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component dirac spinor ψ with each particle these spinors transform under lorentz transformations according to the law ψ exp 1 8 ω μ ν γ μ γ ν ψ displaystyle psi exp left tfrac 1 8 omega _ mu nu gamma _ mu gamma _ nu right psi where γ ν are gamma matrices and ω μν is an antisymmetric 4 4 matrix parametrizing the transformation it can be shown that the scalar product ψ ϕ ψ ϕ ψ γ 0 ϕ displaystyle 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 measurement of spin along the x y or z axes edit each of the hermitian pauli matrices of spin 1 2 particles has two eigenvalues 1 and 1 the corresponding normalized eigenvectors are ψ x 1 2 1 2 x 1 2 1 1 ψ x 1 2 1 2 x 1 2 1 1 ψ y 1 2 1 2 y 1 2 1 i ψ y 1 2 1 2 y 1 2 1 i ψ z 1 2 1 2 z 1 0 ψ z 1 2 1 2 z 0 1 displaystyle begin array lclc psi _ x left frac 1 2 frac 1 2 right rangle _ x displaystyle frac 1 sqrt 2 begin pmatrix 1 1 end pmatrix psi _ x left frac 1 2 frac 1 2 right rangle _ x displaystyle frac 1 sqrt 2 begin pmatrix 1 1 end pmatrix psi _ y left frac 1 2 frac 1 2 right rangle _ y displaystyle frac 1 sqrt 2 begin pmatrix 1 i end pmatrix psi _ y left frac 1 2 frac 1 2 right rangle _ y displaystyle frac 1 sqrt 2 begin pmatrix 1 i end pmatrix psi _ z left frac 1 2 frac 1 2 right rangle _ z begin pmatrix 1 0 end pmatrix psi _ z left frac 1 2 frac 1 2 right rangle _ z begin pmatrix 0 1 end pmatrix end array because any eigenvector multiplied by a constant is still an eigenvector there is ambiguity about the overall sign in this article the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity the present convention is used by software such as sympy while many physics textbooks such as sakurai and griffiths prefer to make it real and positive by the postulates of quantum mechanics an experiment designed to measure the electron spin on the x y or z axis can only yield an eigenvalue of the corresponding spin operator s x s y or s z on that axis i e ħ 2 or ħ 2 the quantum state of a particle with respect to spin can be represented by a two component spinor ψ a b i c d i displaystyle 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 ħ 2 is just ψ x ψ 2 displaystyle big langle psi _ x psi rangle big 2 correspondingly the probability that its spin will be measured as ħ 2 is just ψ x ψ 2 displaystyle big langle psi _ x psi rangle big 2 following the measurement the spin state of the particle collapses into the corresponding eigenstate as a result if the particle s spin along a given axis has been measured to have a given eigenvalue all measurements will yield the same eigenvalue since ψ x ψ x 2 1 displaystyle big langle psi _ x psi _ x rangle big 2 1 etc provided that no measurements of the spin are made along other axes measurement of spin along an arbitrary axis edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed may 2021 learn how and when to remove this message 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 s u ℏ 2 u x σ x u y σ y u z σ z displaystyle s_ u frac hbar 2 u_ x sigma _ x u_ y sigma _ y u_ z sigma _ z the operator s u has eigenvalues of ħ 2 just like the usual spin matrices this method of finding the operator for spin in an arbitrary direction generalizes to higher spin states one takes the dot product of the direction with a vector of the three operators for the three x y z axis directions a normalized spinor for spin 1 2 in the u x u y u z direction which works for all spin states except spin down where it will give 0 0 is 1 2 2 u z 1 u z u x i u y displaystyle frac 1 sqrt 2 2u_ z begin pmatrix 1 u_ z u_ x iu_ y end pmatrix the above spinor is obtained in the usual way by diagonalizing the σ u matrix and finding the eigenstates corresponding to the eigenvalues in quantum mechanics vectors are termed normalized when multiplied by a normalizing factor which results in the vector having a length of unity compatibility of spin measurements edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed may 2021 learn how and when to remove this message since the pauli matrices do not commute measurements of spin along different axes are incompatible that is a measurement along one axis may change the result of a subsequent measurement along a different axis for example if we measure along the x axis our state will become ψ x displaystyle psi _ x pm rangle each of which is an even mixture of ψ z displaystyle psi _ z rangle and ψ z displaystyle psi _ z rangle this means that after the measurement along the x axis a measurement along the z axis has an even chance of being ψ z displaystyle psi _ z rangle or ψ z displaystyle psi _ z rangle regardless of the prior state higher spins edit see also 3d rotation group a note on lie algebras the spin 1 2 operator s ħ 2 σ forms the fundamental representation of su 2 by taking kronecker products of this representation with itself repeatedly one may construct all higher irreducible representations that is the resulting spin operators for higher spin systems in three spatial dimensions can be calculated for arbitrarily large s using this spin operator and ladder operators for example taking the kronecker product of two spin 1 2 yields a four dimensional representation which is separable into a 3 dimensional spin 1 triplet states and a 1 dimensional spin 0 representation singlet state the resulting irreducible representations yield the following spin matrices and eigenvalues in the z basis for spin 1 they are s x ℏ 2 0 1 0 1 0 1 0 1 0 1 1 x 1 2 1 2 1 1 0 x 1 2 1 0 1 1 1 x 1 2 1 2 1 s y ℏ 2 0 i 0 i 0 i 0 i 0 1 1 y 1 2 1 i 2 1 1 0 y 1 2 1 0 1 1 1 y 1 2 1 i 2 1 s z ℏ 1 0 0 0 0 0 0 0 1 1 1 z 1 0 0 1 0 z 0 1 0 1 1 z 0 0 1 displaystyle begin aligned s_ x frac hbar sqrt 2 begin pmatrix 0 1 0 1 0 1 0 1 0 end pmatrix left 1 1 right rangle _ x frac 1 2 begin pmatrix 1 sqrt 2 1 end pmatrix left 1 0 right rangle _ x frac 1 sqrt 2 begin pmatrix 1 0 1 end pmatrix left 1 1 right rangle _ x frac 1 2 begin pmatrix 1 sqrt 2 1 end pmatrix s_ y frac hbar sqrt 2 begin pmatrix 0 i 0 i 0 i 0 i 0 end pmatrix left 1 1 right rangle _ y frac 1 2 begin pmatrix 1 i sqrt 2 1 end pmatrix left 1 0 right rangle _ y frac 1 sqrt 2 begin pmatrix 1 0 1 end pmatrix left 1 1 right rangle _ y frac 1 2 begin pmatrix 1 i sqrt 2 1 end pmatrix s_ z hbar begin pmatrix 1 0 0 0 0 0 0 0 1 end pmatrix left 1 1 right rangle _ z begin pmatrix 1 0 0 end pmatrix left 1 0 right rangle _ z begin pmatrix 0 1 0 end pmatrix left 1 1 right rangle _ z begin pmatrix 0 0 1 end pmatrix end aligned for spin 3 2 they are s x ℏ 2 0 3 0 0 3 0 2 0 0 2 0 3 0 0 3 0 3 2 3 2 x 1 2 2 1 3 3 1 3 2 1 2 x 1 2 2 3 1 1 3 3 2 1 2 x 1 2 2 3 1 1 3 3 2 3 2 x 1 2 2 1 3 3 1 s y ℏ 2 0 i 3 0 0 i 3 0 2 i 0 0 2 i 0 i 3 0 0 i 3 0 3 2 3 2 y 1 2 2 i 3 i 3 1 3 2 1 2 y 1 2 2 i 3 1 i 3 3 2 1 2 y 1 2 2 i 3 1 i 3 3 2 3 2 y 1 2 2 i 3 i 3 1 s z ℏ 2 3 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3 3 2 3 2 z 1 0 0 0 3 2 1 2 z 0 1 0 0 3 2 1 2 z 0 0 1 0 3 2 3 2 z 0 0 0 1 displaystyle begin array lclc s_ x frac hbar 2 begin pmatrix 0 sqrt 3 0 0 sqrt 3 0 2 0 0 2 0 sqrt 3 0 0 sqrt 3 0 end pmatrix left frac 3 2 frac 3 2 right rangle _ x frac 1 2 sqrt 2 begin pmatrix 1 sqrt 3 sqrt 3 1 end pmatrix left frac 3 2 frac 1 2 right rangle _ x frac 1 2 sqrt 2 begin pmatrix sqrt 3 1 1 sqrt 3 end pmatrix left frac 3 2 frac 1 2 right rangle _ x frac 1 2 sqrt 2 begin pmatrix sqrt 3 1 1 sqrt 3 end pmatrix left frac 3 2 frac 3 2 right rangle _ x frac 1 2 sqrt 2 begin pmatrix 1 sqrt 3 sqrt 3 1 end pmatrix s_ y frac hbar 2 begin pmatrix 0 i sqrt 3 0 0 i sqrt 3 0 2i 0 0 2i 0 i sqrt 3 0 0 i sqrt 3 0 end pmatrix left frac 3 2 frac 3 2 right rangle _ y frac 1 2 sqrt 2 begin pmatrix i sqrt 3 i sqrt 3 1 end pmatrix left frac 3 2 frac 1 2 right rangle _ y frac 1 2 sqrt 2 begin pmatrix i sqrt 3 1 i sqrt 3 end pmatrix left frac 3 2 frac 1 2 right rangle _ y frac 1 2 sqrt 2 begin pmatrix i sqrt 3 1 i sqrt 3 end pmatrix left frac 3 2 frac 3 2 right rangle _ y frac 1 2 sqrt 2 begin pmatrix i sqrt 3 i sqrt 3 1 end pmatrix s_ z frac hbar 2 begin pmatrix 3 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3 end pmatrix left frac 3 2 frac 3 2 right rangle _ z begin pmatrix 1 0 0 0 end pmatrix left frac 3 2 frac 1 2 right rangle _ z begin pmatrix 0 1 0 0 end pmatrix left frac 3 2 frac 1 2 right rangle _ z begin pmatrix 0 0 1 0 end pmatrix left frac 3 2 frac 3 2 right rangle _ z begin pmatrix 0 0 0 1 end pmatrix end array for spin 5 2 they are s x ℏ 2 0 5 0 0 0 0 5 0 2 2 0 0 0 0 2 2 0 3 0 0 0 0 3 0 2 2 0 0 0 0 2 2 0 5 0 0 0 0 5 0 s y ℏ 2 0 i 5 0 0 0 0 i 5 0 2 i 2 0 0 0 0 2 i 2 0 3 i 0 0 0 0 3 i 0 2 i 2 0 0 0 0 2 i 2 0 i 5 0 0 0 0 i 5 0 s z ℏ 2 5 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 5 displaystyle begin aligned boldsymbol s _ x frac hbar 2 begin pmatrix 0 sqrt 5 0 0 0 0 sqrt 5 0 2 sqrt 2 0 0 0 0 2 sqrt 2 0 3 0 0 0 0 3 0 2 sqrt 2 0 0 0 0 2 sqrt 2 0 sqrt 5 0 0 0 0 sqrt 5 0 end pmatrix boldsymbol s _ y frac hbar 2 begin pmatrix 0 i sqrt 5 0 0 0 0 i sqrt 5 0 2i sqrt 2 0 0 0 0 2i sqrt 2 0 3i 0 0 0 0 3i 0 2i sqrt 2 0 0 0 0 2i sqrt 2 0 i sqrt 5 0 0 0 0 i sqrt 5 0 end pmatrix boldsymbol s _ z frac hbar 2 begin pmatrix 5 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 5 end pmatrix end aligned the generalization of these matrices for arbitrary spin s is s x a b ℏ 2 δ a b 1 δ a 1 b s 1 a b 1 a b s y a b i ℏ 2 δ a b 1 δ a 1 b s 1 a b 1 a b s z a b ℏ s 1 a δ a b ℏ s 1 b δ a b displaystyle begin aligned left s_ x right _ ab frac hbar 2 left delta _ a b 1 delta _ a 1 b right sqrt s 1 a b 1 ab left s_ y right _ ab frac i hbar 2 left delta _ a b 1 delta _ a 1 b right sqrt s 1 a b 1 ab left s_ z right _ ab hbar s 1 a delta _ a b hbar s 1 b delta _ a b end aligned where indices a b displaystyle a b are integer numbers such that 1 a 2 s 1 1 b 2 s 1 displaystyle 1 leq a leq 2s 1 quad 1 leq b leq 2s 1 also useful in the quantum mechanics of multiparticle systems the general pauli group g n is defined to consist of all n fold tensor products of pauli matrices the analog formula of euler s formula in terms of the pauli matrices r θ n e i θ 2 n σ i cos θ 2 i n σ sin θ 2 displaystyle hat r theta hat mathbf n e i frac theta 2 hat mathbf n cdot boldsymbol sigma i cos frac theta 2 i left hat mathbf n cdot boldsymbol sigma right sin frac theta 2 for higher spins is tractable but less simple 31 parity edit main article parity physics in tables of the spin quantum number s for nuclei or particles the spin is often followed by a or citation needed this refers to the parity with for even parity wave function unchanged by spatial inversion and for odd parity wave function negated by spatial inversion for example see the isotopes of bismuth in which the list of isotopes includes the column nuclear spin and parity for bi 209 the longest lived isotope the entry 9 2 means that the nuclear spin is 9 2 and the parity is odd measuring spin edit the nuclear spin of atoms can be determined by sophisticated improvements to the original stern gerlach experiment 32 a single energy monochromatic molecular beam of atoms in an inhomogeneous magnetic field will split into beams representing each possible spin quantum state for an atom with electronic spin s and nuclear spin i there are 2 s 1 2 i 1 spin states for example neutral na atoms which have s 1 2 were passed through a series of inhomogeneous magnetic fields that selected one of the two electronic spin states and separated the nuclear spin states from which four beams were observed thus the nuclear spin for 23 na atoms was found to be i 3 2 33 34 the spin of pions a type of elementary particle was determined by the principle of detailed balance applied to those collisions of protons that produced charged pions and deuterium p p π d displaystyle p p rightarrow pi d the known spin values for protons and deuterium allows analysis of the collision cross section to show that π displaystyle pi has spin s π 0 displaystyle s_ pi 0 a different approach is needed for neutral pions in that case the decay produced two gamma ray photons with spin one π 0 2 γ displaystyle pi 0 rightarrow 2 gamma this result supplemented with additional analysis leads to the conclusion that the neutral pion also has spin zero 35 66 applications edit spin has important theoretical implications and practical applications well established direct applications of spin include nuclear magnetic resonance nmr spectroscopy in chemistry electron spin resonance esr or epr spectroscopy in chemistry and physics magnetic resonance imaging mri in medicine a type of applied nmr which relies on proton spin density giant magnetoresistive gmr drive head technology in modern hard disks electron spin plays an important role in magnetism with applications for instance in computer memories the manipulation of nuclear spin by radio frequency waves nuclear magnetic resonance is important in chemical spectroscopy and medical imaging spin orbit coupling leads to the fine structure of atomic spectra which is used in atomic clocks and in the modern definition of the second precise measurements of the g factor of the electron have played an important role in the development and verification of quantum electrodynamics photon spin is associated with the polarization of light photon polarization an emerging application of spin is as a binary information carrier in spin transistors the original concept proposed in 1990 is known as datta das spin transistor 36 electronics based on spin transistors are referred to as spintronics the manipulation of spin in dilute magnetic semiconductor materials such as metal doped zno or tio 2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics 37 there are many indirect applications and manifestations of spin and the associated pauli exclusion principle starting with the periodic table of chemistry history edit see also history of quantum mechanics wolfgang pauli lecturing spin was first discovered in the context of the emission spectrum of alkali metals starting around 1910 many experiments on different atoms produced a collection of relationships involving quantum numbers for atomic energy levels partially summarized in bohr s model for the atom 38 106 transitions between levels obeyed selection rules and the rules were known to be correlated with even or odd atomic number additional information was known from changes to atomic spectra observed in strong magnetic field...
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