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and therefore a kind of pressure sometimes known as degeneracy pressure of electrons acts to resist the fermions being overly close elementary fermions with other spins 3 2 5 2 etc are not known to exist elementary particles which are thought of as carrying forces are all bosons with spin 1 they include the photon which carries the electromagnetic force the gluon strong force and the w and z bosons weak force the ability of bosons to occupy the same quantum state is used in the laser which aligns many photons having the same quantum number the same direction and frequency superfluid liquid helium resulting from helium 4 atoms being bosons and superconductivity where pairs of electrons which individually are fermions act as single composite bosons elementary bosons with other spins 0 2 3 etc were not historically known to exist although they have received considerable theoretical treatment and are well established within their respective mainstream theories in particular theoreticians have proposed the graviton predicted to exist by some quantum gravity theories with spin 2 and the higgs boson explaining electroweak symmetry breaking with spin 0 since 2013 the higgs boson with spin 0 has been considered proven to exist 14 it is the first scalar elementary particle spin 0 known to exist in nature atomic nuclei have nuclear spin which may be either half integer or integer so that the nuclei may be either fermions or bosons spin statistics theorem edit main article spin statistics theorem the spin statistics theorem splits particles into two groups bosons and fermions where bosons obey bose einstein statistics and fermions obey fermi dirac statistics and therefore the pauli exclusion principle specifically the theorem requires that particles with half integer spins obey the pauli exclusion principle while particles with integer spin do not as an example electrons have half integer spin and are fermions that obey the pauli exclusion principle while photons have integer spin and do not the theorem was derived by wolfgang pauli in 1940 it relies on both quantum mechanics and the theory of special relativity pauli described this connection between spin and statistics as one of the most important applications of the special relativity theory 15 magnetic moments edit schematic diagram depicting the spin of the neutron as the black arrow and magnetic field lines associated with the neutron magnetic moment the neutron has a negative magnetic moment while the spin of the neutron is upward in this diagram the magnetic field lines at the center of the dipole are downward particles with spin can possess a magnetic dipole moment just like a rotating electrically charged body in classical electrodynamics these magnetic moments can be experimentally observed in several ways e g by the deflection of particles by inhomogeneous magnetic fields in a stern gerlach experiment or by measuring the magnetic fields generated by the particles themselves the intrinsic magnetic moment μ of a spin 1 2 particle with charge q mass m and spin angular momentum s is 16 μ g s q 2 m s displaystyle boldsymbol mu frac g_ text s q 2m mathbf s 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 2 002 319 304 360 92 36 with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation 17 the value of 2 arises from the dirac equation a fundamental equation connecting the electron s spin with its electromagnetic properties and the deviation from 2 arises from the electron s interaction with the surrounding quantum fields including its own electromagnetic field and virtual particles 18 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 19 20 21 μ ν 3 10 19 μ b m ν c 2 ev displaystyle mu _ nu approx 3 times 10 19 mu _ text b frac m_ nu c 2 text ev where the 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 are known to be at most about 1 ev c 2 fine tuning would be necessary in order to prevent large contributions to the neutrino mass via radiative corrections 22 the measurement of neutrino magnetic moments is an active area of research experimental results have put the neutrino magnetic moment at less than 1 2 10 10 times the electron s magnetic moment on the other hand elementary particles with spin but without electric charge such as the photon and z boson do not have a magnetic moment direction edit further information angular momentum operator spin projection quantum number and multiplicity edit in classical mechanics the angular momentum of a particle possesses not only a magnitude how fast the body is rotating but also a direction either up or down on the axis of rotation of the particle quantum mechanical spin also contains information about direction but in a more subtle form quantum mechanics states that the component of angular momentum for a spin s particle measured along any direction can only take on the values 23 s i ℏ s i s i s s 1 s 1 s displaystyle s_ i hbar s_ i quad s_ i in s s 1 dots s 1 s where s i is the spin component along the i th axis either x y or z s i is the spin projection quantum number along the i th axis and s is the principal spin quantum number discussed in the previous section conventionally the direction chosen is the z axis s z ℏ s z s z s s 1 s 1 s displaystyle s_ z hbar s_ z quad s_ z in s s 1 dots s 1 s where 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 2 s 1 possible values of s z the number 2 s 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 component is pointing in the z or z directions respectively and are often referred to as spin up and spin down for a spin 3 2 particle like a delta baryon the possible values are 3 2 1 2 1 2 3 2 vector edit for a given quantum state one could think of a spin vector s textstyle langle s rangle whose components are the expectation values of the spin components along each axis i e s s x s y s z textstyle langle s rangle langle s_ x rangle langle s_ y rangle langle s_ z rangle this vector then would describe the direction in which the spin is pointing corresponding to the classical concept of the axis of rotation it turns out that the spin vector is not very useful in actual quantum mechanical calculations because it cannot be measured directly 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 probability drops off smoothly as the angle between the spin vector and the detector increases until at an angle of 180 that is for detectors oriented in the opposite direction to the spin vector the expectation of detecting particles from the collection reaches a minimum of 0 as a qualitative concept the spin vector is often handy because it is easy to picture classically for instance quantum mechanical spin can exhibit phenomena analogous to classical gyroscopic effects for example one can exert a kind of torque on an electron by putting it in a magnetic field the field acts upon the electron s intrinsic magnetic dipole moment see the following section the result is that the spin vector undergoes precession just like a classical gyroscope this phenomenon is known as electron spin resonance esr the equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance nmr spectroscopy and imaging mathematically quantum mechanical spin states are described by vector like objects known as spinors there are subtle differences between the behavior of spinors and vectors under coordinate rotations for example rotating a spin 1 2 particle by 360 does not bring it back to the same quantum state but to the state with the opposite quantum phase this is detectable in principle with interference experiments to return the particle to its exact original state one needs a 720 rotation the plate trick and möbius strip give non quantum analogies a spin zero particle can only have a single quantum state even after torque is applied rotating a spin 2 particle 180 can bring it back to the same quantum state and a spin 4 particle should be rotated 90 to bring it back to the same quantum state the spin 2 particle can be analogous to a straight stick that looks the same even after it is rotated 180 and a spin 0 particle can be imagined as sphere which looks the same after whatever angle it is turned through mathematical formulation edit definitions edit intrinsic angular momentum edit a particle is said to possess intrinsic spin s displaystyle s if the quantum mechanical states of that particle in its own rest frame are eigenstates of the operator j 2 displaystyle mathbf j 2 with eigenvalue ℏ 2 s s 1 displaystyle hbar 2 s s 1 24 25 relativistic quantum theory edit in relativistic quantum theory spin is defined as an invariant label of a time like unitary irreducible representation of the poincaré group for time like representations corresponding to massive particles the unitary irreducible representations are characterized by the mass m displaystyle m and the spin s displaystyle s where s displaystyle s labels an irreducible projective representation of the rotation group s o 3 displaystyle so 3 26 27 operator edit spin obeys commutation relations 28 analogous to those of the orbital angular momentum s j s k i ℏ ε j k l s l displaystyle left hat s _ j hat s _ k right i hbar varepsilon _ jkl hat s _ l where ε jkl is the levi civita symbol it follows as with angular momentum that the eigenvectors of s 2 displaystyle hat s 2 and s z displaystyle hat s _ z expressed as kets in the total s basis are 2 166 s 2 s m s ℏ 2 s s 1 s m s s z s m s ℏ m s s m s displaystyle begin aligned hat s 2 s m_ s rangle hbar 2 s s 1 s m_ s rangle hat s _ z s m_ s rangle hbar m_ s s m_ s rangle end aligned the spin raising and lowering operators acting on these eigenvectors give s s m s ℏ s s 1 m s m s 1 s m s 1 displaystyle hat s _ pm s m_ s rangle hbar sqrt s s 1 m_ s m_ s pm 1 s m_ s pm 1 rangle where s s x i s y displaystyle hat s _ pm hat s _ x pm i hat s _ y 2 166 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 s all quantum mechanical particles possess an intrinsic spin s displaystyle s though this value may be equal to zero the projection of the spin s displaystyle s on any axis is quantized in units of the reduced planck constant such that the state function of the particle is say not ψ ψ r displaystyle psi psi mathbf r but ψ ψ r s z displaystyle psi psi mathbf r s_ z where s z displaystyle s_ z can take only the values of the following discrete set s z s ℏ s 1 ℏ s 1 ℏ s ℏ displaystyle s_ z in s hbar s 1 hbar dots s 1 hbar s hbar one distinguishes bosons integer spin and fermions half integer spin the total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin pauli matrices edit main article pauli matrices the quantum mechanical operators associated with spin 1 2 observables are s ℏ 2 σ displaystyle hat mathbf s frac hbar 2 boldsymbol sigma where in cartesian components s x ℏ 2 σ x s y ℏ 2 σ y s z ℏ 2 σ z displaystyle s_ x frac hbar 2 sigma _ x quad s_ y frac hbar 2 sigma _ y quad s_ z frac hbar 2 sigma _ z for the special case of spin 1 2 particles σ x σ y and σ z are the three pauli matrices σ x 0 1 1 0 σ y 0 i i 0 σ z 1 0 0 1 displaystyle sigma _ x begin pmatrix 0 1 1 0 end pmatrix quad sigma _ y begin pmatrix 0 i i 0 end pmatrix quad sigma _ z begin pmatrix 1 0 0 1 end pmatrix pauli exclusion principle edit the pauli exclusion principle states that the wavefunction ψ r 1 σ 1 r n σ n displaystyle psi mathbf r _ 1 sigma _ 1 dots mathbf r _ n sigma _ n for a system of n identical particles having spin s must change upon interchanges of any two of the n particles as ψ r i σ i r j σ j 1 2 s ψ r j σ j r i σ i displaystyle psi dots mathbf r _ i sigma _ i dots mathbf r _ j sigma _ j dots 1 2s psi dots mathbf r _ j sigma _ j dots mathbf r _ i sigma _ i dots thus for bosons the prefactor 1 2 s will reduce to 1 for fermions to 1 this permutation postulate for n particle state functions has most important consequences in daily life e g the periodic table of the chemical elements rotations edit see also symmetry in quantum mechanics as described above quantum mechanics states that components of angular momentum measured along any direction can only take a number of discrete values the most convenient quantum mechanical description of particle s spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis for instance for a spin 1 2 particle we would need two numbers a 1 2 giving amplitudes of finding it with projection of angular momentum equal to ħ 2 and ħ 2 satisfying the requirement a 1 2 2 a 1 2 2 1 displaystyle a_ 1 2 2 a_ 1 2 2 1 for a generic particle with spin s we would need 2 s 1 such parameters since these numbers depend on the choice of the axis they...
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