. They are not functions of {{mvar|Î¸}} and {{mvar|Ï†}}. There is also no reason to exclude half-integer values of {{mvar|s}} and {{mvar|ms}}.In addition to their other properties, all quantum mechanical particles possess an intrinsic spin (though this value may be equal to zero). The spin is quantized in units of the reduced Planck constant, such that the state function of the particle is, say, not {{math|

*Ïˆ*{{=}}

*Ïˆ*(

**r**)}}, but {{math|

*Ïˆ*{{=}}

*Ïˆ*(

**r**,

*Ïƒ*)}} where {{mvar|Ïƒ}} is out of the following discrete set of values:

sigma in {-shbar, -(s - 1)hbar, cdots, +(s - 1)hbar, +shbar}.

One distinguishes bosons (integer spin) and fermions (half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.### Pauli matrices

The quantum mechanical operators associated with spin-{{sfrac|1|2}} observables are:
hat{mathbf{S}}=frac{hbar}{2}boldsymbol{sigma}

where in Cartesian components:
S_x = {hbar over 2} sigma_x,quad S_y = {hbar over 2} sigma_y,quad S_z = {hbar over 2} sigma_z ,.

For the special case of spin-{{sfrac|1|2}} particles, {{mvar|Ïƒx}}, {{mvar|Ïƒy}} and {{mvar|Ïƒz}} are the three Pauli matrices, given by:
### Pauli exclusion principle

For systems of {{mvar|N}} identical particles this is related to the Pauli exclusion principle, which states that by interchanges of any two of the {{mvar|N}} particles one must have
psi ( cdots mathbf r_i,sigma_icdots mathbf r_j,sigma_jcdots ) = (-1)^{2s}psi ( cdots mathbf r_j,sigma_jcdots mathbf r_i,sigma_icdots ) .

Thus, for bosons the prefactor {{math|(âˆ’1)2*s*}} will reduce to +1, for fermions to âˆ’1. In quantum mechanics all particles are either bosons or fermions. In some speculative relativistic quantum field theories "supersymmetric" particles also exist, where linear combinations of bosonic and fermionic components appear. In two dimensions, the prefactor {{math|(âˆ’1)2

*s*}} can be replaced by any complex number of magnitude 1 such as in the anyon.

### Rotations

{{see also|symmetries in quantum mechanics}}As described above, quantum mechanics states that 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-{{sfrac|1|2}} particle, we would need two numbers {{math|*a*Â±{{sfrac|1|2}}}}, giving amplitudes of finding it with projection of angular momentum equal to {{math|{{sfrac|

*Ä§*|2}}}} and {{math|âˆ’{{sfrac|

*Ä§*|2}}}}, satisfying the requirement

left|a_frac{1}{2}right|^2 + left|a_{-frac{1}{2}}right|^2 , = 1.

For a generic particle with spin {{mvar|s}}, we would need {{math|2*s*+ 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:

begin{align}

sum_{m=-j}^{j} a_m^* b_m &= sum_{m=-j}^j left(sum_{n=-j}^j U_{nm} a_nright)^* left(sum_{k=-j}^j U_{km} b_kright)

sum_{n=-j}^j sum_{k=-j}^j U_{np}^* U_{kq} &= delta_{pq}.

end{align}Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2).BOOK
sum_{n=-j}^j sum_{k=-j}^j U_{np}^* U_{kq} &= delta_{pq}.

, B.C. Hall

, Quantum Theory for Mathematicians

, Springer

, 354â€“358

, 2013

, There is one {{mvar|n}}-dimensional irreducible representation of SU(2) for each dimension, though this representation is {{mvar|n}}-dimensional real for odd {{mvar|n}} and {{mvar|n}}-dimensional complex for even {{mvar|n}} (hence of real dimension {{math|2, Quantum Theory for Mathematicians

, Springer

, 354â€“358

, 2013

*n*}}). For a rotation by angle {{mvar|Î¸}} in the plane with normal vector hat{boldsymbol{theta}}, {{mvar|U}} can be written

U = e^{-frac{i}{hbar} boldsymbol{theta} cdot mathbf{S}},

where boldsymbol{theta} = theta hat{boldsymbol{theta}}, and {{math|**S**}} is the vector of spin operators.{{hidden|(Click "show" at right to see a proof or "hide" to hide it.)|2=

Working in the coordinate system where hat{theta} = hat{z}, we would like to show that {{mvar|Sx}} and {{mvar|Sy}} are rotated into each other by the angle {{mvar|Î¸}}. Starting with {{mvar|Sx}}. Using units where {{math|

*Ä§*{{=}} 1}}:

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) left[S_z, S_xright] + left(frac{1}{2!}right) (i theta)^2 left[S_z, left[S_z, S_xright]right] + left(frac{1}{3!}right) (i theta)^3 left[S_z, left[S_z, left[S_z, S_xright]right]right] + cdots

end{align}Using the spin operator commutation relations, we see that the commutators evaluate to {{mvar|i Sy}} for the odd terms in the series, and to {{mvar|Sx}} for all of the even terms. Thus:
&{} = S_x + (i theta) left[S_z, S_xright] + left(frac{1}{2!}right) (i theta)^2 left[S_z, left[S_z, S_xright]right] + left(frac{1}{3!}right) (i theta)^3 left[S_z, left[S_z, left[S_z, S_xright]right]right] + cdots

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 costheta - S_y sintheta

end{align}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 {{mvar|s}}).&{} = S_x costheta - S_y sintheta

*Modern Quantum Mechanics*, by J. J. Sakurai, p159

}}A generic rotation in 3-dimensional space can be built by compounding operators of this type using Euler angles:

mathcal{R}(alpha,beta,gamma) = e^{-ialpha S_z}e^{-ibeta S_y}e^{-igamma S_z}

An irreducible representation of this group of operators is furnished by the Wigner D-matrix:
D^s_{m'm}(alpha,beta,gamma) equiv

leftlangle sm' left| mathcal{R}(alpha,beta,gamma) right| sm rightrangle =

e^{-im'alpha} d^s_{m'm}(beta)e^{-i mgamma},

where
leftlangle sm' left| mathcal{R}(alpha,beta,gamma) right| sm rightrangle =

e^{-im'alpha} d^s_{m'm}(beta)e^{-i mgamma},

d^s_{m'm}(beta)= leftlangle sm' left|e^{-ibeta s_y} right| sm rightrangle

is Wigner's small d-matrix. Note that for {{math|*Î³*{{=}} 2Ï€}} and {{math|

*Î±*{{=}}

*Î²*{{=}} 0}}; i.e., a full rotation about the {{mvar|z}}-axis, the Wigner D-matrix elements become

D^s_{m'm}(0, 0, 2pi) = d^s_{m'm}(0) e^{-i m 2 pi} = delta_{m'm} (-1)^{2m}.

Recalling that a generic spin state can be written as a superposition of states with definite {{mvar|m}}, we see that if {{mvar|s}} is an integer, the values of {{mvar|m}} are all integers, and this matrix corresponds to the identity operator. However, if {{mvar|s}} is a half-integer, the values of {{mvar|m}} are also all half-integers, giving {{math|(âˆ’1)2*m*{{=}} âˆ’1}} for all {{mvar|m}}, and hence upon rotation by 2{{pi}} the state picks up a minus sign. This fact is a crucial element of the proof of the spin-statistics theorem.

### Lorentz transformations

We could try the same approach to determine the behavior of spin under general Lorentz transformations, but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact and therefore does not have any faithful, unitary, finite-dimensional representations.In case of spin-{{sfrac|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 {{mvar|Ïˆ}} with each particle. These spinors transform under Lorentz transformations according to the law
psi' = exp{left(tfrac{1}{8} omega_{munu} [gamma_{mu}, gamma_{nu}]right)} psi

where {{mvar|Î³Î½}} are gamma matrices and {{mvar|Ï‰Î¼Î½}} is an antisymmetric 4 Ã— 4 matrix parametrizing the transformation. It can be shown that the scalar product
langlepsi|phirangle = bar{psi}phi = psi^{dagger}gamma_0phi

is preserved. It is not, however, positive definite, so the representation is not unitary.### Measurement of spin along the {{mvar|x}}-, {{mvar|y}}-, or {{mvar|z}}-axes

Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and âˆ’1. The corresponding normalized eigenvectors are:
begin{array}{lclc}

psi_{x+} = left|frac{1}{2}, frac{+1}{2}rightrangle_x =

displaystylefrac{1}{sqrt{2}} !!!!! & begin{pmatrix}{1}{1}end{pmatrix}, &

psi_{x-} = left|frac{1}{2}, frac{-1}{2}rightrangle_x =

displaystylefrac{1}{sqrt{2}} !!!!! & begin{pmatrix}{-1}{1}end{pmatrix},

psi_{y+} = left|frac{1}{2}, frac{+1}{2}rightrangle_y =

displaystylefrac{1}{sqrt{2}} !!!!! & begin{pmatrix}{1}{i}end{pmatrix}, &

psi_{y-} = left|frac{1}{2}, frac{-1}{2}rightrangle_y =

displaystylefrac{1}{sqrt{2}} !!!!! & begin{pmatrix}{1}{-i}end{pmatrix},

psi_{z+} = left|frac{1}{2}, frac{+1}{2}rightrangle_z = &

begin{pmatrix}{1}{0}end{pmatrix}, &

psi_{z-} = left|frac{1}{2}, frac{-1}{2}rightrangle_z = &

begin{pmatrix}{0}{1}end{pmatrix}.

end{array}(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 as sympy; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.)By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the {{mvar|x}}-, {{mvar|y}}-, or {{mvar|z}}-axis can only yield an eigenvalue of the corresponding spin operator ({{mvar|Sx}}, {{mvar|Sy}} or {{mvar|Sz}}) on that axis, i.e. {{math|{{sfrac|displaystylefrac{1}{sqrt{2}} !!!!! & begin{pmatrix}{1}{1}end{pmatrix}, &

psi_{x-} = left|frac{1}{2}, frac{-1}{2}rightrangle_x =

displaystylefrac{1}{sqrt{2}} !!!!! & begin{pmatrix}{-1}{1}end{pmatrix},

psi_{y+} = left|frac{1}{2}, frac{+1}{2}rightrangle_y =

displaystylefrac{1}{sqrt{2}} !!!!! & begin{pmatrix}{1}{i}end{pmatrix}, &

psi_{y-} = left|frac{1}{2}, frac{-1}{2}rightrangle_y =

displaystylefrac{1}{sqrt{2}} !!!!! & begin{pmatrix}{1}{-i}end{pmatrix},

psi_{z+} = left|frac{1}{2}, frac{+1}{2}rightrangle_z = &

begin{pmatrix}{1}{0}end{pmatrix}, &

psi_{z-} = left|frac{1}{2}, frac{-1}{2}rightrangle_z = &

begin{pmatrix}{0}{1}end{pmatrix}.

*Ä§*|2}}}} or {{math|â€“{{sfrac|

*Ä§*|2}}}}. The quantum state of a particle (with respect to spin), can be represented by a two component spinor:

psi = begin{pmatrix} a + bi c + di end{pmatrix}.

When the spin of this particle is measured with respect to a given axis (in this example, the {{mvar|x}}-axis), the probability that its spin will be measured as {{math|{{sfrac|*Ä§*|2}}}} is just leftvert langle psi_{x+} vert psi rangle rightvert ^2. Correspondingly, the probability that its spin will be measured as {{math|â€“{{sfrac|

*Ä§*|2}}}} is just leftvert leftlangleleft. psi_{x-} rightvert psi rightrangle rightvert ^2. Following the measurement, the spin state of the particle will collapse into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to have a given eigenvalue, all measurements will yield the same eigenvalue (since leftvert leftlangleleft. psi_{x+} rightvert psi_{x+} rightrangle rightvert ^2 = 1 , etc), provided that no measurements of the spin are made along other axes.

### Measurement of spin along an arbitrary axis

The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let {{math|*u*{{=}} (

*ux*,

*uy*,

*uz*)}} be an arbitrary unit vector. Then the operator for spin in this direction is simply

S_u = frac{hbar}{2}(u_xsigma_x + u_ysigma_y + u_zsigma_z).

The operator {{mvar|Su}} has eigenvalues of {{math|Â±{{sfrac|*Ä§*|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 {{mvar|x}}-, {{mvar|y}}-, {{mvar|z}}-axis directions.A normalized spinor for spin-{{sfrac|1|2}} in the {{math|(

*ux*,

*uy*,

*uz*)}} direction (which works for all spin states except spin down where it will give {{sfrac|0|0}}), is:

frac{1}{sqrt{2+2u_z}}begin{pmatrix} 1+u_z u_x+iu_y end{pmatrix}.

The above spinor is obtained in the usual way by diagonalizing the {{mvar|Ïƒu}} 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 commute, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the {{mvar|x}}-axis, and we then measure the spin along the {{mvar|y}}-axis, we have invalidated our previous knowledge of the {{mvar|x}}-axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that:
leftvert langle psi_{xpm} mid psi_{ypm} rangle rightvert ^ 2 = leftvert langle psi_{xpm} mid psi_{zpm} rangle rightvert ^ 2 = leftvert langle psi_{ypm} mid psi_{zpm} rangle rightvert ^ 2 = tfrac{1}{2}.

So when physicists measure the spin of a particle along the {{mvar|x}}-axis as, for example, {{math|{{sfrac|*Ä§*|2}}}}, the particle's spin state collapses into the eigenstate mid psi_{x+} rangle. When we then subsequently measure the particle's spin along the {{mvar|y}}-axis, the spin state will now collapse into either mid psi_{y+} rangle or mid psi_{y-} rangle, each with probability {{sfrac|1|2}}. Let us say, in our example, that we measure {{math|âˆ’{{sfrac|

*Ä§*|2}}}}. When we now return to measure the particle's spin along the {{mvar|x}}-axis again, the probabilities that we will measure {{math|{{sfrac|

*Ä§*|2}}}} or {{math|âˆ’{{sfrac|

*Ä§*|2}}}} are each {{sfrac|1|2}} (i.e. they are leftvert langle psi_{x+} mid psi_{y-} rangle rightvert ^ 2 and leftvert langle psi_{x-} mid psi_{y-} rangle rightvert ^ 2 respectively). This implies that the original measurement of the spin along the x-axis is no longer valid, since the spin along the {{mvar|x}}-axis will now be measured to have either eigenvalue with equal probability.

### Higher spins

The spin-{{sfrac|1|2}} operator {{math|**S**{{=}} {{sfrac|

*Ä§*|2}}

**Ïƒ**}} forms the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large {{mvar|s}}, can be calculated using this spin operator and ladder operators.The resulting spin matrices and eigenvalues in the z-basis

- For spin 1 they are
- :begin{align}

S_x &= frac{hbar}{sqrt{2}}

begin{pmatrix}

0 & 1 & 0

1 & 0 & 1

0 & 1 & 0

end{pmatrix}, &

left|1, +1rightrangle_x &= frac{1}{2} begin{pmatrix}{1}{sqrt{2}}{1}end{pmatrix}, &

left|1, 0rightrangle_x &= frac{1}{sqrt{2}} begin{pmatrix}{-1}{0}{1}end{pmatrix}, &

left|1, -1rightrangle_x &= frac{1}{2} begin{pmatrix}{1}{-sqrt{2}}{1}end{pmatrix}

S_y &= frac{hbar}{sqrt{2}}

begin{pmatrix}

0 & -i & 0

i & 0 & -i

0 & i & 0

end{pmatrix}, &

left|1, +1rightrangle_y &= frac{1}{2} begin{pmatrix} -1 -isqrt{2} 1 end{pmatrix}, &

left|1, 0rightrangle_y &= frac{1}{sqrt{2}} begin{pmatrix} 1 0 1 end{pmatrix}, &

left|1, -1rightrangle_y &= frac{1}{2} begin{pmatrix} -1 isqrt{2} 1 end{pmatrix}

S_z &= hbar

begin{pmatrix}

1 & 0 & 0

0 & 0 & 0

0 & 0 & -1

end{pmatrix}, &

left|1, +1rightrangle_z &= begin{pmatrix} 1 0 0 end{pmatrix}, &

left|1, 0rightrangle_z &= begin{pmatrix} 0 1 0 end{pmatrix}, &

left|1, -1rightrangle_z &= begin{pmatrix} 0 0 1 end{pmatrix}

end{align} begin{pmatrix}

0 & 1 & 0

1 & 0 & 1

0 & 1 & 0

end{pmatrix}, &

left|1, +1rightrangle_x &= frac{1}{2} begin{pmatrix}{1}{sqrt{2}}{1}end{pmatrix}, &

left|1, 0rightrangle_x &= frac{1}{sqrt{2}} begin{pmatrix}{-1}{0}{1}end{pmatrix}, &

left|1, -1rightrangle_x &= frac{1}{2} begin{pmatrix}{1}{-sqrt{2}}{1}end{pmatrix}

S_y &= frac{hbar}{sqrt{2}}

begin{pmatrix}

0 & -i & 0

i & 0 & -i

0 & i & 0

end{pmatrix}, &

left|1, +1rightrangle_y &= frac{1}{2} begin{pmatrix} -1 -isqrt{2} 1 end{pmatrix}, &

left|1, 0rightrangle_y &= frac{1}{sqrt{2}} begin{pmatrix} 1 0 1 end{pmatrix}, &

left|1, -1rightrangle_y &= frac{1}{2} begin{pmatrix} -1 isqrt{2} 1 end{pmatrix}

S_z &= hbar

begin{pmatrix}

1 & 0 & 0

0 & 0 & 0

0 & 0 & -1

end{pmatrix}, &

left|1, +1rightrangle_z &= begin{pmatrix} 1 0 0 end{pmatrix}, &

left|1, 0rightrangle_z &= begin{pmatrix} 0 1 0 end{pmatrix}, &

left|1, -1rightrangle_z &= begin{pmatrix} 0 0 1 end{pmatrix}

- For spin {{sfrac|3|2}} they are
- :

S_x = frachbar2

begin{pmatrix}

0 &sqrt{3} &0 &0

sqrt{3} &0 &2 &0

0 &2 &0 &sqrt{3}

0 &0 &sqrt{3} &0

end{pmatrix}, !!!!! &|frac{3}{2}, frac{+3}{2}rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{1}{sqrt{3}}{sqrt{3}}{1}end{pmatrix}, !!!!! & |frac{3}{2}, frac{+1}{2}rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{-sqrt{3}}{-1}{1}{sqrt{3}}end{pmatrix}, !!!!! & |frac{3}{2}, frac{-1}{2}rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{sqrt{3}}{-1}{-1}{sqrt{3}}end{pmatrix}, !!!!! & |frac{3}{2}, frac{-3}{2}rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{-1}{sqrt{3}}{-sqrt{3}}{1}end{pmatrix}

S_y = frachbar2

begin{pmatrix}

0 &-isqrt{3} &0 &0

isqrt{3} &0 &-2i &0

0 &2i &0 &-isqrt{3}

0 &0 &isqrt{3} &0

end{pmatrix}, !!!!! &|frac{3}{2}, frac{+3}{2}rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{i}{-sqrt{3}}{-isqrt{3}}{1}end{pmatrix}, !!!!! &|frac{3}{2}, frac{+1}{2}rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{-isqrt{3}}{1}{-i}{sqrt{3}}end{pmatrix}, !!!!! &|frac{3}{2}, frac{-1}{2}rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{isqrt{3}}{1}{i}{sqrt{3}}end{pmatrix}, !!!!! &|frac{3}{2}, frac{-3}{2}rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{-i}{-sqrt{3}}{isqrt{3}}{1}end{pmatrix}

S_z = frachbar2

begin{pmatrix}

3 &0 &0 &0

0 &1 &0 &0

0 &0 &-1 &0

0 &0 &0 &-3

end{pmatrix}, !!!!! &|frac{3}{2}, frac{+3}{2}rangle_z =!!!!! & begin{pmatrix}{1}{0}{0}{0}end{pmatrix}, !!!!! &|frac{3}{2}, frac{+1}{2}rangle_z =!!!!! & begin{pmatrix}{0}{1}{0}{0}end{pmatrix}, !!!!! &|frac{3}{2}, frac{-1}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{1}{0}end{pmatrix}, !!!!! &|frac{3}{2}, frac{-3}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{1}end{pmatrix}

end{array} begin{pmatrix}

0 &sqrt{3} &0 &0

sqrt{3} &0 &2 &0

0 &2 &0 &sqrt{3}

0 &0 &sqrt{3} &0

end{pmatrix}, !!!!! &|frac{3}{2}, frac{+3}{2}rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{1}{sqrt{3}}{sqrt{3}}{1}end{pmatrix}, !!!!! & |frac{3}{2}, frac{+1}{2}rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{-sqrt{3}}{-1}{1}{sqrt{3}}end{pmatrix}, !!!!! & |frac{3}{2}, frac{-1}{2}rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{sqrt{3}}{-1}{-1}{sqrt{3}}end{pmatrix}, !!!!! & |frac{3}{2}, frac{-3}{2}rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{-1}{sqrt{3}}{-sqrt{3}}{1}end{pmatrix}

S_y = frachbar2

begin{pmatrix}

0 &-isqrt{3} &0 &0

isqrt{3} &0 &-2i &0

0 &2i &0 &-isqrt{3}

0 &0 &isqrt{3} &0

end{pmatrix}, !!!!! &|frac{3}{2}, frac{+3}{2}rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{i}{-sqrt{3}}{-isqrt{3}}{1}end{pmatrix}, !!!!! &|frac{3}{2}, frac{+1}{2}rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{-isqrt{3}}{1}{-i}{sqrt{3}}end{pmatrix}, !!!!! &|frac{3}{2}, frac{-1}{2}rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{isqrt{3}}{1}{i}{sqrt{3}}end{pmatrix}, !!!!! &|frac{3}{2}, frac{-3}{2}rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{-i}{-sqrt{3}}{isqrt{3}}{1}end{pmatrix}

S_z = frachbar2

begin{pmatrix}

3 &0 &0 &0

0 &1 &0 &0

0 &0 &-1 &0

0 &0 &0 &-3

end{pmatrix}, !!!!! &|frac{3}{2}, frac{+3}{2}rangle_z =!!!!! & begin{pmatrix}{1}{0}{0}{0}end{pmatrix}, !!!!! &|frac{3}{2}, frac{+1}{2}rangle_z =!!!!! & begin{pmatrix}{0}{1}{0}{0}end{pmatrix}, !!!!! &|frac{3}{2}, frac{-1}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{1}{0}end{pmatrix}, !!!!! &|frac{3}{2}, frac{-3}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{1}end{pmatrix}

- For spin 2 they are
- :

S_x = frachbar2

begin{pmatrix}

0 &2 &0 &0 &0

2 &0 &sqrt{6} &0 &0

0 &sqrt{6} &0 &sqrt{6} &0

0 &0 &sqrt{6} &0 &2

0 &0 &0 &2 &0

end{pmatrix}, !!!!! &|2, +2rangle_x =!!!!! & frac{1}{4} begin{pmatrix}{1}{2}{sqrt{6}}{2}{1}end{pmatrix}, !!!!! & |2, +1rangle_x =!!!!! & frac{1}{2} begin{pmatrix}{-1}{-1}{0}{1}{1}end{pmatrix}, !!!!! & |2, 0rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{sqrt{3}}{0}{-sqrt{2}}{0}{sqrt{3}}end{pmatrix}, !!!!! & |2, -1rangle_x =!!!!! & frac{1}{2} begin{pmatrix}{-1}{1}{0}{-1}{1}end{pmatrix}, !!!!! & |2, -2rangle_x =!!!!! & frac{1}{4} begin{pmatrix}{1}{-2}{sqrt{6}}{-2}{1}end{pmatrix}

S_y = frachbar2

begin{pmatrix}

0 &-2i &0 &0 &0

2i &0 &-sqrt{6}i&0 &0

0 &sqrt{6}i&0 &-sqrt{6}i&0

0 &0 &sqrt{6}i&0 &-2i

0 &0 &0 &2i &0

end{pmatrix}, !!!!! &|2, +2rangle_y =!!!!! & frac{1}{4} begin{pmatrix}{1}{2i}{-sqrt{6}}{-2i}{1}end{pmatrix}, !!!!! & |2, +1rangle_y =!!!!! & frac{1}{2} begin{pmatrix}{-1}{-i}{0}{-i}{1}end{pmatrix}, !!!!! & |2, 0rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{sqrt{3}}{0}{sqrt{2}}{0}{sqrt{3}}end{pmatrix}, !!!!! & |2, -1rangle_y =!!!!! & frac{1}{2} begin{pmatrix}{-1}{i}{0}{i}{1}end{pmatrix}, !!!!! & |2, -2rangle_y =!!!!! & frac{1}{4} begin{pmatrix}{1}{-2i}{-sqrt{6}}{2i}{1}end{pmatrix}

S_z = hbar

begin{pmatrix}

2 &0 &0 &0 &0

0 &1 &0 &0 &0

0 &0 &0 &0 &0

0 &0 &0 &-1 &0

0 &0 &0 &0 &-2

end{pmatrix}, !!!!! &|2, +2rangle_z =!!!!! & begin{pmatrix}{1}{0}{0}{0}{0}end{pmatrix}, !!!!! & |2, +1rangle_z =!!!!! & begin{pmatrix}{0}{1}{0}{0}{0}end{pmatrix}, !!!!! & |2, 0rangle_z =!!!!! & begin{pmatrix}{0}{0}{1}{0}{0}end{pmatrix}, !!!!! & |2, -1rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{1}{0}end{pmatrix}, !!!!! & |2, -2rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{0}{1}end{pmatrix}

end{array} begin{pmatrix}

0 &2 &0 &0 &0

2 &0 &sqrt{6} &0 &0

0 &sqrt{6} &0 &sqrt{6} &0

0 &0 &sqrt{6} &0 &2

0 &0 &0 &2 &0

end{pmatrix}, !!!!! &|2, +2rangle_x =!!!!! & frac{1}{4} begin{pmatrix}{1}{2}{sqrt{6}}{2}{1}end{pmatrix}, !!!!! & |2, +1rangle_x =!!!!! & frac{1}{2} begin{pmatrix}{-1}{-1}{0}{1}{1}end{pmatrix}, !!!!! & |2, 0rangle_x =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{sqrt{3}}{0}{-sqrt{2}}{0}{sqrt{3}}end{pmatrix}, !!!!! & |2, -1rangle_x =!!!!! & frac{1}{2} begin{pmatrix}{-1}{1}{0}{-1}{1}end{pmatrix}, !!!!! & |2, -2rangle_x =!!!!! & frac{1}{4} begin{pmatrix}{1}{-2}{sqrt{6}}{-2}{1}end{pmatrix}

S_y = frachbar2

begin{pmatrix}

0 &-2i &0 &0 &0

2i &0 &-sqrt{6}i&0 &0

0 &sqrt{6}i&0 &-sqrt{6}i&0

0 &0 &sqrt{6}i&0 &-2i

0 &0 &0 &2i &0

end{pmatrix}, !!!!! &|2, +2rangle_y =!!!!! & frac{1}{4} begin{pmatrix}{1}{2i}{-sqrt{6}}{-2i}{1}end{pmatrix}, !!!!! & |2, +1rangle_y =!!!!! & frac{1}{2} begin{pmatrix}{-1}{-i}{0}{-i}{1}end{pmatrix}, !!!!! & |2, 0rangle_y =!!!!! & frac{1}{2sqrt{2}} begin{pmatrix}{sqrt{3}}{0}{sqrt{2}}{0}{sqrt{3}}end{pmatrix}, !!!!! & |2, -1rangle_y =!!!!! & frac{1}{2} begin{pmatrix}{-1}{i}{0}{i}{1}end{pmatrix}, !!!!! & |2, -2rangle_y =!!!!! & frac{1}{4} begin{pmatrix}{1}{-2i}{-sqrt{6}}{2i}{1}end{pmatrix}

S_z = hbar

begin{pmatrix}

2 &0 &0 &0 &0

0 &1 &0 &0 &0

0 &0 &0 &0 &0

0 &0 &0 &-1 &0

0 &0 &0 &0 &-2

end{pmatrix}, !!!!! &|2, +2rangle_z =!!!!! & begin{pmatrix}{1}{0}{0}{0}{0}end{pmatrix}, !!!!! & |2, +1rangle_z =!!!!! & begin{pmatrix}{0}{1}{0}{0}{0}end{pmatrix}, !!!!! & |2, 0rangle_z =!!!!! & begin{pmatrix}{0}{0}{1}{0}{0}end{pmatrix}, !!!!! & |2, -1rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{1}{0}end{pmatrix}, !!!!! & |2, -2rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{0}{1}end{pmatrix}

- For spin {{sfrac|5|2}} they are
- :

S_x = frachbar2

begin{pmatrix}

0 &sqrt{5} &0 &0 &0 &0

sqrt{5} &0 &2sqrt{2} &0 &0 &0

0 &2sqrt{2} &0 &3 &0 &0

0 &0 &3 &0 &2sqrt{2} &0

0 &0 &0 &2sqrt{2} &0 &sqrt{5}

0 &0 &0 &0 &sqrt{5} &0

end{pmatrix}, !!!!! & |frac{5}{2}, frac{+5}{2}rangle_x =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{1}{sqrt{5}}{sqrt{10}}{sqrt{10}}{sqrt{5}}{1}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+3}{2}rangle_x =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{-sqrt{5}}{-3}{-sqrt{2}}{sqrt{2}}{3}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+1}{2}rangle_x =!!!!! & frac{1}{4} begin{pmatrix}{sqrt{5}}{1}{-sqrt{2}}{-sqrt{2}}{1}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-1}{2}rangle_x =!!!!! & frac{1}{4} begin{pmatrix}{-sqrt{5}}{1}{sqrt{2}}{-sqrt{2}}{-1}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-3}{2}rangle_x =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{sqrt{5}}{-3}{sqrt{2}}{sqrt{2}}{-3}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-5}{2}rangle_x =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{-1}{sqrt{5}}{-sqrt{10}}{sqrt{10}}{-sqrt{5}}{1}end{pmatrix}

S_y = frachbar2

begin{pmatrix}

0 &-isqrt{5} &0 &0 &0 &0

isqrt{5} &0 &-2isqrt{2} &0 &0 &0

0 &2isqrt{2} &0 &-3i &0 &0

0 &0 &3i &0 &-2isqrt{2} &0

0 &0 &0 &2isqrt{2} &0 &-isqrt{5}

0 &0 &0 &0 &isqrt{5} &0

end{pmatrix}, !!!!! & |frac{5}{2}, frac{+5}{2}rangle_y =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{-i}{sqrt{5}}{isqrt{10}}{-sqrt{10}}{-isqrt{5}}{1}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+3}{2}rangle_y =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{isqrt{5}}{-3}{-isqrt{2}}{-sqrt{2}}{-3i}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+1}{2}rangle_y =!!!!! & frac{1}{4} begin{pmatrix}{-isqrt{5}}{1}{-isqrt{2}}{sqrt{2}}{-i}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-1}{2}rangle_y =!!!!! & frac{1}{4} begin{pmatrix}{isqrt{5}}{1}{isqrt{2}}{sqrt{2}}{i}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-3}{2}rangle_y =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{-isqrt{5}}{-3}{isqrt{2}}{-sqrt{2}}{3i}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-5}{2}rangle_y =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{i}{sqrt{5}}{-isqrt{10}}{-sqrt{10}}{isqrt{5}}{1}end{pmatrix}

S_z = frachbar2

begin{pmatrix}

5 &0 &0 &0 &0 &0

0 &3 &0 &0 &0 &0

0 &0 &1 &0 &0 &0

0 &0 &0 &-1 &0 &0

0 &0 &0 &0 &-3 &0

0 &0 &0 &0 &0 &-5

end{pmatrix}, !!!!! & |frac{5}{2}, frac{+5}{2}rangle_z =!!!!! & begin{pmatrix}{1}{0}{0}{0}{0}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+3}{2}rangle_z =!!!!! & begin{pmatrix}{0}{1}{0}{0}{0}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+1}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{1}{0}{0}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-1}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{1}{0}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-3}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{0}{1}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-5}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{0}{0}{1}end{pmatrix}

end{array} begin{pmatrix}

0 &sqrt{5} &0 &0 &0 &0

sqrt{5} &0 &2sqrt{2} &0 &0 &0

0 &2sqrt{2} &0 &3 &0 &0

0 &0 &3 &0 &2sqrt{2} &0

0 &0 &0 &2sqrt{2} &0 &sqrt{5}

0 &0 &0 &0 &sqrt{5} &0

end{pmatrix}, !!!!! & |frac{5}{2}, frac{+5}{2}rangle_x =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{1}{sqrt{5}}{sqrt{10}}{sqrt{10}}{sqrt{5}}{1}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+3}{2}rangle_x =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{-sqrt{5}}{-3}{-sqrt{2}}{sqrt{2}}{3}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+1}{2}rangle_x =!!!!! & frac{1}{4} begin{pmatrix}{sqrt{5}}{1}{-sqrt{2}}{-sqrt{2}}{1}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-1}{2}rangle_x =!!!!! & frac{1}{4} begin{pmatrix}{-sqrt{5}}{1}{sqrt{2}}{-sqrt{2}}{-1}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-3}{2}rangle_x =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{sqrt{5}}{-3}{sqrt{2}}{sqrt{2}}{-3}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-5}{2}rangle_x =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{-1}{sqrt{5}}{-sqrt{10}}{sqrt{10}}{-sqrt{5}}{1}end{pmatrix}

S_y = frachbar2

begin{pmatrix}

0 &-isqrt{5} &0 &0 &0 &0

isqrt{5} &0 &-2isqrt{2} &0 &0 &0

0 &2isqrt{2} &0 &-3i &0 &0

0 &0 &3i &0 &-2isqrt{2} &0

0 &0 &0 &2isqrt{2} &0 &-isqrt{5}

0 &0 &0 &0 &isqrt{5} &0

end{pmatrix}, !!!!! & |frac{5}{2}, frac{+5}{2}rangle_y =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{-i}{sqrt{5}}{isqrt{10}}{-sqrt{10}}{-isqrt{5}}{1}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+3}{2}rangle_y =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{isqrt{5}}{-3}{-isqrt{2}}{-sqrt{2}}{-3i}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+1}{2}rangle_y =!!!!! & frac{1}{4} begin{pmatrix}{-isqrt{5}}{1}{-isqrt{2}}{sqrt{2}}{-i}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-1}{2}rangle_y =!!!!! & frac{1}{4} begin{pmatrix}{isqrt{5}}{1}{isqrt{2}}{sqrt{2}}{i}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-3}{2}rangle_y =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{-isqrt{5}}{-3}{isqrt{2}}{-sqrt{2}}{3i}{sqrt{5}}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-5}{2}rangle_y =!!!!! & frac{1}{4sqrt{2}} begin{pmatrix}{i}{sqrt{5}}{-isqrt{10}}{-sqrt{10}}{isqrt{5}}{1}end{pmatrix}

S_z = frachbar2

begin{pmatrix}

5 &0 &0 &0 &0 &0

0 &3 &0 &0 &0 &0

0 &0 &1 &0 &0 &0

0 &0 &0 &-1 &0 &0

0 &0 &0 &0 &-3 &0

0 &0 &0 &0 &0 &-5

end{pmatrix}, !!!!! & |frac{5}{2}, frac{+5}{2}rangle_z =!!!!! & begin{pmatrix}{1}{0}{0}{0}{0}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+3}{2}rangle_z =!!!!! & begin{pmatrix}{0}{1}{0}{0}{0}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{+1}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{1}{0}{0}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-1}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{1}{0}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-3}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{0}{1}{0}end{pmatrix}, !!!!! & |frac{5}{2}, frac{-5}{2}rangle_z =!!!!! & begin{pmatrix}{0}{0}{0}{0}{0}{1}end{pmatrix}

- The generalization of these matrices for arbitrary spin {{mvar|s}} is
- : begin{align}

left(S_xright)_{ab} & = frac{hbar}{2} left(delta_{a,b+1} + delta_{a+1,b}right) sqrt{(s + 1)(a + b - 1) - ab} ,

left(S_yright)_{ab} & = frac{ihbar}{2} left(delta_{a,b+1} - delta_{a+1,b}right) sqrt{(s + 1)(a + b - 1) - ab}

left(S_zright)_{ab} & = hbar (s + 1 - a) delta_{a,b} = hbar (s + 1 - b) delta_{a,b}

end{align} left(S_yright)_{ab} & = frac{ihbar}{2} left(delta_{a,b+1} - delta_{a+1,b}right) sqrt{(s + 1)(a + b - 1) - ab}

left(S_zright)_{ab} & = hbar (s + 1 - a) delta_{a,b} = hbar (s + 1 - b) delta_{a,b}

- : where indices a, b are integer numbers such that
- :: 1 le a le 2s + 1 and
- :: 1 le b le 2s + 1

e^{i thetaleft(hat{mathbf{n}} cdot boldsymbol{sigma}right)} =

Icos(theta) + i left(hat{mathbf{n}} cdot boldsymbol{sigma}right) sin(theta)

for higher spins is tractable, but less simple.JOURNAL, 10.3842/SIGMA.2014.084, Curtright, Fairlie, Zachos, T L, D B, C K, Thomas Curtright, David Fairlie, Cosmas Zachos, 2014, A compact formula for rotations as spin matrix polynomials, SIGMA, 10, 084, 1402.3541, 2014SIGMA..10..084C, Icos(theta) + i left(hat{mathbf{n}} cdot boldsymbol{sigma}right) sin(theta)

## Parity

In tables of the spin quantum number {{mvar|s}} for nuclei or particles, the spin is often followed by a "+" or "âˆ’". This refers to the parity with "+" for even parity (wave function unchanged by spatial inversion) and "âˆ’" for odd parity (wave function negated by spatial inversion). For example, see the isotopes of bismuth in which the List of isotopes includes the column Nuclear spin and parity. For Bi-209, the only stable isotope, the entry 9/2â€“ means that the nuclear spin is 9/2 and the parity is odd.## Applications

Spin has important theoretical implications and practical applications. Well-established*direct*applications of spin include:

- Nuclear magnetic resonance (NMR) spectroscopy in chemistry;
- Electron spin resonance spectroscopy in chemistry and physics;
- Magnetic resonance imaging (MRI) in medicine, a type of applied NMR, which relies on proton spin density;
- Giant magnetoresistive (GMR) drive head technology in modern hard disks.

*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 clocks and in the modern definition of the second. Precise measurements of the {{mvar|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 of light.An emerging application of spin is as a binary information carrier in spin transistors. The original concept, proposed in 1990, is known as Datta-Das spin transistor.JOURNAL, 10.1063/1.102730, Datta. S and B. Das, Electronic analog of the electrooptic modulator, Applied Physics Letters, 56, 665â€“667, 1990, 7, 1990ApPhL..56..665D, Electronics based on spin transistors are referred to as spintronics. The manipulation of spin in dilute magnetic semiconductor materials, such as metal-doped ZnO or TiO2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.JOURNAL, Assadi, M.H.N, Hanaor, D.A.H, Theoretical study on copper's energetics and magnetism in TiO2 polymorphs, Journal of Applied Physics, 2013, 113, 23, 233913â€“233913â€“5, 10.1063/1.4811539, 1304.1854, 2013JAP...113w3913A, There are many

*indirect*applications and manifestations of spin and the associated Pauli exclusion principle, starting with the periodic table of chemistry.

## History

File:Wolfgang Pauli young.jpg|thumb|Wolfgang PauliWolfgang PauliSpin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "two-valuedness not describable classically"-WEB,weblink Wolfgang Pauli, Exclusion Principle and Quantum Mechanics, December 13, 1946, Nobel Lecture, Nobel Prize, associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can have the same quantum state in the same quantum system.The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of LandÃ©'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 at Leiden University. 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 results). This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.Mathematically speaking, a fiber bundle description is needed. The tangent bundle effect is additive and relativistic; that is, it vanishes if {{mvar|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, known to Ludwik Silberstein in 1914).Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics invented by SchrÃ¶dinger and Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators, and introduced a two-component spinor wave-function.Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac published the Dirac equation, which described the relativistic electron. In the Dirac equation, a four-component spinor (known as a "Dirac spinor") was used for the electron wave-function. Relativistic spin explained gyromagnetic anomaly, which was (in retrospect) first observed by Samuel Jackson Barnett in 1914 (see Einsteinâ€“de Haas effect). In 1940, Pauli proved the*spin-statistics theorem*, which states that fermions have half-integer spin and bosons have integer spin.In retrospect, the first direct experimental evidence of the electron spin was the Sternâ€“Gerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1927.JOURNAL

, B. Friedrich, D. Herschbach

, Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics

, Physics Today

, 56, 12, 53

, 2003

, 10.1063/1.1650229, 2003PhT....56l..53F,

, Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics

, Physics Today

, 56, 12, 53

, 2003

, 10.1063/1.1650229, 2003PhT....56l..53F,

## See also

{{div col|colwidth=18em}}- Einsteinâ€“de Haas effect
- Spin-orbital
- Chirality (physics)
- Dynamic nuclear polarisation
- Helicity (particle physics)
- Holsteinâ€“Primakoff transformation
- Kramers theorem
- Pauli equation
- Pauliâ€“Lubanski pseudovector
- Raritaâ€“Schwinger equation
- Representation theory of SU(2)
- Spin-{{sfrac|1|2}}
- Spin-flip
- Spin isomers of hydrogen
- Spin tensor
- Spin wave
- Spin engineering
- Yrast
- Zitterbewegung

## References

{{reflist|group=note}}{{reflist|30em}}## Further reading

- BOOK, Cohen-Tannoudji, Claude, Diu, Bernard, LaloÃ«, Franck, 2006, Quantum Mechanics, John Wiley & Sons, 978-0-471-56952-7, 2 volume set,
- BOOK, E. U., Condon, G. H., Shortley, 1935, The Theory of Atomic Spectra, Cambridge University Press, 978-0-521-09209-8, Especially Chapter 3,
- JOURNAL, J. A., Hipple, H., Sommer, Thomas, H.A., 1949, A precise method of determining the faraday by magnetic resonance, Physical Review, 76, 12, 1877â€“1878, 10.1103/PhysRev.76.1877.2, 1949PhRv...76.1877H, weblink
- BOOK, Edmonds, A. R., 1957, Angular Momentum in Quantum Mechanics, Princeton University Press, 978-0-691-07912-7,
- BOOK, Jackson, John David, 1998, Classical Electrodynamics, 3rd, John Wiley & Sons, 978-0-471-30932-1,
- BOOK, Serway, Raymond A., Jewett, John W., Physics for Scientists and Engineers, 6th, Brooks/Cole, 2004, 978-0-534-40842-8,
- BOOK, Thompson, William J., 1994, Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems, Wiley, 978-0-471-55264-2,
- BOOK, Tipler, Paul, Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics, 5th, W. H. Freeman, 2004, 978-0-7167-0809-4,
- Sin-Itiro Tomonaga, The Story of Spin, 1997

## External links

- Goudsmit on the discovery of electron spin.
*Nature*: " Milestones in 'spin' since 1896."- ECE 495N Lecture 36: Spin Online lecture by S. Datta

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