spin

In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles (hadrons), and atomic nuclei.^{[note 1]}

All elementary particles of a given kind have the same spin quantum number, an important part of the quantum state of a particle. 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. The spin direction (also called spin for short) of a particle is an important intrinsic degree of freedom.

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 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.

There are two types of angular momentum in quantum mechanics: Orbital angular momentum, which is a generalization of angular momentum in classical mechanics (L=r×p), and spin, which has no analogue in classical mechanics.^[1][2] Since spin is a type of angular momentum, it has the same dimensions: J•s in SI units. In practice, however, SI units are never used to describe spin: instead, it is written as a multiple of the reduced Planck constant ħ. In natural units, the ħ is omitted, so spin is written as a unitless number. The spin quantum numbers are always unitless numbers by definition.

The head-on collision of a quark (the red ball) from one proton (the orange ball) with a gluon (the green ball) from another proton with opposite spin; spin is represented by the blue arrows circling the protons and the quark. The blue question marks circling the gluon represent the question: Are gluons polarised? The particles ejected from the collision are a shower of quarks and one photon of light (the purple ball).

Spin 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 spins obey the same mathematical laws as quantized angular momenta do. On the other hand, spins have some peculiar properties that distinguish them from orbital angular momenta:

• Spin quantum numbers may take on half-odd-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 g-factor differing from 1. This could only occur classically if the internal charge of the particle were distributed differently from its mass.

Elementary particles

Elementary particles are particles for which there is no known method of division into smaller units. Theoretical and experimental studies have shown that the spin possessed by such particles cannot be explained by postulating that they are made up of even smaller particles rotating about a common center of mass (see classical electron radius); as far as can be determined, these elementary particles have no inner structure. The spin of an elementary particle is a truly intrinsic physical property, akin to the particle's electric charge and rest mass.

Father and Mother of the series Spin Family (2009) by physicist-turned-sculptor Julian Voss-Andreae. The two pictured objects illustrate the geometry of a spin 5/2 object (blue 'male' on the left) and spin 2 object (pink 'female' on the right). Spin Family, on display in the art exhibition "Quantum Objects" playfully equates fermions with the male and bosons with the female gender, depicting the first spin 1/2, 1, 3/2, 2, and 5/2 objects as a family of five.^[3]

The conventional definition of the spin quantum number s 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 quantised. 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, even values of n. That is why rather than was defined as the quantum mechanical unit of angular momentum. When spin was discovered it was too late to change.

All known matter is ultimately composed of elementary particles called fermions, and all elementary fermions have s = 1/2. Examples of fermions are the electron and positron, the quarks making up protons and neutrons, and the neutrinos. Elementary particles emit and receive one or more particles called bosons. This boson exchange gives rise to the three fundamental interactions ("forces") of the Standard model of particle physics; hence bosons are also called force carriers. These bosons have s=1. A basic example of a boson is the photon. Electromagnetism is the force that results when charged particles exchange photons.^{[citation needed]}

Theory predicts the existence of two bosons whose s differs from 1. The force carrier for gravity is the hypothetical graviton; theory suggests that it has s = 2. The Higgs mechanism predicts that elementary particles acquire nonzero rest mass by exchanging hypothetical Higgs bosons with an all-pervasive Higgs field. Theory predicts that the Higgs boson has s = 0. If so, it would be the only elementary particle for which this is the case.

Composite particles

The spin of composite particles, such as protons, neutrons, and atomic nuclei is usually understood to mean the total angular momentum. This is the sum of the spins and orbital angular momenta of the constituent particles. Such a composite spin is subject to the same quantization condition as any other angular momentum.

Composite particles are often referred to as having a definite spin, just like elementary particles; for example, the proton is a spin-1/2 particle. This is understood to refer to the spin of the lowest-energy internal state of the composite particle (i.e., a given spin and orbital configuration of the constituents).^[4]

It is not always easy to deduce the spin of a composite particle from first principles. For example, even though we know that the proton is a spin-1/2 particle, the question of how this spin is distributed among the three internal valence quarks and the surrounding sea quarks and gluons is an active area of research.

Delta baryons, which decay into protons and neutrons, have spin 3/2. All the three quarks inside a Delta baryon (Δ) have their spin axis pointing in the same direction, unlike the nearly identical proton and neutron (called "nucleons") in which the intrinsic spin of one of the three constituent quarks is always opposite the spin of the other two. This difference in spin alignment is the only quantum number distinction between the Δ⁺ and Δ⁰ and ordinary nucleons.

Atoms and molecules

The spin of atoms and molecules is the sum of the spins of unpaired electrons, which may be parallel or antiparallel. It is responsible for paramagnetism.

The spin-statistics theorem

The spin of a particle has crucial consequences for its properties in statistical mechanics. Particles with half-integer spin obey Fermi-Dirac statistics, and are known as fermions. They are required to occupy antisymmetric quantum states (see the article on identical particles.) This property forbids fermions from sharing quantum states – a restriction known as the Pauli exclusion principle. Particles with integer spin, on the other hand, obey Bose-Einstein statistics, and are known as bosons. These particles occupy "symmetric states", and can therefore share quantum states. The proof of this is known as the spin-statistics theorem, which relies on both quantum mechanics and the theory of special relativity. In fact, "the connection between spin and statistics is one of the most important applications of the special relativity theory".^[5]

Magnetic moments

Magnetic field lines around a magnetostatic dipole; the magnetic dipole itself is in the center and is seen from the side.

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-½ particle with charge q, mass m, and spin angular momentum S, is^[6]

where the dimensionless quantity g_s is called the spin g-factor. For exclusively orbital rotations it would be 1 (assuming that the mass and the charge occupy spheres of equal radius).

The electron, being a charged elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g-factor, which has been experimentally determined to have the value −2.0023193043622(15), with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation.^[7] 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.^[8] 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:^[9][10][11]

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.^[12]

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, 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.

Spin direction

Spin projection quantum number and spin 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 ^[13]

where S_i is the spin component along the i-axis (either x, y, or z), s_i is the spin projection quantum number along the i-axis, and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z-axis:

where S_z is the spin component along the z-axis, s_z is the spin projection quantum number along the z-axis.

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 spin-1/2 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 spin-3/2 particle, like a delta baryon, the possible values are +3/2, +1/2, −1/2, −3/2.

Spin vector

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 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 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 used in nuclear magnetic resonance sensing.

Mathematically, quantum mechanical spin is not described by vectors as in classical angular momentum, but by 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 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 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 of spin

Spin operator

Spin obeys commutation relations analogous to those of the orbital angular momentum:

where is the Levi-Civita symbol. It follows (as with angular momentum) that the eigenvectors of S² and S_z (expressed as kets in the total S basis) are:

The spin raising and lowering operators acting on these eigenvectors give:

, where

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 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 (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 and spin operators

The quantum mechanical operators associated with spin observables are:

In the special case of spin-1/2 particles, σ_x, σ_y and σ_z are the three Pauli matrices, given by:

Spin and the Pauli exclusion principle

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 (see Anyon).

Electrons are fermions with s = 1/2; quanta of light ("photons") are bosons with s = 1. This shows also explicitly that the property spin cannot be fully explained as a classical intrinsic orbital angular momentum, e.g., similar to that of a "spinning top", since orbital angular rotations would lead to integer values of s. Instead one is dealing with an essential legacy of relativity. The photon, in contrast, is always relativistic (velocity v ≈ c), and the corresponding classical theory, that of Maxwell, is also relativistic.

The above permutation postulate for N-particle state functions has most-important consequences in daily life, e.g. the periodic table of the chemists or biologists.

Spin and rotations

As described above, quantum mechanics states that component 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 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:

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). There is one n-dimensional irreducible representation of SU(2) for each dimension, though this representation is n-dimensional real for odd n and n-dimensional 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 and is the vector of spin operators.

(Click "show" at right to see a proof or "hide" to hide it.)

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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).^[14]

---

A generic rotation in 3-dimensional 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 D-matrix:

where

is Wigner's small d-matrix. Note that for γ = 2π and α = β = 0, i.e. a full rotation about the z-axis, the Wigner D-matrix 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 half-integer, the values of m are also all half-integers, 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 spin-statistics theorem.

Spin and Lorentz transformations

We could try the same approach to determine the behavior of spin under general Lorentz transformations, but we'd 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 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 with each particle. These spinors transform under Lorentz transformations according to the law

where are gamma matrices and is an antisymmetric 4x4 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.

Measuring spin along the x, y, and z axes

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 x-axis), 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 (see compatibility section below).

Measuring spin along an arbitrary axis

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 spin-1/2 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.

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 x-axis, and we then measure the spin along the y-axis, we have invalidated our previous knowledge of the x-axis 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 x-axis 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 y-axis, 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 x-axis 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 x-axis is no longer valid, since the spin along the x-axis will now be measured to have either eigenvalue with equal probability.

Spin and parity

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.

Applications

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

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

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.

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 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.

A possible future direct application of spin is as a binary information carrier in spin transistors. Original concept proposed in 1990 is known as Datta-Das spin transistor.^[15] Electronics based on spin transistors is called spintronics, which includes the manipulation of spins in semiconductor devices.

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

History

Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "two-valued 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.

Wolfgang Pauli

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 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. 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 ones). 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 discovered by Schrödinger 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. In 1940, Pauli proved the spin-statistics theorem, which states that fermions have half-integer 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.^[16]

See also

Notes

References

External links

• "Spintronics. Feature Article" in Scientific American, June 2002.
• Goudsmit on the discovery of electron spin.
• Nature: "Milestones in 'spin' since 1896."
• ECE 495N Lecture 36: Spin Online lecture by S. Datta