Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory. The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The relativistic spin–statistics theorem connects electron spin quantization to the Pauli exclusion principle: observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion. Spin is described mathematically as a vector for some particles such as photons, and as a spinor for other particles such as electrons. Spinors behave similarly to vectors: they have definite magnitudes and change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change. These are indicated by assigning the particle a spin quantum number. The SI units of spin are the same as classical angular momentum (i.e., N·m·s, J·s, or kg·m2·s−1). In quantum mechanics, angular momentum and spin angular momentum take discrete values proportional to the Planck constant. In practice, spin is usually given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ. Often, the "spin quantum number" is simply called "spin".
Models
Rotating charged mass The earliest models for electron spin imagined a rotating charged mass, but this model fails when examined in detail. The required space distribution does not match limits on the electron radius, and the required rotation speed exceeds the speed of light. In the Standard Model, the fundamental particles are all considered "point-like", and they have their effects through the field that surrounds them. Any model for spin based on mass rotation would need to be consistent with that model.
Pauli's "classically non-describable two-valuedness" 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. This approach allowed Pauli to develop a proof of his fundamental Pauli exclusion principle, a proof now called the spin-statistics theorem. 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.
Circulation of classical fields 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. By applying Frederik Belinfante's approach to calculating the angular momentum of a field, Hans C. Ohanian showed that "spin is essentially a wave property ... generated by a circulating flow of charge in the wave field of the electron". This same concept of spin can be applied to gravity waves in water: "spin is generated by subwavelength circular motion of water particles". Unlike classical wavefield circulation, which allows continuous values of angular momentum, quantum wavefields allow only discrete values. Consequently, energy transfer to or from spin states always occurs in fixed quantum steps. Only a few steps are allowed: for many qualitative purposes, the complexity of the spin quantum wavefields can be ignored and the system properties can be discussed in terms of "integer" or "half-integer" spin models as discussed in quantum numbers below.
In Bohmian mechanics Spin can be understood differently depending on the interpretations of quantum mechanics. In the de Broglie–Bohm interpretation, particles have definitive trajectories but their motion is driven by the wave function or pilot wave. In this interpretation, the spin is a property of the pilot wave and not of the particle itself.
Dirac's relativistic electron Relativistic calculations of spin properties for electrons requires the Dirac equation.
Relation to orbital angular momentum As the name suggests, spin was originally conceived as the rotation of a particle around some axis. Historically orbital angular momentum related to particle orbits. While the names based on mechanical models have survived, the physical explanation has not. Quantization fundamentally alters the character of both spin and orbital angular momentum. Since elementary particles are point-like, self-rotation is not well-defined for them. However, spin implies that the phase of the particle depends on the angle as e i S θ , {\displaystyle e^{iS\theta }\ ,} for rotation of angle θ around the axis parallel to the spin S. This is equivalent to the quantum-mechanical interpretation of momentum as phase dependence in the position, and of orbital angular momentum as phase dependence in the angular position. For fermions, the picture is less clear: From the Ehrenfest theorem, the angular velocity is equal to the derivative of the Hamiltonian to its conjugate momentum, which is the total angular momentum operator J = L + S . Therefore, if the Hamiltonian H has any dependence on the spin S, then ∂ H/ ∂ S must be non-zero; consequently, for classical mechanics, the existence of spin in the Hamiltonian will produce an actual angular velocity, and hence an actual physical rotation – that is, a change in the phase-angle, θ, over time. However, whether this holds true for free electrons is ambiguous, since for an electron, | S |² is a constant 1/ 2 ℏ , and one might decide that since it cannot change, no partial (∂) can exist. Therefore it is a matter of interpretation whether the Hamiltonian must include such a term, and whether this aspect of classical mechanics extends into quantum mechanics (any particle's intrinsic spin angular momentum, S, is a quantum number arising from a "spinor" in the mathematical solution to the Dirac equation, rather than being a more nearly physical quantity, like orbital angular momentum L). Nevertheless, spin appears in the Dirac equation, and thus the relativistic Hamiltonian of the electron, treated as a Dirac field, can be interpreted as including a dependence in the spin S.
Quantum number
Spin obeys the mathematical laws of angular momentum quantization. The specific properties of spin angular momenta include:
Spin quantum numbers may take either half-integer or integer values. Although the direction of its spin can be changed, the magnitude of the spin of an elementary particle cannot be changed. The spin of a charged particle is associated with a magnetic dipole moment with a g-factor that differs from 1. (In the classical context, this would imply the internal charge and mass distributions differing for a rotating nonconductive object.) The conventional definition of the spin quantum number is s = n/2, where n can be any non-negative integer. Hence the allowed values of s are 0, 1/2, 1, 3/2, 2, etc. The value of s for an elementary particle depends only on the type of particle and cannot be altered in any known way (in contrast to the spin direction described below). The spin angular momentum S of any physical system is quantized. The allowed values of S are
S = ℏ s ( s + 1 ) = h 2 π n 2 ( n + 2 ) 2 = h 4 π n ( n + 2 ) , {\displaystyle S=\hbar \,{\sqrt {s(s+1)}}={\frac {h}{2\pi }}\,{\sqrt {{\frac {n}{2}}{\frac {(n+2)}{2}}}}={\frac {h}{4\pi }}\,{\sqrt {n(n+2)}},}
where h is the Planck constant, and ℏ = h 2 π {\textstyle \hbar ={\frac {h}{2\pi }}} is the reduced Planck constant. In contrast, orbital angular momentum can only take on integer values of s; i.e., even-numbered values of n.
Fermions and bosons Those particles with half-integer spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer spins, such as 0, 1, 2, are known as bosons. The two families of particles obey different rules and broadly have different roles in the world around us. A key distinction between the two families is that fermions obey the Pauli exclusion principle: that is, there cannot be two identical fermions simultaneously having the same quantum numbers (meaning, roughly, having the same position, velocity and spin direction). Fermions obey the rules of Fermi–Dirac statistics. In contrast, bosons obey the rules of Bose–Einstein statistics and have no such restriction, so they may "bunch together" in identical states. Also, composite particles can have spins different from their component particles. For example, a helium-4 atom in the ground state has spin 0 and behaves like a boson, even though the quarks and electrons which make it up are all fermions. This has some profound consequences:
Quarks and leptons (including electrons and neutrinos), which make up what is classically known as matter, are all fermions with spin 1/2. The common idea that "matter takes up space" actually comes from the Pauli exclusion principle acting on these particles to prevent the fermions from being in the same quantum state. Further compaction would require electrons to occupy the same energy states, and therefore a kind of pressure (sometimes known as degeneracy pressure of electrons) acts to resist the fermions being overly close. Elementary fermions with other spins (3/2, 5/2, etc.) are not known to exist. Elementary particles which are thought of as carrying forces are all bosons with spin 1. They include the photon, which carries the electromagnetic force, the gluon (strong force), and the W and Z bosons (weak force). The ability of bosons to occupy the same quantum state is used in the laser, which aligns many photons having the same quantum number (the same direction and frequency), superfluid liquid helium resulting from helium-4 atoms being bosons, and superconductivity, where pairs of electrons (which individually are fermions) act as single composite bosons. Elementary bosons with other spins (0, 2, 3, etc.) were not historically known to exist, although they have received considerable theoretical treatment and are well established within their respective mainstream theories. In particular, theoreticians have proposed the graviton (predicted to exist by some quantum gravity theories) with spin 2, and the Higgs boson (explaining electroweak symmetry breaking) with spin 0. Since 2013, the Higgs boson with spin 0 has been considered proven to exist. It is the first scalar elementary particle (spin 0) known to exist in nature. Atomic nuclei have nuclear spin which may be either half-integer or integer, so that the nuclei may be either fermions or bosons.
Spin–statistics theorem
The spin–statistics theorem splits particles into two groups: bosons and fermions, where bosons obey Bose–Einstein statistics, and fermions obey Fermi–Dirac statistics (and therefore the Pauli exclusion principle). Specifically, the theorem requires that particles with half-integer spins obey the Pauli exclusion principle while particles with integer spin do not. As an example, electrons have half-integer spin and are fermions that obey the Pauli exclusion principle, while photons have integer spin and do not. The theorem was derived by Wolfgang Pauli in 1940; it relies on both quantum mechanics and the theory of special relativity. Pauli described this connection between spin and statistics as "one of the most important applications of the special relativity theory".
Magnetic moments
Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern–Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves. The intrinsic magnetic moment μ of a spin-1/2 particle with charge q, mass m, and spin angular momentum S is
μ = g s q 2 m S , {\displaystyle {\boldsymbol {\mu }}={\frac {g_{\text{s}}q}{2m}}\mathbf {S} ,}
where the dimensionless quantity gs is called the spin g-factor. For exclusively orbital rotations, it would be 1 (assuming that the mass and the charge occupy spheres of equal radius). The electron, being a charged elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g-factor, which has been experimentally determined to have the value −2.00231930436092(36), with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation. The value of 2 arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties; and the deviation from 2 arises from the electron's interaction with the surrounding quantum fields, including its own electromagnetic field and virtual particles. 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:
μ ν ≈ 3 × 10 − 19 μ B m ν c 2 eV , {\displaystyle \mu _{\nu }\approx 3\times 10^{-19}\mu _{\text{B}}{\frac {m_{\nu }c^{2}}{\text{eV}}},}
where the μν are the neutrino magnetic moments, mν are the neutrino masses, and μB 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 μB are "unnatural" because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses are known to be at most about 1 eV/c2, fine-tuning would be necessary in order to prevent large contributions to the neutrino mass via radiative corrections. The measurement of neutrino magnetic moments is an active area of research. Experimental results have put the neutrino magnetic moment at less than 1.2×10−10 times the electron's magnetic moment. On the other hand, elementary particles with spin but without electric charge, such as the photon and Z boson, do not have a magnetic moment.
Direction
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 for a spin-s particle measured along any direction can only take on the values
S i = ℏ s i , s i ∈ { − s , − ( s − 1 ) , … , s − 1 , s } , {\displaystyle S_{i}=\hbar s_{i},\quad s_{i}\in \{-s,-(s-1),\dots ,s-1,s\},}
where Si is the spin component along the i-th axis (either x, y, or z), si is the spin projection quantum number along the i-th axis, and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z axis:
S z = ℏ s z , s z ∈ { − s , − ( s − 1 ) , … , s − 1 , s } , {\displaystyle S_{z}=\hbar s_{z},\quad s_{z}\in \{-s,-(s-1),\dots ,s-1,s\},}
where Sz is the spin component along the z axis, sz is the spin projection quantum number along the z axis. One can see that there are 2s + 1 possible values of sz. The number "2s + 1" is the multiplicity of the spin system. For example, there are only two possible values for a spin-1/2 particle: sz = +1/2 and sz = −1/2. These correspond to quantum 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 a delta baryon, the possible values are +3/2, +1/2, −1/2, −3/2.
Vector For a given quantum state, one could think of a spin vector ⟨ S ⟩ {\textstyle \langle S\rangle } whose components are the expectation values of the spin components along each axis, i.e., ⟨ S ⟩ = [ ⟨ S x ⟩ , ⟨ S y ⟩ , ⟨ S z ⟩ ] {\textstyle \langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]} . 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: sx, sy and 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 Stern–Gerlach apparatus, the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin-1/2 particles, this 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 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 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-1/2 particle by 360° 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° rotation. (The plate trick and Möbius 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.
Mathematical formulation
Definitions
Intrinsic angular momentum A particle is said to possess intrinsic spin s {\displaystyle s} if the quantum-mechanical states of that particle in its own rest frame are eigenstates of the operator J 2 {\displaystyle \mathbf {J} ^{2}} with eigenvalue ℏ 2 s ( s + 1 ) {\displaystyle \hbar ^{2}s(s+1)} .
Relativistic quantum theory In relativistic quantum theory, spin is defined as an invariant label of a time-like unitary irreducible representation of the Poincaré group. For time-like representations, corresponding to massive particles, the unitary irreducible representations are characterized by the mass M {\displaystyle M} and the spin s {\displaystyle s} , where s {\displaystyle s} labels an irreducible projective representation of the rotation group S O ( 3 ) {\displaystyle SO(3)} .
Operator Spin obeys commutation relations analogous to those of the orbital angular momentum:
[ S ^ j , S ^ k ] = i ℏ ε j k l S ^ l , {\displaystyle \left[{\hat {S}}_{j},{\hat {S}}_{k}\right]=i\hbar \varepsilon _{jkl}{\hat {S}}_{l},}
where εjkl is the Levi-Civita symbol. It follows (as with angular momentum) that the eigenvectors of S ^ 2 {\displaystyle {\hat {S}}^{2}} and S ^ z {\displaystyle {\hat {S}}_{z}} (expressed as kets in the total S basis) are
S ^ 2 | s , m s ⟩ = ℏ 2 s ( s + 1 ) | s , m s ⟩ , S ^ z | s , m s ⟩ = ℏ m s | s , m s ⟩ . {\displaystyle {\begin{aligned}{\hat {S}}^{2}|s,m_{s}\rangle &=\hbar ^{2}s(s+1)|s,m_{s}\rangle ,\\{\hat {S}}_{z}|s,m_{s}\rangle &=\hbar m_{s}|s,m_{s}\rangle .\end{aligned}}}
The spin raising and lowering operators acting on these eigenvectors give
S ^ ± | s , m s ⟩ = ℏ s ( s + 1 ) − m s ( m s ± 1 ) | s , m s ± 1 ⟩ , {\displaystyle {\hat {S}}_{\pm }|s,m_{s}\rangle =\hbar {\sqrt {s(s+1)-m_{s}(m_{s}\pm 1)}}|s,m_{s}\pm 1\rangle ,}
where S ^ ± = S ^ x ± i S ^ y {\displaystyle {\hat {S}}_{\pm }={\hat {S}}_{x}\pm i{\hat {S}}_{y}} . But unlike orbital angular momentum, the eigenvectors are not spherical harmonics. They are not functions of θ and φ. There is also no reason to exclude half-integer values of s and ms. All quantum-mechanical particles possess an intrinsic spin s {\displaystyle s} (though this value may be equal to zero). The projection of the spin s {\displaystyle s} on any axis is quantized in units of the reduced Planck constant, such that the state function of the particle is, say, not ψ = ψ ( r ) {\displaystyle \psi =\psi (\mathbf {r} )} , but ψ = ψ ( r , s z ) {\displaystyle \psi =\psi (\mathbf {r} ,s_{z})} , where s z {\displaystyle s_{z}} can take only the values of the following discrete set:
s z ∈ { − s ℏ , − ( s − 1 ) ℏ , … , + ( s − 1 ) ℏ , + s ℏ } . {\displaystyle s_{z}\in \{-s\hbar ,-(s-1)\hbar ,\dots ,+(s-1)\hbar ,+s\hbar \}.}
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-1/2 observables are
S ^ = ℏ 2 σ , {\displaystyle {\hat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }},}
where in Cartesian components
S x = ℏ 2 σ x , S y = ℏ 2 σ y , S z = ℏ 2 σ z . {\displaystyle S_{x}={\frac {\hbar }{2}}\sigma _{x},\quad S_{y}={\frac {\hbar }{2}}\sigma _{y},\quad S_{z}={\frac {\hbar }{2}}\sigma _{z}.}
For the special case of spin-1/2 particles, σx, σy and σz are the three Pauli matrices:
σ x = ( 0 1 1 0 ) , σ y = ( 0 − i i 0 ) , σ z = ( 1 0 0 − 1 ) . {\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}
Pauli exclusion principle The Pauli exclusion principle states that the wavefunction
ψ ( r 1 , σ 1 , … , r N , σ N ) {\displaystyle \psi (\mathbf {r} _{1},\sigma _{1},\dots ,\mathbf {r} _{N},\sigma _{N})} for a system of N identical particles having spin s must change upon interchanges of any two of the N particles as
ψ ( … , r i , σ i , … , r j , σ j , … ) = ( − 1 ) 2 s ψ ( … , r j , σ j , … , r i , σ i , … ) . {\displaystyle \psi (\dots ,\mathbf {r} _{i},\sigma _{i},\dots ,\mathbf {r} _{j},\sigma _{j},\dots )=(-1)^{2s}\psi (\dots ,\mathbf {r} _{j},\sigma _{j},\dots ,\mathbf {r} _{i},\sigma _{i},\dots ).}
Thus, for bosons the prefactor (−1)2s will reduce to +1, for fermions to −1. This permutation postulate for N-particle state functions has most important consequences in daily life, e.g. the periodic table of the chemical elements.
Rotations
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
| a + 1 / 2 | 2 + | a − 1 / 2 | 2 = 1. {\displaystyle |a_{+1/2}|^{2}+|a_{-1/2}|^{2}=1.}
For a generic particle with spin s, we would need 2s + 1 such parameters. Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It 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:
∑ m = − j j a m ∗ b m = ∑ m = − j j ( ∑ n = − j j U n m