# Spin

Spin Type Particle (examples)
${\ displaystyle 0}$ Boson Higgs boson
${\ displaystyle {\ tfrac {1} {2}} \ hbar}$ Fermion Electron , neutrino , quarks
${\ displaystyle 1 \ hbar}$ Boson Photon , gluon , W boson and Z boson
${\ displaystyle {\ tfrac {3} {2}} \ hbar}$ Fermion supersymmetric particles (hypothetical)
${\ displaystyle 2 \ hbar}$ Boson Graviton (hypothetical)

In particle physics, spin (from English spin 'rotation') is the intrinsic angular momentum of particles. In the case of fundamental particles , like mass, it is an unchangeable internal particle property . It is a half or whole number multiple ( spin quantum number ) of the reduced Planckian quantum of action . Apart from the fact that it is not caused by the (rotary) movement of a mass, it has all the properties of a classical mechanical intrinsic angular momentum, in particular with regard to conservation of angular momentum and coordinate transformations , and is therefore also an axial vector . Spin can only be understood in terms of quantum mechanics. The spin statistics theorem connects the spin of a particle with the kind of statistical description of several identical particles: particles with a half - integer spin quantum number obey the Fermi-Dirac statistics and are called fermions , particles with an integer spin quantum number obey the Bose-Einstein statistics and are called bosons . ${\ displaystyle \ textstyle \ hbar}$

Fundamental particles with spins are known so far (see table on the right). Fundamental particles with the spins have been postulated, but so far not proven. ${\ displaystyle 0 \, \ hbar, {\ tfrac {1} {2}} \ hbar, 1 \, \ hbar}$${\ displaystyle {\ tfrac {3} {2}} \ hbar, 2 \, \ hbar}$

In composite systems, e.g. B. In the case of protons , neutrons , atomic nuclei , atoms , molecules , excitons , hadrons and particles , the spin is obtained by adding the spins and orbital angular momenta of the components according to the rules of quantum mechanical angular momentum addition . ${\ displaystyle \ Omega ^ {-}}$

A spin was first assigned to the electron in 1925 in order to be able to consistently explain a number of incomprehensible details of the optical spectra of atoms with a single concept (for the discovery and reception of the spin, see electron spin ). Spin has been attributed to the proton since 1928 because an anomaly in the specific heat of hydrogen gas cannot be explained otherwise. ${\ displaystyle {\ tfrac {1} {2}} \ hbar}$${\ displaystyle {\ tfrac {1} {2}} \ hbar}$

The half-integer spin cannot be explained clearly or semiclassically by a rotary movement. A formal justification was discovered in 1928 in relativistic quantum mechanics . The half-integer spin of the electrons and quarks leads via the spin statistics theorem to the Pauli principle , which is fundamental for the structure of atomic nuclei and the atomic shells . The Pauli principle also determines the chemical behavior of atoms , as expressed in the periodic table of the elements. According to this, the half-integer spin plays a decisive role in the structure of matter through to its macroscopic properties.

Stephen Hawking uses an arrow analogy in his book A Brief History of Time to illustrate the spin: “A particle with spin 0 is a point: It looks the same from all directions. A particle with spin 1, on the other hand, is like an arrow: It looks different from different directions. The particle only looks the same again with one complete revolution (360 degrees). A particle with spin 2 is like an arrow with a point at each end. It looks the same again after half a turn (180 degrees). Correspondingly, particles with a higher spin look the same again if one rotates by smaller fractions of a complete revolution. [In addition] there are] particles [...] that do not look the same again after one revolution: Rather, two complete revolutions are required! The spin of such particles is given as ½. "

Important experiments on spin are mostly based on the fact that a charged particle with spin also has a magnetic moment . In the Einstein-de-Haas effect , the change in the direction of the electron spins in an iron rod causes it to rotate macroscopically. In the Stern-Gerlach experiment , the electron spin enabled the first direct evidence of directional quantization . The effects of nuclear magnetic resonance or electron spin resonance are used in chemistry ( nuclear magnetic resonance spectroscopy, NMR), biology and medicine ( magnetic resonance tomography, MRT) for detailed investigations of materials, tissues and processes.

Unlike the half-integer spin of the leptons, the integer spin of the photon ( light quantum ) results from the long-known existence of electromagnetic waves with circular polarization . Direct experimental evidence was obtained in 1936 using the rotational movement of a macroscopic object after interacting with photons.

## Spin operator, eigenvalues ​​and quantum numbers

The spin operator obeys the same three commutation relations as orbital angular momentum operator and total angular momentum: ${\ displaystyle {\ hat {\ vec {s}}} = ({\ hat {s}} _ {x}, \, {\ hat {s}} _ {y}, \, {\ hat {s} } _ {z})}$

${\ displaystyle [{\ hat {s}} _ {x}, {\ hat {s}} _ {y}] = i \ hbar {\ hat {s}} _ {z}}$(also for cyclically swapped )${\ displaystyle x, y, z}$

Therefore all other general rules of the quantum mechanical angular momentum also apply here. During the orbital angular momentum due to integer multiples of the quantum of action may occur as eigenvalues, as eigenvalues for spin and half-integer multiples are possible. ${\ displaystyle {\ hat {\ vec {l}}}}$${\ displaystyle {\ hat {\ vec {l}}} \ cdot {\ hat {\ vec {p}}} = 0}$

Since the three components are not interchangeable, one chooses as the maximum possible set of interchangeable operators, analogous to the orbital angular momentum, the square of the magnitude ,, and its -component, (the projection onto the -axis) . An eigenstate of the particle zu has the eigenvalue ; the set of values ​​for the spin quantum number is included . For abbreviation, a particle with spin quantum number is often referred to as a “particle with spin ”. ${\ displaystyle {\ hat {\ vec {s}}} ^ {2}}$${\ displaystyle z}$${\ displaystyle {\ hat {s}} _ {z}}$${\ displaystyle z}$${\ displaystyle {\ hat {\ vec {s}}} ^ {2}}$${\ displaystyle s {\ mathord {(}} {\ mathord {s}} + {\ mathord {1}}) \, \ hbar ^ {2}}$${\ displaystyle \, s}$${\ displaystyle s = 0, \, {\ tfrac {1} {2}}, \, 1, \, {\ tfrac {3} {2}} \; \ dots}$${\ displaystyle \, s}$${\ displaystyle \, s}$

The eigenvalues ​​for are denoted by. In this, the magnetic spin quantum number has one of the values , all of which together are either half-integer (then in an even number) or only in whole numbers (then in an odd number). ${\ displaystyle {\ hat {s}} _ {z}}$${\ displaystyle \, m_ {s} \ hbar}$${\ displaystyle \, ({\ mathord {2}} {\ mathord {s}} + {\ mathord {1}})}$${\ displaystyle \, m_ {s} = - s, \, - (s - {\ mathord {1}}), \, \ dots, \, + s}$${\ displaystyle \, s}$

Observed values ​​for the spin quantum number of elementary particles are

The rules for adding two angular momentum apply exactly the same for orbital angular momentum and spin. Therefore, adding two half-integer angular impulses results in an integer (as with two integer ones too), while a half-integer and an integer angular momentum add up to a half-integer angular momentum. A system of bosons and fermions therefore has a half-integer total angular momentum if and only if it contains an odd number of fermions.

Even with many composite particles and quasiparticles, the angular momentum around the center of gravity is referred to as spin in the everyday language of physics (e.g. for protons, neutrons, atomic nuclei, atoms, ...). Here it can also have different values ​​for the same type of particle, depending on the excited state of the particle. In these composite systems, the angular momentum is formed from the spins and orbital angular momenta of their fundamental components according to the generally applicable rules of quantum mechanical addition. They are not considered further here.

## Boson, fermion, particle number conservation

The spin leads to the fundamental and unchangeable classification of the elementary particles into bosons (spin whole-number) and fermions (spin half- integer ). This is a basis of the Standard Model . Thus the total angular momentum of a fermion is half-integer in every conceivable state, that of a boson is an integer. It also follows that a system that contains an odd number of fermions in addition to any number of bosons can only have a half-integer total angular momentum, and with an even number of fermions only an integer total angular momentum.

From the theorem of the conservation of the total angular momentum of a system in all possible processes follows the restriction - which is consistent with the observation - that the fermions can only be generated or annihilated in pairs , never individually, because otherwise the total angular momentum changes from an integral to a half-integer Value or vice versa. Bosons, on the other hand, can also be created or destroyed individually.

## Exchange symmetry, statistics, Pauli principle

The class division into bosons (spin integer) and fermions (spin half integer) has strong effects on the possible states and processes of a system in which several particles of the same type are present. Since, because of the indistinguishability of similar particles, exchanging two of them creates the same physical state of the system, the state vector (or the wave function ) can only remain the same or change its sign during this exchange. All observations show that the first case always applies to bosons ( symmetry of the wave function when exchanged), but always the second applies to fermions ( antisymmetry of the wave function when exchanged). The direct consequence of the antisymmetry is the Pauli principle , according to which there can be no system that contains two identical fermions in the same single-particle state. This principle determines z. B. the structure of the atomic shell and is therefore one of the foundations for the physical explanation of the properties of macroscopic matter (e.g. the chemical behavior of the elements in the periodic table and the (approximate) incompressibility of liquids and solids). The fact that there are two different commutation symmetries explains the great differences between many-body systems made up of fermions and bosons. Examples are the electron gas in metal (fermions) or the photons in cavity radiation (bosons), but also all of astrophysics . When treated with statistical methods, fermions follow the Fermi-Dirac statistics , and bosons follow the Bose-Einstein statistics . The spin statistics theorem provides a profound reason for this connection . Although the forces emanating from the spins are mostly negligible (magnetic dipole interaction!) And are usually completely neglected in the theoretical description, the mere property of the particles to have a half or whole number spin thus has far-reaching consequences in the macroscopic world.

## Spin operator and basis states for spin ½

The spin operator has three components that each have exactly two eigenvalues . Since the three components have the same commutation relations as in every angular momentum operator , there are no common eigen-states. If you choose (as usual) the alignment along the -axis, then the two eigenstates are designated with the quantum numbers as "parallel" or "antiparallel" to the -axis. and then have the expectation values ​​zero. ${\ displaystyle {\ hat {\ vec {s}}} = ({\ hat {s}} _ {x}, \, {\ hat {s}} _ {y}, \, {\ hat {s} } _ {z})}$${\ displaystyle s = {\ tfrac {1} {2}}}$${\ displaystyle \ pm {\ tfrac {\ hbar} {2}}}$${\ displaystyle z}$${\ displaystyle {\ hat {s}} _ {z}}$${\ displaystyle m_ {s} = \ pm {\ tfrac {1} {2}}}$${\ displaystyle z}$${\ displaystyle {\ hat {s}} _ {x}}$${\ displaystyle {\ hat {s}} _ {y}}$

In addition to the general properties of the quantum mechanical angular momentum, spin also has special properties. They are based on having only two eigenvalues. Therefore, the double application gives the up or dump operator always zero: . ${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle {\ hat {s}} _ {z}}$ ${\ displaystyle {\ hat {s}} _ {\ pm} = {\ hat {s}} _ {x} \ pm i {\ hat {s}} _ {y}}$${\ displaystyle {\ hat {s}} _ {\ pm} ^ {2} = 0}$

To simplify the formulas, Wolfgang Pauli carried out

${\ displaystyle {\ hat {s}} _ {i} = {\ tfrac {\ hbar} {2}} {\ hat {\ sigma}} _ {i}}$(for )${\ displaystyle i = x, \, y, \, z}$

the three Pauline spin operators introduced. From then follows (for ) ${\ displaystyle \ sigma _ {x}, \ sigma _ {y}, \ sigma _ {z}}$${\ displaystyle {\ hat {s}} _ {\ pm} ^ {2} = 0}$${\ displaystyle i, j = x, y, z; \ \ i \ neq j}$

${\ displaystyle {\ hat {\ sigma}} _ {i} ^ {2} = 1 \, \ quad {\ hat {\ sigma}} _ {j} {\ hat {\ sigma}} _ {i} = - {\ hat {\ sigma}} _ {i} {\ hat {\ sigma}} _ {j} \;, \ quad ({\ hat {\ vec {\ sigma}}} \ cdot {\ hat {\ vec {p}}}) ^ {2} = {\ hat {\ vec {p}}} \; ^ {2}}$.

The last equation is valid for every other vector operator, the components of which are interchangeable with and with each other . ${\ displaystyle {\ hat {\ vec {p}}}}$${\ displaystyle {\ hat {\ vec {s}}}}$

The obscure conclusions:

• Because is . In other words, in every conceivable state a spin particle has a well-defined and always the same value to the square of the component of its spin in any direction, the largest that is possible at all. In the two states of "(anti-) parallel" alignment to the axis, the two components perpendicular to the square of the magnitude are therefore twice as large as the component along the alignment axis . A normal vector with these properties is not parallel to the -axis, but rather closer to the xy-plane perpendicular to it .${\ displaystyle {\ hat {\ sigma}} _ {i} ^ {2} = 1}$${\ displaystyle {\ hat {s}} _ {x} ^ {2} = {\ hat {s}} _ {y} ^ {2} = {\ hat {s}} _ {z} ^ {2} = ({\ tfrac {\ hbar} {2}}) ^ {2}}$${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle z}$${\ displaystyle z}$
• The component of the vector in the direction of the spin always has the same magnitude as the vector itself.${\ displaystyle {\ hat {\ vec {p}}}}$

The two states (in the parlance "spin parallel or anti-parallel to the axis", often also referred to with the descriptive symbols or ) form a basis in the two-dimensional complex state space for the spin degree of freedom of a spin particle. The state in which the spin is aligned parallel to any other direction is also a linear combination of these two basis vectors with certain complex coefficients. For the state with spin parallel to the axis z. B. both coefficients have the same magnitude, also for the state parallel to the -axis , but with a different complex phase. Even if the spatial directions and are perpendicular to each other, the correspondingly aligned states are not orthogonal (the only state that is too orthogonal ). ${\ displaystyle | m_ {s} \ rangle = \ left | \ pm {\ tfrac {1} {2}} \ right \ rangle}$${\ displaystyle z}$${\ displaystyle \ left | \ uparrow \ right \ rangle}$${\ displaystyle \ left | \ downarrow \ right \ rangle}$${\ displaystyle \ mathbb {C} ^ {2}}$${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \ left | {+ {\ tfrac {1} {2}}} \ right \ rangle}$${\ displaystyle \ in \ mathbb {C} ^ {2}}$${\ displaystyle \ left | {- {\ tfrac {1} {2}}} \ right \ rangle}$

Note: The matrix representation of the Pauli spin operators are the Pauli matrices . Mathematically, the smallest representations of the spinalgebra are the spinors .

## Spin ½ and three-dimensional vector

The expected value of the angular momentum vector has, among all possible values ​​of the angular momentum quantum number (0, 1/2, 1, 3/2, ...) only for spin ½ the two properties that are clearly associated with a vector in three-dimensional space: It has in every possible one State always the same length and always a well-defined direction. ${\ displaystyle \ langle {\ hat {\ vec {s}}} \ rangle = (\ langle {\ hat {s}} _ {x} \ rangle, \, \ langle {\ hat {s}} _ {y } \ rangle, \, \ langle {\ hat {s}} _ {z} \ rangle)}$${\ displaystyle \ vert \ langle {\ hat {\ vec {s}}} \ rangle \ vert = {\ tfrac {1} {2}} \ hbar}$

Because for any spin state (normalized with ) is ${\ displaystyle \ vert \ chi \ rangle = \ alpha \ left | \ uparrow \ right \ rangle + \ beta \ left | \ downarrow \ right \ rangle}$${\ displaystyle \ vert \ alpha \ vert ^ {2} + \ vert \ beta \ vert ^ {2} = 1}$

${\ displaystyle \ vert \ langle {\ hat {\ vec {s}}} \ rangle \ vert ^ {2} = \ langle \ chi \ vert {\ hat {s}} _ {x} \ vert \ chi \ rangle ^ {2} + \ langle \ chi \ vert {\ hat {s}} _ {y} \ vert \ chi \ rangle ^ {2} + \ langle \ chi \ vert {\ hat {s}} _ {z} \ vert \ chi \ rangle ^ {2} = {\ tfrac {1} {4}} \ hbar ^ {2} (\ vert \ alpha \ vert ^ {2} + \ vert \ beta \ vert ^ {2}) ^ {2} \ equiv ({\ tfrac {1} {2}} \ hbar) ^ {2} \.}$

Furthermore, for any spin state (i.e. for any linear combination of and ) there is exactly one direction in three-dimensional space to which the spin is then as parallel as in the state to the -axis. For the linear combination , the polar angle and azimuth angle of the orientation direction can be taken from the equation . This corresponds to the idea of ​​a normal vector in three-dimensional space, which one can always use to define the axis. ${\ displaystyle \ left | {+ {\ tfrac {1} {2}}} \ right \ rangle}$${\ displaystyle \ left | {- {\ tfrac {1} {2}}} \ right \ rangle}$${\ displaystyle \ left | {+ {\ tfrac {1} {2}}} \ right \ rangle}$${\ displaystyle z}$${\ displaystyle \ left \ vert \ chi \ right \ rangle = \ alpha \ left | \ uparrow \ right \ rangle + \ beta \ left | \ downarrow \ right \ rangle}$${\ displaystyle \ theta}$${\ displaystyle \ phi}$${\ displaystyle {\ tfrac {\ alpha} {\ beta}} = {\ tfrac {\ cos (\ theta / 2)} {\ exp {(i \ phi) \, \ sin (\ theta / 2)}} }}$${\ displaystyle z}$

Both of these only apply to the quantum number among all angular momenta possible in quantum mechanics . In this respect, of all quantum mechanical angular momentum, the spin comes closest to the idea of ​​a vector. The vector operator, on the other hand, has some highly unusual properties (see previous section). ${\ displaystyle s = {\ tfrac {1} {2}}}$${\ displaystyle {\ tfrac {1} {2}}}$ ${\ displaystyle {\ hat {\ vec {s}}} = ({\ hat {s}} _ {x}, \, {\ hat {s}} _ {y}, \, {\ hat {s} } _ {z})}$

## Spin ½ as the equivalent of all 2-state systems

If a physical system only has two basic states (at least in an approximate view, e.g. with two neighboring energy levels, while the others, more distant, are neglected), it is formally an exact image of the 2-state system for the spin . For this system three operators can be defined regardless of their physical meaning: An upgrade operator and a dismount operator converts the second base state into the first or vice versa, and otherwise results in zero. The third operator gives the first base state the eigenvalue and the second . If these operators are named in sequence , they satisfy the same equations as the operators of the same name for the spin . They can also be rewritten in the vector operator , which, like every angular momentum operator, describes the infinitesimal rotations in an (abstract) three-dimensional space due to its commutation relations. ${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle + {\ tfrac {1} {2}}}$${\ displaystyle - {\ tfrac {1} {2}}}$${\ displaystyle {\ hat {s}} _ {+}, \, {\ hat {s}} _ {-}, {\ hat {s}} _ {z}}$${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle {\ hat {\ vec {s}}} = ({\ hat {s}} _ {x}, \, {\ hat {s}} _ {y}, {\ hat {s}} _ {z})}$

The mathematical background of this equivalence is the fact that the basic transformations in the two-dimensional complex Hilbert space form a representation of the group SU (2) which is “twice as large” as the group SO (3) of the rotations in real three-dimensional space. The difference to the "normal" rotations in three-dimensional space is that the rotation generated by the spin operator with the rotation angle 360 ​​° is not represented by the identity matrix , but by . The physical state goes into itself, but the state vector turns into its negative. One is compatible with the other because state vectors that differ by only one complex factor describe the same state . Only a 720 ° rotation produces the same state vector again. ${\ displaystyle \ mathbf {1}}$${\ displaystyle - \ mathbf {1}}$

If one takes different elementary particles for the two basic states, e.g. proton and neutron , or electron and electron-neutrino , the physical quantity defined by this procedure is called the isospin of the particle. This has also proven itself for multi-particle systems, i.e. H. their states can be classified according to how the isospins of their individual particles add up to the total isospin, whereby the rules of addition of quantum mechanical angular momenta are fully valid. This isospin concept played an important role in the development of elementary particle physics .

## Two particles with spin ½

The total spin can have the values and here. With the designation for the base states of each of the particles, the two-particle states are formed with the quantum numbers and so: ${\ displaystyle \, S = 1}$${\ displaystyle \, S = 0}$${\ displaystyle \ left | \ uparrow \ right \ rangle \, \ left | \ downarrow \ right \ rangle}$${\ displaystyle S}$${\ displaystyle M_ {S}}$

${\ displaystyle \ {\, \ left | {\ uparrow \ uparrow} \ right \ rangle \, \ {\ tfrac {1} {\ sqrt {2}}} (\ left | {\ uparrow \ downarrow} \ right \ rangle + \ left | {\ downarrow \ uparrow} \ right \ rangle) \, \ \ left | {\ downarrow \ downarrow} \ right \ rangle \, \}}$for ( triplet )${\ displaystyle \, S = 1 \;, \ M_ {S} = + 1, \, 0, \, - 1}$
${\ displaystyle {\ tfrac {1} {\ sqrt {2}}} (\ left | \ uparrow \ downarrow \ right \ rangle - \ left | \ downarrow \ uparrow \ right \ rangle)}$for ( singlet )${\ displaystyle \, S = 0, \; M_ {S} = 0}$

The two cases (i.e. the component of the total spin is zero) are the simplest examples of an entangled state consisting of two summands. Here, in each of the two summands and the -components of the two individual spins together result in zero. This no longer applies if, instead of the (equally large) spins, other vector operators are considered that have different sizes for the two particles. For example, the magnetic moments of electron and proton in the H atom differ by a factor of approx. 700. If for clarification or is written for the electron with its large magnetic moment , the two states are called . While every single one of the summands shows a magnetic moment almost the same size as the electron, aligned in ( ) -direction or in ( ) -direction, the entire magnetic moment of the atom in such an entangled state has the -component zero. This shows that both summands and must be present at the same time so that this can result. ${\ displaystyle M_ {S} = 0}$${\ displaystyle z}$${\ displaystyle \ left | \ uparrow \ downarrow \ right \ rangle}$${\ displaystyle \ left | \ downarrow \ uparrow \ right \ rangle}$${\ displaystyle z}$${\ displaystyle \ left | \ Uparrow \ right \ rangle}$${\ displaystyle \ left | \ Downarrow \ right \ rangle}$${\ displaystyle (M_ {S} = 0)}$${\ displaystyle {\ tfrac {1} {\ sqrt {2}}} (\ left | \ Uparrow \ downarrow \ right \ rangle \ pm \ left | \ Downarrow \ uparrow \ right \ rangle)}$${\ displaystyle + z}$${\ displaystyle -z}$${\ displaystyle z}$${\ displaystyle \ left | \ Uparrow \ downarrow \ right \ rangle}$${\ displaystyle \ left | \ Downarrow \ uparrow \ right \ rangle}$

## Two equal particles with spin ½

### Exchange symmetry in spin and position coordinates

The triplet state is symmetric, the singlet state antisymmetric with regard to the spins, because the interchanging of the two particles means here to write the two arrows for their spin state in the above formulas in reverse order. Since the complete state vector of two identical fermions changes the sign when all their coordinates are exchanged , the position-dependent part that exists in addition to the spin component must also have a defined symmetry, antisymmetric in the triplet, symmetric in the singlet. If the spatial coordinates are exchanged, the charge distributions of both electrons are simply exchanged, but their shape remains exactly the same as before. However, if the charge distributions overlap, two different values ​​arise for the electrostatic repulsion energy: In the antisymmetrically entangled spatial state, the amount of energy is smaller than in the symmetrical one, because the probability of both electrons being at the same place in the antisymmetrical spatial state is certainly zero, in the symmetrical state it is not (im Overlap area). This purely quantum mechanical effect is called the exchange interaction . It explains the strong influence of the total spin of the electrons on the energy level of their atom, although the spins themselves have no electrostatic and only a slight magnetic interaction. ${\ displaystyle | \ psi ({\ vec {r}} _ {1}, {\ vec {r}} _ {2}) \ rangle}$

### The spherically symmetric singlet state

If one forms the state vector for the singlet state not with the spin states aligned in -direction but with those aligned in -direction , the state is still one and the same (because there is only one): ${\ displaystyle z}$${\ displaystyle \ left | \ uparrow \ right \ rangle \, \ left | \ downarrow \ right \ rangle}$${\ displaystyle x}$${\ displaystyle \ left | \ leftarrow \ right \ rangle \, \ left | \ rightarrow \ right \ rangle}$

${\ displaystyle {\ tfrac {1} {\ sqrt {2}}} \; (\, \ left | \ uparrow \ downarrow \ right \ rangle - \ left | \ downarrow \ uparrow \ right \ rangle \,) \ quad \ equiv \ quad {\ tfrac {1} {\ sqrt {2}}} \; (\, \ left | \ leftarrow \, \ rightarrow \ right \ rangle - \ left | \ rightarrow \, \ leftarrow \ right \ rangle \,) \ cdot}$

Formally, this is a sequence of and . ${\ displaystyle \ left | {\ rightarrow} \ right \ rangle = {\ tfrac {1} {\ sqrt {2}}} (\, \ left | {\ uparrow} \ right \ rangle + \ left | {\ downarrow } \ right \ rangle \,)}$${\ displaystyle \ left | {\ leftarrow} \ right \ rangle = {\ tfrac {1} {\ sqrt {2}}} (\, \ left | {\ uparrow} \ right \ rangle - \ left | {\ downarrow } \ right \ rangle \,)}$

There is a thought experiment for this which illuminates the difficulties of perception in understanding the superposition of indivisible particles:

1. In a He + ion with the one 1s electron in the state , the yield with which an electron in the state can be extracted is measured . Answer: 50%.${\ displaystyle \ left | {\ leftarrow} \ right \ rangle}$${\ displaystyle \ left | {\ uparrow} \ right \ rangle}$
2. The He + ion now traps a second electron in the 1s state. Because both electrons have the same spatial wave functions, the state is symmetrical in terms of location and antisymmetric in terms of spin. The new electron does not simply set its spin opposite to the existing one ( ), but the correct entanglement for the singlet is automatically formed (according to the formula above). This singlet state is (although the vector looks different) the same one that would have formed from two electrons in the states .${\ displaystyle \ left | {\ leftarrow \, \ rightarrow} \ right \ rangle}$${\ displaystyle \ left | \ uparrow \ right \ rangle, \ left | \ downarrow \ right \ rangle}$
3. As a result, the same measurement as in No. 1 (extraction of ) now (i.e. after step 2) shows a yield of 100%. This apparent contradiction "per se" is only compatible with the view trained on macroscopic conditions if both electrons could have "split up" and reassembled with the correct halves.${\ displaystyle \ left | {\ uparrow} \ right \ rangle}$

## Spin and Dirac equation, anomalous magnetic moment

The theoretical justification of the spin is based on the Dirac equation , discovered by Paul Dirac in 1928 , which, as a relativistically correct wave equation, takes the place of the non-relativistic Schrödinger equation. A condition for the relativistic invariance of the associated equation for the energy is that energy and momentum appear linearly in it. This is in the Schrödinger equation is not the case, because it is based on according to classical mechanics , in operators: . Dirac found in${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle E = {\ tfrac {p ^ {2}} {2m}}}$${\ displaystyle {\ hat {H}} = {\ tfrac {{\ hat {p}} ^ {2}} {2m}}}$

${\ displaystyle {\ hat {\ vec {\ sigma}}} \ cdot {\ hat {\ vec {p}}} = {\ hat {| {\ vec {p}} |}}}$.

the linear operator we are looking for for the magnitude of the momentum. In the further formulation of this approach, the Paulische matrices had to be according to ${\ displaystyle 2 {\ times} 2}$${\ displaystyle {\ hat {\ vec {\ sigma}}}}$

${\ displaystyle {\ hat {\ vec {\ alpha}}} = {\ begin {pmatrix} 0 & {\ hat {\ vec {\ sigma}}} \\ {\ hat {\ vec {\ sigma}}} & 0 \ end {pmatrix}}}$

can be expanded to form matrices. This showed that for a free particle, for which one has to assume conservation of angular momentum, the orbital angular momentum is not a constant of motion, but the quantity identified as the total angular momentum . The constant additional term is the spin. ${\ displaystyle 4 {\ times} 4}$${\ displaystyle {\ hat {\ vec {l}}} = {\ hat {\ vec {r}}} \ times {\ hat {\ vec {p}}}}$${\ displaystyle {\ hat {\ vec {j}}} = {\ tfrac {\ hbar} {2}} {\ hat {\ vec {\ sigma}}}} + {\ hat {\ vec {r}}} \ times {\ hat {\ vec {p}}}}$${\ displaystyle {\ hat {\ vec {s}}} = {\ tfrac {\ hbar} {2}} {\ hat {\ vec {\ sigma}}}}$

If the effect of a static magnetic field is added to the Dirac equation, the result is an additional energy similar to that of a magnetic dipole . This dipole is parallel to the spin, just like the magnetic dipole of a circular current is parallel to its orbital angular momentum. However, compared to the orbital angular momentum of the circular current, it has exactly twice the strength. The anomalous magnetic moment of the Dirac particle is thus larger by the anomalous spin-g-factor than can be understood classically. ${\ displaystyle g_ {s} = 2}$

Experimentally, however, the electron has a value of approximately 2.00232. This deviation of the spin-g-factor of the electron${\ displaystyle g_ {e}}$ is explained by quantum electrodynamics .

## Remarks

1. A rolling cone ball has an angular momentum of approx.${\ displaystyle 3 \ cdot 10 ^ {33} \; \ hbar}$
2. Mathematically, the SU (2) is the superposition group of the SO (3)

## Individual evidence

1. GE Uhlenbeck, S. Goudsmit: Replacement of the hypothesis of non-mechanical compulsion by a requirement regarding the internal behavior of each individual electron . In: Natural Sciences . tape 13 , no. 47 , 1925, pp. 953-954 , doi : 10.1007 / BF01558878 .
2. ^ DM Dennison: A Note on the Specific Heat of the Hydrogen Molecule . In: Proceedings of the Royal Society of London Series A . tape 115 , no. 771 , 1927, pp. 483-486 , doi : 10.1098 / rspa.1927.0105 . Why exactly a macroscopically measurable property of the H 2 molecule leads to the spin of the atomic nuclei is described in detail in Jörn Bleck-Neuhaus: Elementare Particles. Modern physics from the atoms to the standard model (=  Springer textbook ). Springer-Verlag, Berlin 2010, ISBN 978-3-540-85299-5 , chap. 7 , doi : 10.1007 / 978-3-540-85300-8_7 .
3. ^ Richard Beth: Mechanical Detection and Measurement of the Angular Momentum of Light . In: Physical Review . tape 50 , 1936, pp. 115-125 , doi : 10.1103 / PhysRev.50.115 .
4. Cornelius Noack : Comments on the quantum theory of orbital angular momentum . In: Physical sheets . tape 41 , no. 8 , 1985, pp. 283–285 ( see homepage [PDF; 154 kB ; accessed on November 26, 2012]).
5. W. Pauli: On the quantum mechanics of the magnetic electron , Zeitschrift für Physik Vol. 43, p. 601 (1927)
6. Sakurai, Modern Quantum Mechanics , chap. 3.4)
7. a b See e.g. BU Krey and A. Owen: Basic Theoretical Physics - A Concise Overview , Berlin, Springer 2007, ISBN 978-3-540-36804-5 , in particular the chapter on Einstein-Podolski-Rosen paradoxes
8. A simple representation in uni-bremen.de: State of identical fermions
9. ^ PAM Dirac: The Quantum Theory of the Electron . In: Proceedings of the Royal Society of London. Series A . tape 117 , no. 778 , 1928, pp. 610-624 , doi : 10.1098 / rspa.1928.0023 .