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In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.^{[1]}^{[2]}
Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. Orbital angular momentum operator is the quantummechanical counterpart to the classical notion of angular momentum: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus).^{[3]}^{[4]} The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.^{[5]}
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.^{[2]}
The SI unit of spin is the joulesecond, just as with classical angular momentum. In practice, however, it is written as a multiple of the reduced Planck constant ħ, usually in natural units, where the ħ is omitted, resulting in a unitless number. Spin quantum numbers are unitless numbers by definition.
When combined with the spinstatistics theorem, the spin of electrons results in the Pauli exclusion principle, which in turn underlies the periodic table of chemical elements.
Wolfgang Pauli was the first to propose the concept of spin, but he did not name it. In 1925, Ralph Kronig, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested a physical interpretation of particles spinning around their own axis. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it.
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:
The conventional definition of the spin quantum number, s, is s = n/2, where n can be any nonnegative integer. Hence the allowed values of s are 0, 1/2, 1, 3/2, 2, etc. The value of s for an elementary particle depends only on the type of particle, and cannot be altered in any known way (in contrast to the spin direction described below). The spin angular momentum, S, of any physical system is quantized. The allowed values of S are:
where h is the Planck constant. In contrast, orbital angular momentum can only take on integer values of s; i.e., evennumbered values of n.
Those particles with halfinteger spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer spins, such as 0, 1, 2, are known as bosons. The two families of particles obey different rules and broadly have different roles in the world around us. A key distinction between the two families is that fermions obey the Pauli exclusion principle; that is, there cannot be two identical fermions simultaneously having the same quantum numbers (meaning, roughly, having the same position, velocity and spin direction). In contrast, bosons obey the rules of Bose–Einstein statistics and have no such restriction, so they may "bunch together" even if in identical states. Also, composite particles can have spins different from the particles which comprise them. For example, a helium atom can have spin 0 and therefore can behave like a boson even though the quarks and electrons which make it up are all fermions.
This has profound practical applications:
Theoretical and experimental studies have shown that the spin possessed by elementary particles cannot be explained by postulating that they are made up of even smaller particles rotating about a common center of mass analogous to a classical electron radius; as far as can be presently determined, these elementary particles have no inner structure. The spin of an elementary particle is therefore seen as a truly intrinsic physical property, akin to the particle's electric charge and rest mass.
The proof that particles with halfinteger spin (fermions) obey Fermi–Dirac statistics and the Pauli Exclusion Principle, and particles with integer spin (bosons) obey Bose–Einstein statistics, occupy "symmetric states", and thus can share quantum states, is known as the spinstatistics theorem. 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".^{[6]}
Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern–Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves.
The intrinsic magnetic moment μ of a spin1/2 particle with charge q, mass m, and spin angular momentum S, is^{[7]}
where the dimensionless quantity g_{s} is called the spin gfactor. 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 gfactor, which has been experimentally determined to have the value 2.0023193043622(15), with the digits in parentheses denoting −measurement uncertainty in the last two digits at one standard deviation.^{[8]} 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 0.002319304... arises from the electron's interaction with the surrounding electromagnetic field, including its own field.^{[9]} Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a nonzero 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 nonzero neutrino masses predicts neutrino magnetic moments of:^{[10]}^{[11]}^{[12]}
where the μ_{ν} are the neutrino magnetic moments, m_{ν} are the neutrino masses, and μ_{B} is the Bohr magneton. New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments. It can be shown in a model independent way that neutrino magnetic moments larger than about 10^{−14} μ_{B} are unnatural, because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses cannot exceed about 1 eV, these radiative corrections must then be assumed to be fine tuned to cancel out to a large degree.^{[13]}
The measurement of neutrino magnetic moments is an active area of research. As of 2001^{[update]}, the latest experimental results have put the neutrino magnetic moment at less than 1.2×10^{−10} times the electron's 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. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, producing a macroscopic, nonzero 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.
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 ^{[14]}
where S_{i} is the spin component along the iaxis (either x, y, or z), s_{i} is the spin projection quantum number along the iaxis, and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the zaxis:
where S_{z} is the spin component along the zaxis, s_{z} is the spin projection quantum number along the zaxis.
One can see that there are 2s+1 possible values of s_{z}. The number "2s + 1" is the multiplicity of the spin system. For example, there are only two possible values for a spin1/2 particle: s_{z} = +1/2 and s_{z} = −1/2. These correspond to quantum states in which the spin is pointing in the +z or −z directions respectively, and are often referred to as "spin up" and "spin down". For a spin3/2 particle, like a delta baryon, the possible values are +3/2, +1/2, −1/2, −3/2.
For a given quantum state, one could think of a spin vector whose components are the expectation 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 the axis of rotation. It turns out that the spin vector is not very useful in actual quantum mechanical calculations, because it cannot be measured directly: s_{x}, s_{y} and s_{z} 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 Stern–Gerlach apparatus, the spin vector does have a welldefined 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 spin1/2 particles, this maximum probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180 degrees—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%.
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 moment—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 vectorlike objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin1/2 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. A spinzero particle can only have a single quantum state, even after torque is applied. Rotating a spin2 particle 180 degrees can bring it back to the same quantum state and a spin4 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.
Spin obeys commutation relations analogous to those of the orbital angular momentum:
where is the LeviCivita symbol. It follows (as with angular momentum) that the eigenvectors of S^{2} and S_{z} (expressed as kets in the total S basis) are:
The spin raising and lowering operators acting on these eigenvectors give:
But unlike orbital angular momentum the eigenvectors are not spherical harmonics. They are not functions of θ and φ. There is also no reason to exclude halfinteger values of s and m.
In addition to their other properties, all quantum mechanical particles possess an intrinsic spin (though it may have the intrinsic spin 0, too). The spin is quantized in units of the reduced Planck constant, such that the state function of the particle is, say, not , but where is out of the following discrete set of values:
One distinguishes bosons (integer spin) and fermions (halfinteger spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.
The quantum mechanical operators associated with spin observables are:
where in Cartesian components:
For the special case of spin1/2 particles, σ_{x}, σ_{y} and σ_{z} are the three Pauli matrices, given by:
For systems of N identical particles this is related to the Pauli exclusion principle, which states that by interchanges of any two of the N particles one must have
Thus, for bosons the prefactor (−1)^{2s} will reduce to +1, for fermions to −1. In quantum mechanics all particles are either bosons or fermions. In some speculative relativistic quantum field theories "supersymmetric" particles also exist, where linear combinations of bosonic and fermionic components appear. In two dimensions, the prefactor (−1)^{2s} can be replaced by any complex number of magnitude 1 such as in the Anyon.
The above permutation postulate for Nparticle state functions has mostimportant consequences in daily life, e.g. the periodic table of the chemists or biologists.
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 1/2 particle, we would need two numbers a_{±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 spin s, we would need 2s + 1 such parameters. Since these numbers depend on the choice of the axis, they transform into each other nontrivially 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:
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).^{[15]} There is one ndimensional irreducible representation of SU(2) for each dimension, though this representation is ndimensional real for odd n and ndimensional complex for even n (hence of real dimension 2n). For a rotation by angle θ in the plane with normal vector , U can be written
where is a and S is the vector of spin operators.
(Click "show" at right to see a proof or "hide" to hide it.)
Working in the coordinate system where , we would like to show that S_{x} and S_{y} are rotated into each other by the angle θ. Starting with S_{x}. Using units where ħ = 1:
Using the spin operator commutation relations, we see that the commutators evaluate to iS_{y} for the odd terms in the series, and to S_{x} 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 number s).^{[16]}
A generic rotation in 3dimensional space can be built by compounding operators of this type using Euler angles:
An irreducible representation of this group of operators is furnished by the Wigner Dmatrix:
where
is Wigner's small dmatrix. Note that for γ = 2π and α = β = 0; i.e., a full rotation about the zaxis, the Wigner Dmatrix elements become
Recalling that a generic spin state can be written as a superposition of states with definite m, we see that if s is an integer, the values of m are all integers, and this matrix corresponds to the identity operator. However, if s is a halfinteger, the values of m are also all halfintegers, giving (−1)^{2m} = −1 for all m, and hence upon rotation by 2π the state picks up a minus sign. This fact is a crucial element of the proof of the spinstatistics theorem.
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 noncompact and therefore does not have any faithful, unitary, finitedimensional representations.
In case of spin 1/2 particles, it is possible to find a construction that includes both a finitedimensional representation and a scalar product that is preserved by this representation. We associate a 4component Dirac spinor with each particle. These spinors transform under Lorentz transformations according to the law
where are gamma matrices and is an antisymmetric 4×4 matrix parametrizing the transformation. It can be shown that the scalar product
is preserved. It is not, however, positive definite, so the representation is not unitary.
Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:
By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the x, y or z axis can only yield an eigenvalue of the corresponding spin operator (S_{x}, S_{y} or S_{z}) on that axis, i.e. ħ/2 or –ħ/2. The quantum state of a particle (with respect to spin), can be represented by a two component spinor:
When the spin of this particle is measured with respect to a given axis (in this example, the xaxis), the probability that its spin will be measured as ħ/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 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 , 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. Let u = (u_{x}, u_{y}, u_{z}) be an arbitrary unit vector. Then the operator for spin in this direction is simply
The operator S_{u} has eigenvalues of ±ħ/2, just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three x, y, z axis directions.
A normalized spinor for spin1/2 in the (u_{x}, u_{y}, u_{z}) direction (which works for all spin states except spin down where it will give 0/0), is:
The above spinor is obtained in the usual way by diagonalizing the 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 not commute, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the xaxis, and we then measure the spin along the yaxis, we have invalidated our previous knowledge of the xaxis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that:
So when physicists measure the spin of a particle along the xaxis as, for example, ħ/2, the particle's spin state collapses into the eigenstate . When we then subsequently measure the particle's spin along the yaxis, the spin state will now collapse into either or , each with probability 1/2. Let us say, in our example, that we measure –ħ/2. When we now return to measure the particle's spin along the xaxis again, the probabilities that we will measure ħ/2 or –ħ/2 are each 1/2 (i.e. they are and respectively). This implies that the original measurement of the spin along the xaxis is no longer valid, since the spin along the xaxis will now be measured to have either eigenvalue with equal probability.
The spin1/2 operator S = ħ/2σ form 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 s, can be calculated using this spin operator and ladder operators.
The resulting spin matrices for spin 1 are:
for spin 3/2 they are
and for spin 5/2 they are
The generalization of these matrices for arbitrary s is
Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_{n} is defined to consist of all nfold tensor products of Pauli matrices.
The analog formula of Euler's formula in terms of the Pauli matrices:
for higher spins is tractable, but less simple.^{[17]}
In tables of the spin quantum number s for nuclei or particles, the spin is often followed by a "+" or "−". This refers to the parity with "+" for even parity (wave function unchanged by spatial inversion) and "−" for odd parity (wave function negated by spatial inversion). For example, see the isotopes of bismuth.
Spin has important theoretical implications and practical applications. Wellestablished direct applications of spin include:
Electron spin plays an important role in magnetism, with applications for instance in computer memories. The manipulation of nuclear spin by radiofrequency waves (nuclear magnetic resonance) is important in chemical spectroscopy and medical imaging.
Spinorbit 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 gfactor 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 DattaDas spin transistor.^{[18]} Electronics based on spin transistors are referred to as spintronics. The manipulation of spin in dilute magnetic semiconductor materials, such as metaldoped ZnO or TiO_{2} imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.^{[19]}
There are many indirect applications and manifestations of spin and the associated Pauli exclusion principle, starting with the periodic table of chemistry.
Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "twovalued quantum degree of freedom" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can share the same quantum state at the same time.
The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in early 1925 that it was produced by the selfrotation 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 factoroftwo 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 c goes to infinity. It is one half of the value obtained without regard for the tangent space orientation, but with opposite sign. Thus the combined effect differs from the latter by a factor two (Thomas precession).
Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics invented by Schrödinger and Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators, and introduced a twocomponent spinor wavefunction.
Pauli's theory of spin was nonrelativistic. However, in 1928, Paul Dirac published the Dirac equation, which described the relativistic electron. In the Dirac equation, a fourcomponent spinor (known as a "Dirac spinor") was used for the electron wavefunction. In 1940, Pauli proved the spinstatistics theorem, which states that fermions have halfinteger spin and bosons integer spin.
In retrospect, the first direct experimental evidence of the electron spin was the Stern–Gerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1927.^{[20]}

