Spin (physics) - Citizendia

For other uses, see Spin (disambiguation).

In physics and chemistry, spin has a special meaning, representing a non-classical kind of angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number Although this special property is only explained in the relativistic quantum mechanics of Paul Dirac, it plays a most-important role already in non-relativistic quantum mechanics, e. g. , it essentially determines the structure of atoms.

In classical mechanics, any spin angular momentum of a body is associated with self rotation, e. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects g. , the rotation of the body around its own center of mass. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation For example, the spin of the Earth is associated with its daily rotation about the polar axis. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 A geographical pole, or geographic pole, is either of two fixed points on the surface of a spinning body or Planet, at 90 degrees from the Equator, based On the other hand, the orbital angular momentum of the Earth is associated with its annual motion around the Sun. A year (from Old English gēr) is the time between two recurrences of an event related to the Orbit of the Earth around the Sun The Sun (Sol is the Star at the center of the Solar System.

In fact, in classical theories there is no analogue to the quantum mechanical property meant by the name spin. The concept of this nonclassical property of elementary particles was first proposed in 1925 by Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit; but the name related to the phenomenon of spin in physics is Wolfgang Pauli. Ralph Kronig was a German-American Physicist ( March 10, 1904 — November 16, 1995) George Eugene Uhlenbeck ( December 6 1900, Batavia Dutch East Indies &ndash October 31 1988, Boulder Colorado) Samuel Abraham Goudsmit (born July 11, 1902 Den Haag, The Netherlands, died December 4, 1978 in Reno Nevada

1 Spin in quantum mechanics
2 Spin and the Pauli exclusion principle
3 Broad overview
4 History
5 Spin direction
- 5.1 Spin and rotations
- 5.2 Spin and Lorentz transformations
6 Spin and magnetic moments
7 The spin-statistics connection
8 Mathematical formulation of spin in quantum mechanics
9 Direct and indirect applications
10 See also
11 References
12 External links

Spin in quantum mechanics

In quantum mechanics, spin is an intrinsic property of all elementary particles related to angular momentum. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

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

$[S_i, S_j ] = i \hbar \epsilon_{ijk} S_k$

It follows (as with angular momentum) that the eigenvectors of $S 2$ and $S z$ (expressed as kets in the total $S$ basis) are:

$S^2 |s,m\rangle = \hbar^2 s(s + 1) |s,m\rangle$

$S_z |s,m\rangle = \hbar m |s,m\rangle$

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

$S_\pm |s,m\rangle = \hbar\sqrt{s(s+1)-m(m\pm 1)} |s,m\pm 1 \rangle$ , where $S_\pm = S_x \pm i S_y$

But unlike orbital angular momentum the eigenvectors are not spherical harmonics. In Physics, the canonical commutation relation is the relation between Canonical conjugate quantities (quantities which are related by definition such that one is The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of They are not functions of $θ$ and $φ$ . There is also no reason to exclude half integer values of s and m.

In quantum mechanics, the non-classical property spin is especially important for systems at atomic length scales, such as individual atoms, protons, or electrons. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Such particles and the spin of quantum mechanical systems ("particle spin") possess several non-classical features and for such systems spin angular momentum cannot be associated with rotation but instead refers only to the presence of an 'angular momentum-like' property. (Note that particles are quantum mechanical entities, which also exhibit wave-like behavior due to the so-called wave-particle duality. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and )

Precisely, in addition to their other properties, all quantum mechanical particles possess the above-mentioned non-classical kind of intrinsic "spin". This is quantized in units of the reduced action constant $\hbar$ , such that the state function of the particle is, e. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. g. , not $\psi = \psi(\mathbf r)$ , but $\psi =\psi(\mathbf r,\sigma)\,,$ where $σ$ is out of the following discrete set of values: $\sigma \in \{-S\cdot\hbar , -(S-1)\cdot\hbar , ... ,+(S-1)\cdot\hbar ,+S\cdot\hbar\}$ . One distinguishes bosons (S=0 or 1 or 2 or . In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein . . ) and fermions (S=1/2 or 3/2 or 5/2 or . In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. . . ). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.

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

$\psi ( \,...\, ;\,\mathbf r_i,\sigma_i\,;\, ...\,;\mathbf r_j,\sigma_j\,;\,...) \stackrel{!}{=}(-1)^{2S}\cdot \psi ( \,...\, ;\,\mathbf r_j,\sigma_j\,;\, ...\,;\mathbf r_i,\sigma_i\,;\,...)\,.$

Thus, for bosons the prefactor $( - 1) 2 S$ will reduce to +1, for fermions, in contrast, to (-1). The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In quantum mechanics all particles are either bosons or fermions. In relativistic quantum field theories also "supersymmetric" particles exist, where linear combinations of bosonic and fermionic components appear. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that Only in two dimensions you are allowed to replace the prefactor $( - 1) 2 S$ by any complex number of magnitude 1
(-> anyons). In Mathematics and Physics, an anyon is a type of particle that only occurs in two-dimensional systems

Electrons are fermions with S=1/2; quanta of light ("photons") are bosons with S=1. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena 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 $v\equiv c)$ , and the corresponding classical theory, that of Maxwell, is also relativistic. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric

The above permutation postulate for N-particle state functions has most-important consequences in daily life, e. g. the already mentioned periodic table of the chemists or biologists. The periodic table of the chemical elements is a tabular method of displaying the Chemical elements Although precursors to this table exist its invention is

After this condensed presentation of some essentials, a broad overview is given:

Broad overview

One of the most remarkable discoveries associated with quantum physics is the fact that elementary particles can possess non-zero spin. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made Elementary particles are particles that cannot be divided into any smaller units, such as the photon, the electron, and the various quarks. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. Theoretical and experimental studies have shown that the spin possessed by these 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 are true point particles. The classical electron radius, also known as the Lorentz radius or the Thomson scattering length is based on a classical (i A point particle (or point-like, often spelled pointlike) is an idealized object heavily used in Physics. The spin that they carry is a truly intrinsic physical property, akin to a particle's electric charge and mass.

According to quantum mechanics, the angular momentum of any system is quantized. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. The magnitude of angular momentum, $S$ , can only take on the values according to this relation:

$S = \hbar \, \sqrt{s (s+1)},$

where $\hbar$ is the reduced Planck's constant, and s is a non-negative integer or half-integer (0, 1/2, 1, 3/2, 2, etc. The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a half-integer is a Number of the form n + 1/2 where n is an Integer. ). For instance, electrons (which are elementary particles) are called "spin-1/2" particles because their intrinsic spin angular momentum has s = 1/2. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made

The spin carried by each elementary particle has a fixed s value that depends only on the type of particle, and cannot be altered in any known way (although, as we will see, it is possible to change the direction in which the spin "points". ) Every electron in existence possesses s = 1/2. Other elementary spin-1/2 particles include neutrinos and quarks. Neutrinos are Elementary particles that travel close to the Speed of light, lack an Electric charge, are able to pass through ordinary matter almost In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. On the other hand, photons are spin-1 particles, whereas the hypothetical graviton is a spin-2 particle. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In Physics, the graviton is a hypothetical Elementary particle, a Boson to be exact that mediates the force of Gravity in the framework The hypothetical Higgs boson is unique among elementary particles in having a spin of zero. The Higgs Boson is a hypothetical massive scalar Elementary particle predicted to exist by the Standard Model of Particle physics

The spin of composite particles, such as protons, neutrons, atomic nuclei, and atoms, is made up of the spins of the constituent particles, and their total angular momentum is the sum of their spin and the orbital angular momentum of their motions around one another. In Physics, a bound state is a composite of two or more building blocks ( particles or bodies) that behaves as a single object The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The angular momentum quantization condition applies to both elementary and composite particles. 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). 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. Nucleons ( Protons and Neutrons are spin =1/2 Subatomic particles composed of Quarks In the late 1980s the European Muon Collaboration In Physics, the quark model is a classification scheme for Hadrons in terms of their valence quarks, i In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the

History

Spin was first discovered in the context of the emission spectrum of alkali metals. An element's 'emission spectrum' is the relative intensity of Electromagnetic radiation of each Frequency it emits when it is Heated (or more generally when Trends The alkali metals show a number of trends when moving down the group - for instance decreasing electronegativity increasing reactivity and decreasing melting and boiling In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. An electron shell may be crudely thought of as an Orbit followed by Electrons around an Atom nucleus. 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 Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system.

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. Ralph Kronig was a German-American Physicist ( March 10, 1904 — November 16, 1995) Alfred Landé ( 13 December, 1888 &ndash 30 October 1976) was a German-American physicist known for his contributions to quantum theory 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. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. Largely due to Pauli's criticism, Kronig decided not to publish his idea.

In the fall of 1925, the same thought came to two Dutch physicists, George Uhlenbeck and Samuel Goudsmit. George Eugene Uhlenbeck ( December 6 1900, Batavia Dutch East Indies &ndash October 31 1988, Boulder Colorado) Samuel Abraham Goudsmit (born July 11, 1902 Den Haag, The Netherlands, died December 4, 1978 in Reno Nevada Under the advice of Paul Ehrenfest, they published their results. Paul Ehrenfest ( January 18, 1880 – September 25, 1933) was an Austrian Physicist and Mathematician, who 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). Llewellyn Hilleth Thomas ( 31 October 1903 - 20 April 1992) was a British Physicist and applied mathematician. 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. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. The tangent bundle effect is additive and relativistic; that is, it vanishes if c goes to infinity. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the 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). In Physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction to the precession of a Gyroscope in a rotating

Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics discovered by Schrödinger and Heisenberg. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the He pioneered the use of Pauli matrices as a representation of the spin operators, and introduced a two-component spinor wave-function. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the

Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac published the Dirac equation, which described the relativistic electron. In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J 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. The spin-statistics theorem in Quantum mechanics relates the spin of a particle to the statistics obeyed by that particle In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein

Spin direction

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (the axis of rotation of the particle). A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation 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 (say along the z-axis) can only take on the values

$\hbar s_z, \qquad s_z = - s, - s + 1, \cdots, s - 1, s$

where s is the principal spin quantum number discussed in the previous section. One can see that there are 2s+1 possible values of s_z. 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". In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. See spin-1/2. In Quantum mechanics, spin is an intrinsic property of all elementary particles.

For a given quantum state , it is possible to describe a spin vector $\lang S \rang$ whose components are the expectation values of the spin components along each axis, i. In Quantum mechanics, the expectation value is the predicted mean value of the result of an experiment e. , $\lang S \rang = [\lang s_x \rang, \lang s_y \rang, \lang s_z \rang]$ . This vector describes the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) 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. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain 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. In Quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is an important 1922 experiment on the Deflection 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. A gyroscope is a device for measuring or maintaining orientation, based on the principles of Angular momentum. 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). A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Physics, Astronomy, Chemistry, and Electrical engineering, the term magnetic moment of a system (such as a loop of Electric current The result is that the spin vector undergoes precession, just like a classical gyroscope. Precession refers to a change in the direction of the axis of a rotating object

Mathematically, quantum mechanical spin is not described by a vector as in classical angular momentum. It is described using a family of objects known as spinors. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the There are subtle differences between the behavior of spinors and vectors under coordinate rotations. In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed 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. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern To return the particle to its exact original state, one needs a 720 degree rotation. And a spin zero particle can only have a single quantum state ,even after torque is applied. Rotating a spin-2 particle 180 degree 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 degrees it is turned.

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 $a_{\pm 1/2}$ , giving amplitudes of finding it with projection of angular momentum equal to $\hbar/2$ and $-\hbar/2$ , satisfying the requirement

$|a_{1/2}|^2 + |a_{-1/2}|^2 \, = 1$

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 quantum mechanical inner product, and so should our transformation matrices:

$\sum_{m=-j}^{j} a_m^* b_m = \sum_{m=-j}^{j} (\sum_{n=-j}^j U_{nm} a_n)^* (\sum_{k=-j}^j U_{km} b_k)$

$\sum_{n=-j}^j \sum_{k=-j}^j U_{np}^* U_{kq} = \delta_{pq}$

Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO(3). In the mathematical field of Representation theory, a projective representation of a group G on a Vector space V over a In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of This article is about rotations in three-dimensional Euclidean space Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2). Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n There is one irreducible representation of SU(2) for each dimension. For example, spin 1/2 particles transform under rotations according to a 2-dimensional representation, which is generated by Pauli matrices:

$\begin{pmatrix} a_{1/2}' \\ a_{-1/2}' \end{pmatrix} = \exp{(i \sigma_z \gamma / 2)} \exp{(i \sigma_x \beta / 2)} \exp{(i \sigma_z \alpha / 2)} \begin{pmatrix} a_{1/2} \\ a_{-1/2} \end{pmatrix}$

where $α,β,γ$ are Euler angles. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant

Particles with higher spin transform in a similar way using higher-dimensional representations; see the article on Pauli matrices for matrices generating rotations for spin 1 and 3/2. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices.

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. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to 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 Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting

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 $ψ$ with each particle. In Quantum field theory, Dirac spinor is the Bispinor in the plane-wave solution \psi = \omega_\vec{p}\e^{-ipx} \ of the These spinors transform under Lorentz transformations according to the law

$\psi' = \exp{\left(\frac{1}{8} \omega_{\mu\nu} [\gamma_{\mu}, \gamma_{\nu}]\right)} \psi$

where $γ μ$ are gamma matrices and $ω μν$ is an antisymmetric 4x4 matrix parametrizing the transformation. In Mathematical physics, the gamma matrices, {γ0 γ1 γ2 γ3} also known as the Dirac matrices, form a matrix-valued It can be shown that the scalar product

$\langle\psi|\phi\rangle = \bar{\psi}\phi = \psi^{\dagger}\gamma_0\phi$

is preserved. (It is not, however, positive definite, so the representation is not unitary. )

Spin and magnetic moments

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. In Physics, Astronomy, Chemistry, and Electrical engineering, the term magnetic moment of a system (such as a loop of Electric current Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently 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. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is an important 1922 experiment on the Deflection

The intrinsic magnetic moment μ of a particle with charge q, mass m, and spin S, is

$\mu = g \, \frac{q}{2m}\, S$

where the dimensionless quantity g is called the g-factor. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units For the acceleration-related quantity in mechanics see ''g''-force. For exclusively orbital rotations it would be 1.

The electron, despite being an elementary particle, possesses a nonzero magnetic moment. In Atomic physics, the magnetic dipole moment of an Electron is caused by its intrinsic property of spin within a magnetic field 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. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. In Physics, the Landé g-factor is a particular example of a G-factor, namely for an Electron with both spin and Orbital angular 0023193043768(86), with the first 12 figures certain. 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. In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides 00231456893. . . arises from the electron's interaction with the surrounding electromagnetic field, including its own field. The electromagnetic field is a physical field produced by electrically charged objects.

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. In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. The magnetic moment of the neutron comes from the moments of the individual quarks and their orbital motions.

The neutrinos are both elementary and electrically neutral, and theory indicates that they have zero magnetic moment. Neutrinos are Elementary particles that travel close to the Speed of light, lack an Electric charge, are able to pass through ordinary matter almost The measurement of neutrino magnetic moments is an active area of research. As of 2001, the latest experimental results have put the neutrino magnetic moment at less than 1. Year 2001 ( MMI) was a Common year starting on Monday according to the Gregorian calendar. 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. Materials are physical Substances used as inputs to production or Manufacturing. 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. Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it The Curie point ( Tc) or Curie temperature, is a term in Physics and Materials science, named after Pierre Curie (1859-1906 A magnetic domain describes a region within a material which has uniform Magnetization. These are the ordinary "magnets" with which we are all familiar.

The study of the behavior of such "spin models" is a thriving area of research in condensed matter physics. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. 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. The Ising model, named after the physicist Ernst Ising, is a mathematical model in Statistical mechanics. These models have many interesting properties, which have led to interesting results in the theory of phase transitions. In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another

The spin-statistics connection

The spin of a particle has crucial consequences for its properties in statistical mechanics. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics Particles with half-integer spin obey Fermi-Dirac statistics, and are known as fermions. In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. They are required to occupy antisymmetric quantum states (see the article on identical particles. Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another even in principle ) This property forbids fermions from sharing quantum states - a restriction known as the Pauli exclusion principle. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 Particles with integer spin, on the other hand, obey Bose-Einstein statistics, and are known as bosons. In Statistical mechanics, Bose - Einstein statistics (or more colloquially B-E statistics determines the statistical distribution of In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein 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. The spin-statistics theorem in Quantum mechanics relates the spin of a particle to the statistics obeyed by that particle Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In fact, the connection between spin and statistics is one of the most important and remarkable consequences of special relativity.

Mathematical formulation of spin in quantum mechanics

Pauli matrices and spin operators

The quantum mechanical operators associated with spin observables are:

$S_x = {\hbar \over 2} \sigma_x$

$S_y = {\hbar \over 2} \sigma_y$

$S_z = {\hbar \over 2} \sigma_z$

In the special case of spin-1/2 $σ x$ , $σ y$ and $σ z$ are the three Pauli matrices, given by:

$\sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$

$\sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}$

$\sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$

Measurement of the spin along the x, y, and z axes

Each of the (hermitian) Pauli matrices has two eigenvalues, +1 and -1. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, an operator is a function which operates on (or modifies another function In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. A number of Mathematical entities are named Hermitian, after the Mathematician Charles Hermite: Hermitian adjoint In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes The corresponding normalized eigenvectors are:

$\psi_{x+} = \frac{1}{\sqrt{2}} \begin{pmatrix} {1}\\{1}\end{pmatrix}$ , $\psi_{x-} = \frac{1}{\sqrt{2}} \begin{pmatrix} {1}\\{-1}\end{pmatrix}$ , $\psi_{y+} = \frac{1}{\sqrt{2}} \begin{pmatrix} {1}\\{i}\end{pmatrix}$ , $\psi_{y-} = \frac{1}{\sqrt{2}} \begin{pmatrix} {1}\\{-i}\end{pmatrix}$ , $\psi_{z+} = \begin{pmatrix} 1\\0\end{pmatrix}$ , $\psi_{z-} = \begin{pmatrix} 0\\1\end{pmatrix}$ . In Quantum mechanics, Wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

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 spin operator ( $S x$ , $S y$ , $S z$ ) on that axis, ${\hbar \over 2}$ and ${-\hbar \over 2}$ . The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. 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 x-axis), the probability that its spin will be measured as ${\hbar \over 2}$ is just $\mid \langle \psi \mid \psi_{x+} \rangle \mid ^2$ . In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the Correspondingly, the probability that its spin will be measured as ${-\hbar \over 2}$ is just $\mid \langle \psi \mid \psi_{x-} \rangle \mid ^2$ . Following the measurement, the spin state of the particle will collapse into the corresponding eigenstate. In certain interpretations of quantum mechanics, wave function collapse is one of two processes by which Quantum systems apparently evolve according to the laws of 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 $\mid \langle \psi_{x+} \mid \psi_{x+} \rangle \mid ^2 = 1$ , etc), provided that no measurements of the spin are made along other axes (see section compatibility below).

Measurement of the 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 $\sigma_u = \hbar(u_x\sigma_x + u_y\sigma_y + u_z\sigma_z)/2$ . The operator $σ u$ has eigenvalues of $\pm\hbar/2$ , just like the usual Pauli 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:

$\frac{1}{\sqrt{2+2u_z}}\begin{bmatrix} 1+u_z \\ u_x+iu_y \end{bmatrix}.$

The above spinor is obtained by normalizing the left column of the matrix $1\pm\sigma_u$ where "1" is the 2x2 unit matrix. This trick of writing the eigenvectors of the Pauli matrices depends on certain details of the density matrix representation of quantum states.

Compatibility of spin measurements

Since the Pauli matrices anticommute, 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:

$\mid \langle \psi_{x+/-} \mid \psi_{y+/-} \rangle \mid ^ 2 = \mid \langle \psi_{x+/-} \mid \psi_{z+/-} \rangle \mid ^ 2 = \mid \langle \psi_{y+/-} \mid \psi_{z+/-} \rangle \mid ^ 2 = \frac{1}{2}$

So when we measure the spin of a particle along the x-axis as, for example, ${\hbar \over 2}$ , the particle's spin state collapses into the eigenstate $\mid \psi_{x+} \rangle$ . In certain interpretations of quantum mechanics, wave function collapse is one of two processes by which Quantum systems apparently evolve according to the laws of When we then subsequently measure the particle's spin along the y-axis, the spin state will now collapse into either $\mid \psi_{y+} \rangle$ or $\mid \psi_{y-} \rangle$ , each with probability $\frac{1}{2}$ . Let us say, in our example, that we measure ${-\hbar \over 2}$ . When we now return to measure the particle's spin along the x-axis again, the probabilities that we will measure ${\hbar \over 2}$ or ${-\hbar \over 2}$ are each $\frac{1}{2}$ (i. e. they are $\mid \langle \psi_{x+} \mid \psi_{y-} \rangle \mid ^ 2$ and $\mid \langle \psi_{x-} \mid \psi_{y-} \rangle \mid ^ 2$ ). This implies that our 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.

Direct and indirect applications

Well established direct applications of spin are nuclear magnetic resonance spectroscopy in chemistry; electron spin resonance spectroscopy in chemistry and physics; proton spin density with magnetic resonance imaging (MRI) in medicine; and GMR drive head technology in modern hard disks. Electron paramagnetic resonance (EPR or electron spin resonance (ESR Spectroscopy is a technique for studying Chemical species that have one or more unpaired Giant magnetoresistance (GMR is a quantum mechanical effect a type of Magnetoresistance effect observed in thin film structures composed of alternating Ferromagnetic A hard disk drive ( HDD) commonly referred to as a hard drive, hard disk, or fixed disk drive, is a Non-volatile storage device

A possible application of spin is as a binary information carrier in spin transistors. The magnetically-sensitive transistor (also known as the spin transistor or spintronic transistor--named for Spintronics, the technology which this development spawned originally Electronics based on spin transistors is called spintronics. Spintronics (a Neologism meaning "spin transport electronics" also known as magnetoelectronics is an Emerging technology which exploits the intrinsic

But finally we remind to the many indirect applications based on spin and the Pauli principle, e. g. the periodic table of Dmitri Mendeleev. The periodic table of the chemical elements is a tabular method of displaying the Chemical elements Although precursors to this table exist its invention is Dmitri Ivanovich Mendeleev (sometimes spelled Mendeleyev; Дми́трий Ива́нович Менделе́ев) ( &ndash) was a Russian chemist and

References

Griffiths, David J. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Particle physics, helicity is the projection of the spin \vec S onto the direction of momentum \hat p: h = \vec The Pauli Equation, also known as the Schrödinger-Pauli equation is the formulation of the Schrödinger equation for spin one-half particles which takes into account The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. In Theoretical physics, the Rarita-Schwinger equation is the relativistic Field equation of spin -3/2 Fermions It is similar to the In the study of the Representation theory of Lie groups, the study of representations of SU(2 is fundamental to the study of representations of Semisimple Lie In Quantum mechanics, spin is an intrinsic property of all elementary particles. Basis for spin magnetic moments A spin magnetic moment is induced by all charged particles but the particle most usable by modern technology is a certain Lepton In Quantum mechanics, spin multiplicity denotes the number of possible quantum states of a system with given principal spin quantum number S In Atomic physics, the spin quantum number is a Quantum number that parameterizes the intrinsic Angular momentum (or spin angular momentum or simply In Mathematics and Mathematical physics, the Euclidean group SE(d of Direct isometries is generated by Translations In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the Spintronics (a Neologism meaning "spin transport electronics" also known as magnetoelectronics is an Emerging technology which exploits the intrinsic Yrast (Swedish pronunciation, English) is a technical term in Nuclear physics that refers to a state of a nucleus with a minimum of Energy (2004). Introduction to Quantum Mechanics (2nd ed. ). Prentice Hall. ISBN 0-13-111892-7.

Shankar, R. (1994). "chapter 14-Spin", Principles of Quantum Mechanics (2nd ed. ). Springer. ISBN 0-306-44790-8.
Lawden, Derek (2005). The Mathematical Principles of Quantum Mechanics. Dover. ISBN 0-486-44223-3.

"Spintronics. Feature Article" in Scientific American, June 2002

External links

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