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r 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 transform into each other non trivially when this axis is rotated it is clear that the transformation law must be linear so we can represent it by associating a matrix with each rotation and the product of two transformation matrices corresponding to rotations a and b must be equal up to phase to the matrix representing rotation ab further rotations preserve the quantum mechanical inner product and so should our transformation matrices m j j a m b m m j j n j j u n m a n k j j u k m b k displaystyle sum _ m j j a_ m b_ m sum _ m j j left sum _ n j j u_ nm a_ n right left sum _ k j j u_ km b_ k right n j j k j j u n p u k q δ p q displaystyle 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 each such representation corresponds to a representation of the covering group of so 3 which is su 2 29 there is one n dimensional irreducible representation of su 2 for each dimension though this representation is n dimensional real for odd n and n dimensional complex for even n hence of real dimension 2 n for a rotation by angle θ in the plane with normal vector θ textstyle hat boldsymbol theta u e i ℏ θ s displaystyle u e frac i hbar boldsymbol theta cdot mathbf s where θ θ θ textstyle boldsymbol theta theta hat boldsymbol theta and s is the vector of spin operators proof working in the coordinate system where θ z textstyle hat theta hat z we would like to show that s x and s y are rotated into each other by the angle θ starting with s x using units where ħ 1 s x u s x u e i θ s z s x e i θ s z s x i θ s z s x 1 2 i θ 2 s z s z s x 1 3 i θ 3 s z s z s z s x displaystyle begin aligned s_ x rightarrow u dagger s_ x u e i theta s_ z s_ x e i theta s_ z s_ x i theta left s_ z s_ x right left frac 1 2 right i theta 2 left s_ z left s_ z s_ x right right left frac 1 3 right i theta 3 left s_ z left s_ z left s_ z s_ x right right right cdots end aligned using the spin operator commutation relations we see that the commutators evaluate to i s y for the odd terms in the series and to s x for all of the even terms thus u s x u s x 1 θ 2 2 s y θ θ 3 3 s x cos θ s y sin θ displaystyle begin aligned u dagger s_ x u s_ x left 1 frac theta 2 2 cdots right s_ y left theta frac theta 3 3 cdots right s_ x cos theta s_ y sin theta end aligned as expected note that since we only relied on the spin operator commutation relations this proof holds for any dimension i e for any principal spin quantum number s 30 164 a generic rotation in 3 dimensional space can be built by compounding operators of this type using euler angles r α β γ e i α s x e i β s y e i γ s z displaystyle mathcal r alpha beta gamma e i alpha s_ x e i beta s_ y e i gamma s_ z an irreducible representation of this group of operators is furnished by the wigner d matrix d m m s α β γ s m r α β γ s m e i m α d m m s β e i m γ displaystyle d_ m m s alpha beta gamma equiv langle sm mathcal r alpha beta gamma sm rangle e im alpha d_ m m s beta e im gamma where d m m s β s m e i β s y s m displaystyle d_ m m s beta langle sm e i beta s_ y sm rangle is wigner s small d matrix note that for γ 2π and α β 0 i e a full rotation about the z axis the wigner d matrix elements become d m m s 0 0 2 π d m m s 0 e i m 2 π δ m m 1 2 m displaystyle d_ m m s 0 0 2 pi d_ m m s 0 e im2 pi delta _ m m 1 2m recalling that a generic spin state can be written as a superposition of states with definite m we see that if s is an integer the values of m are all integers and this matrix corresponds to the identity operator however if s is a half integer the values of m are also all half integers giving 1 2 m 1 for all m and hence upon rotation by 2 π the state picks up a minus sign this fact is a crucial element of the proof of the spin statistics theorem lorentz transformations 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 we could try the same approach to determine the behavior of spin under general lorentz transformations but we would immediately discover a major obstacle unlike so 3 the group of lorentz transformations so 3 1 is non compact and therefore does not have any faithful unitary finite dimensional representations in case of spin 1 2 particles it is possible to find a construction that includes both a finite dimensional representation and a scalar product that is preserved by this representation we associate a 4 component dirac spinor 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 ant...
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