Intrinsic quantum property of particles
This article is about the concept in quantum mechanics. For the concept in classical mechanics, see 4️⃣ Rotation
Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, 4️⃣ atomic nuclei, and atoms.[1][2]: 183–184 Spin should not be conceptualized as involving the "rotation" of a particle's "internal mass", as 4️⃣ ordinary use of the word may suggest: spin is a quantized property of waves.[3]
The existence of electron spin angular momentum 4️⃣ is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete 4️⃣ angular momenta despite having no orbital angular momentum.[4] The existence of the electron spin can also be inferred theoretically from 4️⃣ the spin–statistics theorem and from the Pauli exclusion principle—and vice versa, given the particular spin of the electron, one may 4️⃣ derive the Pauli exclusion principle.
Spin is described mathematically as a vector for some particles such as photons, and as spinors 4️⃣ and bispinors for other particles such as electrons. Spinors and bispinors behave similarly to vectors: they have definite magnitudes and 4️⃣ change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude 4️⃣ of spin angular momentum, though its direction may change. These are indicated by assigning the particle a spin quantum number.[2]: 4️⃣ 183–184
The SI unit of spin is the same as classical angular momentum (i.e., N·m·s, J·s, or kg·m2·s−1). In practice, spin 4️⃣ is usually given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant 4️⃣ ħ, which has the same dimensions as angular momentum. Often, the "spin quantum number" is simply called "spin".
Relation to classical 4️⃣ rotation [ edit ]
The very earliest models for electron spin imagined a rotating charged mass, but this model fails when 4️⃣ examined in detail: the required space distribution does not match limits on the electron radius: the required rotation speed exceeds 4️⃣ the speed of light. In the Standard Model, the fundamental particles are all considered "point-like": they have their effects through 4️⃣ the field that surrounds them.[5] Any model for spin based on mass rotation would need to be consistent with that 4️⃣ model.
The classical analog for quantum spin is a circulation of energy or momentum-density in the particle wave field: "spin is 4️⃣ essentially a wave property".[3] This same concept of spin can be applied to gravity waves in water: "spin is generated 4️⃣ by subwavelength circular motion of water particles".[6]
Photon spin is the quantum-mechanical description of light polarization, where spin +1 and spin 4️⃣ −1 represent two opposite directions of circular polarization. Thus, light of a defined circular polarization consists of photons with the 4️⃣ same spin, either all +1 or all −1. Spin represents polarization for other vector bosons as well.
Relation to orbital angular 4️⃣ momentum [ edit ]
As the name suggests, spin was originally conceived as the rotation of a particle around some axis. 4️⃣ Historically orbital angular momentum related to particle orbits.[7]: 131 While the names based on mechanical models have survived, the physical 4️⃣ explanation has not. Quantization fundamentally alters the character of both spin and orbital angular momentum.
Since elementary particles are point-like, self-rotation 4️⃣ is not well-defined for them. However, spin implies that the phase of the particle depends on the angle as e 4️⃣ i S θ {\displaystyle e^{iS\theta }} , for rotation of angle θ around the axis parallel to the spin S. 4️⃣ This is equivalent to the quantum-mechanical interpretation of momentum as phase dependence in the position, and of orbital angular momentum 4️⃣ as phase dependence in the angular position.
For fermions, the picture is less clear. Angular velocity is equal by Ehrenfest theorem 4️⃣ to the derivative of the Hamiltonian to its conjugate momentum, which is the total angular momentum operator J = L 4️⃣ + S. Therefore, if the Hamiltonian H is dependent upon the spin S, dH/dS is non-zero, and the spin causes 4️⃣ angular velocity, and hence actual rotation, i.e. a change in the phase-angle relation over time. However, whether this holds for 4️⃣ free electron is ambiguous, since for an electron, S2 is constant, and therefore it is a matter of interpretation whether 4️⃣ the Hamiltonian includes such a term. Nevertheless, spin appears in the Dirac equation, and thus the relativistic Hamiltonian of the 4️⃣ electron, treated as a Dirac field, can be interpreted as including a dependence in the spin S.[8] Under this interpretation, 4️⃣ free electrons also self-rotate, with the zitterbewegung effect understood as this rotation.
Quantum number [ edit ]
Spin obeys the mathematical laws 4️⃣ of angular momentum quantization. The specific properties of spin angular momenta include:
Spin quantum numbers may take half-integer values.
Although the direction 4️⃣ of its spin can be changed, the magnitude of the spin of an elementary particle cannot be changed.
The spin of 4️⃣ a charged particle is associated with a magnetic dipole moment with a g -factor that differs from 1. (In the 4️⃣ classical context, this would imply the internal charge and mass distributions differing for a rotating object.[9])
The conventional definition of the 4️⃣ spin quantum number is s = n/2, where n can be any non-negative integer. Hence the allowed values of s 4️⃣ are 0, 1/2, 1, 3/2, 2, etc. The value of s for an elementary particle depends only on the type 4️⃣ of particle and cannot be altered in any known way (in contrast to the spin direction described below). The spin 4️⃣ angular momentum S of any physical system is quantized. The allowed values of S are
S = ℏ s ( s 4️⃣ + 1 ) = h 2 π n 2 ( n + 2 ) 2 = h 4 π n 4️⃣ ( n + 2 ) , {\displaystyle S=\hbar \,{\sqrt {s(s+1)}}={\frac {h}{2\pi }}\,{\sqrt {{\frac {n}{2}}{\frac {(n+2)}{2}}}}={\frac {h}{4\pi }}\,{\sqrt {n(n+2)}},}
h
ℏ = h 4️⃣ 2 π {\textstyle \hbar ={\frac {h}{2\pi }}}
s
n
Fermions and bosons [ edit ]
whereis the Planck constant , andis the reduced Planck 4️⃣ constant. In contrast, orbital angular momentum can only take on integer values of; i.e., even-numbered values of
Those particles with half-integer 4️⃣ spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer spins, such as 0, 1, 4️⃣ 2, are known as bosons. The two families of particles obey different rules and broadly have different roles in the 4️⃣ world around us. A key distinction between the two families is that fermions obey the Pauli exclusion principle: that is, 4️⃣ there cannot be two identical fermions simultaneously having the same quantum numbers (meaning, roughly, having the same position, velocity and 4️⃣ spin direction). Fermions obey the rules of Fermi–Dirac statistics. In contrast, bosons obey the rules of Bose–Einstein statistics and have 4️⃣ no such restriction, so they may "bunch together" in identical states. Also, composite particles can have spins different from their 4️⃣ component particles. For example, a helium-4 atom in the ground state has spin 0 and behaves like a boson, even 4️⃣ though the quarks and electrons which make it up are all fermions.
This has some profound consequences:
Spin–statistics theorem [ edit ]
The 4️⃣ spin–statistics theorem splits particles into two groups: bosons and fermions, where bosons obey Bose–Einstein statistics, and fermions obey Fermi–Dirac statistics 4️⃣ (and therefore the Pauli exclusion principle). Specifically, the theory states that particles with an integer spin are bosons, while all 4️⃣ other particles have half-integer spins and are fermions. As an example, electrons have half-integer spin and are fermions that obey 4️⃣ the Pauli exclusion principle, while photons have integer spin and do not. The theorem relies on both quantum mechanics and 4️⃣ the theory of special relativity, and this connection between spin and statistics has been called "one of the most important 4️⃣ applications of the special relativity theory".[11]
Magnetic moments [ edit ]
Schematic diagram depicting the spin of the neutron as the black 4️⃣ arrow and magnetic field lines associated with the neutron magnetic moment. The neutron has a negative magnetic moment. While the 4️⃣ spin of the neutron is upward in this diagram, the magnetic field lines at the center of the dipole are 4️⃣ downward.
Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These 4️⃣ magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in 4️⃣ a Stern–Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves.
The intrinsic magnetic moment μ of a 4️⃣ spin- 1/2 particle with charge q, mass m, and spin angular momentum S, is[12]
μ = g s q 2 m 4️⃣ 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 4️⃣ exclusively orbital rotations it would be 1 (assuming that the mass and the charge occupy spheres of equal radius).
The electron, 4️⃣ being a charged elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics 4️⃣ is its accurate prediction of the electron g-factor, which has been experimentally determined to have the value −2.00231930436256(35), with the 4️⃣ digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation.[13] The value of 2 arises 4️⃣ from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties, and the deviation from −2 4️⃣ arises from the electron's interaction with the surrounding electromagnetic field, including its own field.[14]
Composite particles also possess magnetic moments associated 4️⃣ with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an 4️⃣ early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are 4️⃣ electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital 4️⃣ motions.
Neutrinos are both elementary and electrically neutral. The minimally extended Standard Model that takes into account non-zero neutrino masses predicts 4️⃣ neutrino magnetic moments of:[15][16][17]
μ ν ≈ 3 × 10 − 19 μ B m ν c 2 eV , {\displaystyle 4️⃣ \mu _{
u }\approx 3\times 10^{-19}\mu _{\text{B}}{\frac {m_{
u }c^{2}}{\text{eV}}},}
where the μ ν are the neutrino magnetic moments, m ν are the 4️⃣ neutrino masses, and μ B is the Bohr magneton. New physics above the electroweak scale could, however, lead to significantly 4️⃣ higher neutrino magnetic moments. It can be shown in a model-independent way that neutrino magnetic moments larger than about 10−14 4️⃣ μ B are "unnatural" because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino 4️⃣ masses are known to be at most about 1 eV/c2, fine-tuning would be necessary in order to prevent large contributions 4️⃣ to the neutrino mass via radiative corrections.[18] The measurement of neutrino magnetic moments is an active area of research. Experimental 4️⃣ 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 4️⃣ particles with spin but without electric charge, such as a photon or a Z boson, do not have a magnetic 4️⃣ moment.
Curie temperature and loss of alignment [ edit ]
In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic 4️⃣ fields that cancel one another, because each dipole points in a random direction, with the overall average being very near 4️⃣ zero. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments spontaneously align locally, 4️⃣ producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.
In 4️⃣ paramagnetic materials, the magnetic dipole moments of individual atoms will partially align with an externally applied magnetic field. In diamagnetic 4️⃣ materials, on the other hand, the magnetic dipole moments of individual atoms align oppositely to any externally applied magnetic field, 4️⃣ even if it requires energy to do so.
The study of the behavior of such "spin models" is a thriving area 4️⃣ of research in condensed matter physics. For instance, the Ising model describes spins (dipoles) that have only two possible states, 4️⃣ up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. These models 4️⃣ have many interesting properties, which have led to interesting results in the theory of phase transitions.
Direction [ edit ]
Spin projection 4️⃣ quantum number and multiplicity [ edit ]
In classical mechanics, the angular momentum of a particle possesses not only a magnitude 4️⃣ (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of 4️⃣ the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the 4️⃣ component of angular momentum for a spin-s particle measured along any direction can only take on the values[19]
S i = 4️⃣ ℏ s i , s i ∈ { − s , − ( s − 1 ) , … , 4️⃣ 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 4️⃣ the i-th axis (either x, y, or z), s i is the spin projection quantum number along the i-th axis, 4️⃣ and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z 4️⃣ axis:
S z = ℏ s z , s z ∈ { − s , − ( s − 1 ) 4️⃣ , … , 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 4️⃣ spin component along the z axis, s z is the spin projection quantum number along the z axis.
One can see 4️⃣ that there are 2s + 1 possible values of s z . The number "2s + 1" is the multiplicity 4️⃣ of the spin system. For example, there are only two possible values for a spin- 1/2 particle: s z = 4️⃣ + 1/2 and s z = − 1/2. These correspond to quantum states in which the spin component is pointing 4️⃣ in the +z or −z directions respectively, and are often referred to as "spin up" and "spin down". For a 4️⃣ spin- 3/2 particle, like a delta baryon, the possible values are + 3/2, + 1/2, − 1/2, − 3/2.
Vector [ 4️⃣ edit ]
A single point in space can rotate continuously without becoming tangled. Notice that after a 360-degree rotation, the spiral 4️⃣ flips between clockwise and counterclockwise orientations. It returns to its original configuration after spinning a full 720°.
For a given quantum 4️⃣ state, one could think of a spin vector ⟨ S ⟩ {\textstyle \langle S\rangle } whose components are the expectation 4️⃣ values of the spin components along each axis, i.e., ⟨ S ⟩ = [ ⟨ S x ⟩ , ⟨ 4️⃣ S y ⟩ , ⟨ S z ⟩ ] {\textstyle \langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]} . 4️⃣ This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the 4️⃣ axis of rotation. It turns out that the spin vector is not very useful in actual quantum-mechanical calculations, because it 4️⃣ cannot be measured directly: s x , s y and s z cannot possess simultaneous definite values, because of a 4️⃣ quantum uncertainty relation between them. However, for statistically large collections of particles that have been placed in the same pure 4️⃣ quantum state, such as through the use of a Stern–Gerlach apparatus, the spin vector does have a well-defined experimental meaning: 4️⃣ It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the 4️⃣ maximum possible probability (100%) of detecting every particle in the collection. For spin- 1/2 particles, this probability drops off smoothly 4️⃣ as the angle between the spin vector and the detector increases, until at an angle of 180°—that is, for detectors 4️⃣ oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 4️⃣ 0%.
As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum-mechanical 4️⃣ spin can exhibit phenomena analogous to classical gyroscopic effects. For example, one can exert a kind of "torque" on an 4️⃣ electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole moment—see the following 4️⃣ section). The result is that the spin vector undergoes precession, just like a classical gyroscope. This phenomenon is known as 4️⃣ electron spin resonance (ESR). The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance (NMR) spectroscopy 4️⃣ and imaging.
Mathematically, quantum-mechanical spin states are described by vector-like objects known as spinors. There are subtle differences between the behavior 4️⃣ of spinors and vectors under coordinate rotations. For example, rotating a spin- 1/2 particle by 360° does not bring it 4️⃣ back to the same quantum state, but to the state with the opposite quantum phase; this is detectable, in principle, 4️⃣ with interference experiments. To return the particle to its exact original state, one needs a 720° rotation. (The Plate trick 4️⃣ and Möbius strip give non-quantum analogies.) A spin-zero particle can only have a single quantum state, even after torque is 4️⃣ applied. Rotating a spin-2 particle 180° can bring it back to the same quantum state, and a spin-4 particle should 4️⃣ be rotated 90° to bring it back to the same quantum state. The spin-2 particle can be analogous to a 4️⃣ straight stick that looks the same even after it is rotated 180°, and a spin-0 particle can be imagined as 4️⃣ sphere, which looks the same after whatever angle it is turned through.
Mathematical formulation [ edit ]
Operator [ edit ]
Spin obeys 4️⃣ commutation relations[20] analogous to those of the orbital angular momentum:
[ S ^ j , S ^ k ] = i 4️⃣ ℏ ε j k l S ^ l , {\displaystyle \left[{\hat {S}}_{j},{\hat {S}}_{k}\right]=i\hbar \varepsilon _{jkl}{\hat {S}}_{l},}
ε jkl
S ^ 2 {\displaystyle 4️⃣ {\hat {S}}^{2}}
S ^ z {\displaystyle {\hat {S}}_{z}}
S
: 166
S ^ 2 | s , m s ⟩ = ℏ 2 s 4️⃣ ( s + 1 ) | s , m s ⟩ , S ^ z | s , m s 4️⃣ ⟩ = 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 4️⃣ m_{s}|s,m_{s}\rangle .\end{aligned}}}
whereis the Levi-Civita symbol . It follows (as with angular momentum ) that the eigenvectors ofand(expressed as kets in 4️⃣ the total basis ) are
The spin raising and lowering operators acting on these eigenvectors give
S ^ ± | s , 4️⃣ m s ⟩ = ℏ s ( s + 1 ) − m s ( m s ± 1 ) 4️⃣ | 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 ,}
S ^ 4️⃣ ± = S ^ x ± i S ^ y {\displaystyle {\hat {S}}_{\pm }={\hat {S}}_{x}\pm i{\hat {S}}_{y}}
: 166
where
But unlike orbital 4️⃣ angular momentum, the eigenvectors are not spherical harmonics. They are not functions of θ and φ. There is also no 4️⃣ reason to exclude half-integer values of s and m s .
All quantum-mechanical particles possess an intrinsic spin s {\displaystyle s} 4️⃣ (though this value may be equal to zero). The projection of the spin s {\displaystyle s} on any axis is 4️⃣ quantized in units of the reduced Planck constant, such that the state function of the particle is, say, not ψ 4️⃣ = ψ ( r ) {\displaystyle \psi =\psi (\mathbf {r} )} , but ψ = ψ ( r , s 4️⃣ z ) {\displaystyle \psi =\psi (\mathbf {r} ,s_{z})} , where s z {\displaystyle s_{z}} can take only the values of 4️⃣ the following discrete set:
s z ∈ { − s ℏ , − ( s − 1 ) ℏ , … 4️⃣ , + ( s − 1 ) ℏ , + s ℏ } . {\displaystyle s_{z}\in \{-s\hbar ,-(s-1)\hbar ,\dots ,+(s-1)\hbar 4️⃣ ,+s\hbar \}.}
One distinguishes bosons (integer spin) and fermions (half-integer spin). The total angular momentum conserved in interaction processes is then 4️⃣ the sum of the orbital angular momentum and the spin.
Pauli matrices [ edit ]
The quantum-mechanical operators associated with spin- 1/2 4️⃣ observables are
S ^ = ℏ 2 σ , {\displaystyle {\hat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }},}
S x = ℏ 4️⃣ 2 σ x , S y = ℏ 2 σ y , S z = ℏ 2 σ z . 4️⃣ {\displaystyle S_{x}={\frac {\hbar }{2}}\sigma _{x},\quad S_{y}={\frac {\hbar }{2}}\sigma _{y},\quad S_{z}={\frac {\hbar }{2}}\sigma _{z}.}
where in Cartesian components
For the special case of 4️⃣ spin- 1/2 particles, σ x , σ y and σ z are the three Pauli matrices:
σ x = ( 0 4️⃣ 1 1 0 ) , σ y = ( 0 − i i 0 ) , σ z = ( 4️⃣ 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 ]
For systems 4️⃣ of N identical particles this is related to the Pauli exclusion principle, which states that its wavefunction ψ ( r 4️⃣ 1 , σ 1 , … , r N , σ N ) {\displaystyle \psi (\mathbf {r} _{1},\sigma _{1},\dots ,\mathbf 4️⃣ {r} _{N},\sigma _{N})} must change upon interchanges of any two of the N particles as
ψ ( … , r i 4️⃣ , σ i , … , r j , σ j , … ) = ( − 1 ) 2 4️⃣ s ψ ( … , r j , σ j , … , r i , σ i , … 4️⃣ ) . {\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 4️⃣ {r} _{i},\sigma _{i},\dots ).}
Thus, for bosons the prefactor (−1)2s will reduce to +1, for fermions to −1. In quantum mechanics 4️⃣ all particles are either bosons or fermions. In some speculative relativistic quantum field theories "supersymmetric" particles also exist, where linear 4️⃣ combinations of bosonic and fermionic components appear. In two dimensions, the prefactor (−1)2s can be replaced by any complex number 4️⃣ of magnitude 1 such as in the anyon.
The above permutation postulate for N-particle state functions has most important consequences in 4️⃣ daily life, e.g. the periodic table of the chemical elements.
Rotations [ edit ]
As described above, quantum mechanics states that components 4️⃣ of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum-mechanical description 4️⃣ of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of 4️⃣ projection of its intrinsic angular momentum on a given axis. For instance, for a spin- 1/2 particle, we would need 4️⃣ two numbers a ±1/2 , giving amplitudes of finding it with projection of angular momentum equal to + ħ/2 and 4️⃣ − ħ/2, satisfying the requirement
| a + 1 / 2 | 2 + | a − 1 / 2 | 4️⃣ 2 = 1. {\displaystyle |a_{+1/2}|^{2}+|a_{-1/2}|^{2}=1.}
For a generic particle with spin s, we would need 2s + 1 such parameters. Since 4️⃣ these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. 4️⃣ It is clear that the transformation law must be linear, so we can represent it by associating a matrix with 4️⃣ each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to 4️⃣ phase) to the matrix representing rotation AB. Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices:
∑ 4️⃣ m = − j j a m ∗ b m = ∑ m = − j j ( ∑ n 4️⃣ = − j j U n m a n ) ∗ ( ∑ k = − j j U k 4️⃣ 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 = − 4️⃣ 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, 4️⃣ these matrices furnish a unitary projective representation of the rotation group SO(3). Each such representation corresponds to a representation of 4️⃣ the covering group of SO(3), which is SU(2).[21] There is one n-dimensional irreducible representation of SU(2) for each dimension, though 4️⃣ this representation is n-dimensional real for odd n and n-dimensional complex for even n (hence of real dimension 2n). For 4️⃣ a rotation by angle θ in the plane with normal vector θ ^ {\textstyle {\hat {\boldsymbol {\theta }}}} ,
U = 4️⃣ e − i ℏ θ ⋅ S , {\displaystyle U=e^{-{\frac {i}{\hbar }}{\boldsymbol {\theta }}\cdot \mathbf {S} },}
θ = θ θ 4️⃣ ^ {\textstyle {\boldsymbol {\theta }}=\theta {\hat {\boldsymbol {\theta }}}}
S
Proof Working in the coordinate system where θ ^ = z ^ 4️⃣ {\textstyle {\hat {\theta }}={\hat {z}}} , we would like to show that S x and S y are rotated into 4️⃣ each other by the angle θ. Starting with S x . Using units where ħ = 1: S x → 4️⃣ U † S x U = e i θ S z S x e − i θ S z = 4️⃣ S x + ( i θ ) [ S z , S x ] + ( 1 2 ! ) 4️⃣ ( i θ ) 2 [ S z , [ S z , S x ] ] + ( 1 4️⃣ 3 ! ) ( i θ ) 3 [ S z , [ S z , [ S z , 4️⃣ 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 4️⃣ the spin operator commutation relations, we see that the commutators evaluate to i S y for the odd terms in 4️⃣ the series, and to S x for all of the even terms. Thus: U † S x U = S 4️⃣ x [ 1 − θ 2 2 ! + ⋯ ] − S y [ θ − θ 3 3 4️⃣ ! ⋯ ] = S x cos θ − S y sin θ , {\displaystyle {\begin{aligned}U^{\dagger }S_{x}U&=S_{x}\left[1-{\frac {\theta 4️⃣ ^{2}}{2!}}+\cdots \right]-S_{y}\left[\theta -{\frac {\theta ^{3}}{3!}}\cdots \right]\\&=S_{x}\cos \theta -S_{y}\sin \theta ,\end{aligned}}} s )[22] : 164 as expected. Note that since we 4️⃣ only relied on the spin operator commutation relations, this proof holds for any dimension (i.e., for any principal spin quantum 4️⃣ number
where, andis the vector of spin operators
A generic rotation in 3-dimensional space can be built by compounding operators of this 4️⃣ type using Euler angles:
R ( α , β , γ ) = e − i α S x e − 4️⃣ 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 4️⃣ irreducible representation of this group of operators is furnished by the Wigner D-matrix:
D m ′ m s ( α , 4️⃣ β , γ ) ≡ ⟨ s m ′ | R ( α , β , γ ) | s 4️⃣ m ⟩ = e − i m ′ α d m ′ m s ( β ) e − i 4️⃣ 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 },}
d m ′ 4️⃣ m s ( β ) = ⟨ s m ′ | e − i β s y | s m 4️⃣ ⟩ {\displaystyle d_{m'm}^{s}(\beta )=\langle sm'|e^{-i\beta s_{y}}|sm\rangle }
γ = 2π
α = β = 0
z
D m ′ m s ( 0 , 4️⃣ 0 , 2 π ) = d m ′ m s ( 0 ) e − i m 2 π 4️⃣ = 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}.}
whereis Wigner's small d-matrix 4️⃣ . Note that forand; i.e., a full rotation about theaxis, the Wigner D-matrix elements become
Recalling that a generic spin state 4️⃣ can be written as a superposition of states with definite m, we see that if s is an integer, the 4️⃣ values of m are all integers, and this matrix corresponds to the identity operator. However, if s is a half-integer, 4️⃣ the values of m are also all half-integers, giving (−1)2m = −1 for all m, and hence upon rotation by 4️⃣ 2π the state picks up a minus sign. This fact is a crucial element of the proof of the spin–statistics 4️⃣ theorem.
Lorentz transformations [ edit ]
We could try the same approach to determine the behavior of spin under general Lorentz transformations, 4️⃣ but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact and therefore 4️⃣ does not have any faithful, unitary, finite-dimensional representations.
In case of spin- 1/2 particles, it is possible to find a construction 4️⃣ that includes both a finite-dimensional representation and a scalar product that is preserved by this representation. We associate a 4-component 4️⃣ Dirac spinor ψ with each particle. These spinors transform under Lorentz transformations according to the law
ψ ′ = exp 4️⃣ ( 1 8 ω μ ν [ γ μ , γ ν ] ) ψ , {\displaystyle \psi '=\exp {\left({\tfrac 4️⃣ {1}{8}}\omega _{\mu
u }[\gamma _{\mu },\gamma _{
u }]\right)}\psi ,}
γ ν
ω μν
⟨ ψ | ϕ ⟩ = ψ ¯ ϕ = ψ 4️⃣ † γ 0 ϕ {\displaystyle \langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi }
Measurement of spin along 4️⃣ the x , y , or z axes [ edit ]
whereare gamma matrices , andis an antisymmetric 4 × 4 4️⃣ matrix parametrizing the transformation. It can be shown that the scalar productis preserved. It is not, however, positive-definite, so the 4️⃣ representation is not unitary.
Each of the (Hermitian) Pauli matrices of spin- 1/2 particles has two eigenvalues, +1 and −1. The 4️⃣ corresponding normalized eigenvectors are
ψ x + = | 1 2 , + 1 2 ⟩ x = 1 2 ( 4️⃣ 1 1 ) , ψ x − = | 1 2 , − 1 2 ⟩ x = 1 2 4️⃣ ( 1 − 1 ) , ψ y + = | 1 2 , + 1 2 ⟩ y = 4️⃣ 1 2 ( 1 i ) , ψ y − = | 1 2 , − 1 2 ⟩ y 4️⃣ = 1 2 ( 1 − i ) , ψ z + = | 1 2 , + 1 2 4️⃣ ⟩ z = ( 1 0 ) , ψ z − = | 1 2 , − 1 2 ⟩ 4️⃣ 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 4️⃣ {\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 4️⃣ _{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 4️⃣ overall sign. In this article, the convention is chosen to make the first element imaginary and negative if there is 4️⃣ a sign ambiguity. The present convention is used by software such as SymPy; while many physics textbooks, such as Sakurai 4️⃣ and Griffiths, prefer to make it real and positive.)
By the postulates of quantum mechanics, an experiment designed to measure the 4️⃣ electron spin on the x, y, or z axis can only yield an eigenvalue of the corresponding spin operator (S 4️⃣ x , S y or S z ) on that axis, i.e. ħ/2 or – ħ/2. The quantum state of 4️⃣ a particle (with respect to spin), can be represented by a two-component spinor:
ψ = ( a + b i c 4️⃣ + 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 4️⃣ axis (in this example, the x axis), the probability that its spin will be measured as ħ/2 is just | 4️⃣ ⟨ ψ x + | ψ ⟩ | 2 {\displaystyle {\big |}\langle \psi _{x+}|\psi \rangle {\big |}^{2}} . Correspondingly, the 4️⃣ probability that its spin will be measured as – ħ/2 is just | ⟨ ψ x − | ψ ⟩ 4️⃣ | 2 {\displaystyle {\big |}\langle \psi _{x-}|\psi \rangle {\big |}^{2}} . Following the measurement, the spin state of the particle 4️⃣ collapses into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to 4️⃣ have a given eigenvalue, all measurements will yield the same eigenvalue (since | ⟨ ψ x + | ψ x 4️⃣ + ⟩ | 2 = 1 {\displaystyle {\big |}\langle \psi _{x+}|\psi _{x+}\rangle {\big |}^{2}=1} , etc.), provided that no measurements 4️⃣ of the spin are made along other axes.
Measurement of spin along an arbitrary axis [ edit ]
The operator to measure 4️⃣ spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let u = (u x , 4️⃣ u y , u z ) be an arbitrary unit vector. Then the operator for spin in this direction is 4️⃣ simply
S u = ℏ 2 ( u x σ x + u y σ y + u z σ z 4️⃣ ) . {\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 4️⃣ usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, 4️⃣ one takes the dot product of the direction with a vector of the three operators for the three x-, y-, 4️⃣ z-axis directions.
A normalized spinor for spin- 1/2 in the (u x , u y , u z ) direction (which 4️⃣ works for all spin states except spin down, where it will give 0/0) is
1 2 + 2 u z ( 4️⃣ 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 4️⃣ in the usual way by diagonalizing the σ u matrix and finding the eigenstates corresponding to the eigenvalues. In quantum 4️⃣ mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of 4️⃣ unity.
Compatibility of spin measurements [ edit ]
Since the Pauli matrices do not commute, measurements of spin along the different axes 4️⃣ are incompatible. This means that if, for example, we know the spin along the x axis, and we then measure 4️⃣ the spin along the y axis, we have invalidated our previous knowledge of the x axis spin. This can be 4️⃣ seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that
| ⟨ ψ x ± | ψ 4️⃣ y ± ⟩ | 2 = | ⟨ ψ x ± | ψ z ± ⟩ | 2 = | 4️⃣ ⟨ ψ y ± | ψ z ± ⟩ | 2 = 1 2 . {\displaystyle {\big |}\langle \psi _{x\pm 4️⃣ }|\psi _{y\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{x\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{y\pm }|\psi _{z\pm }\rangle {\big 4️⃣ |}^{2}={\tfrac {1}{2}}.}
So when physicists measure the spin of a particle along the x axis as, for example, ħ/2, the particle's 4️⃣ spin state collapses into the eigenstate | ψ x + ⟩ {\displaystyle |\psi _{x+}\rangle } . When we then subsequently 4️⃣ measure the particle's spin along the y axis, the spin state will now collapse into either | ψ y + 4️⃣ ⟩ {\displaystyle |\psi _{y+}\rangle } or | ψ y − ⟩ {\displaystyle |\psi _{y-}\rangle } , each with probability 1/2. 4️⃣ Let us say, in our example, that we measure − ħ/2. When we now return to measure the particle's spin 4️⃣ along the x axis again, the probabilities that we will measure ħ/2 or − ħ/2 are each 1/2 (i.e. they 4️⃣ are | ⟨ ψ x + | ψ y − ⟩ | 2 {\displaystyle {\big |}\langle \psi _{x+}|\psi _{y-}\rangle {\big 4️⃣ |}^{2}} and | ⟨ ψ x − | ψ y − ⟩ | 2 {\displaystyle {\big |}\langle \psi _{x-}|\psi _{y-}\rangle 4️⃣ {\big |}^{2}} respectively). This implies that the original measurement of the spin along the x axis is no longer valid, 4️⃣ since the spin along the x axis will now be measured to have either eigenvalue with equal probability.
Higher spins [ 4️⃣ edit ]
The spin- 1/2 operator S = ħ/2σ forms the fundamental representation of SU(2). By taking Kronecker products of this 4️⃣ representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher-spin systems 4️⃣ in three spatial dimensions can be calculated for arbitrarily large s using this spin operator and ladder operators. For example, 4️⃣ taking the Kronecker product of two spin- 1/2 yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 (triplet 4️⃣ states) and a 1-dimensional spin-0 representation (singlet state).
The resulting irreducible representations yield the following spin matrices and eigenvalues in the 4️⃣ z-basis:
For spin 1 they are S x = ℏ 2 ( 0 1 0 1 0 1 0 1 0 4️⃣ ) , | 1 , + 1 ⟩ x = 1 2 ( 1 2 1 ) , | 1 4️⃣ , 0 ⟩ x = 1 2 ( − 1 0 1 ) , | 1 , − 1 ⟩ 4️⃣ x = 1 2 ( 1 − 2 1 ) S y = ℏ 2 ( 0 − i 0 4️⃣ i 0 − i 0 i 0 ) , | 1 , + 1 ⟩ y = 1 2 ( 4️⃣ − 1 − i 2 1 ) , | 1 , 0 ⟩ y = 1 2 ( 1 0 4️⃣ 1 ) , | 1 , − 1 ⟩ y = 1 2 ( − 1 i 2 1 ) 4️⃣ S z = ℏ ( 1 0 0 0 0 0 0 0 − 1 ) , | 1 , 4️⃣ + 1 ⟩ z = ( 1 0 0 ) , | 1 , 0 ⟩ z = ( 0 4️⃣ 1 0 ) , | 1 , − 1 ⟩ z = ( 0 0 1 ) {\displaystyle {\begin{aligned}S_{x}&={\frac {\hbar 4️⃣ }{\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 4️⃣ _{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 ( 4️⃣ 0 3 0 0 3 0 2 0 0 2 0 3 0 0 3 0 ) , | 3 4️⃣ 2 , + 3 2 ⟩ x = 1 2 2 ( 1 3 3 1 ) , | 3 4️⃣ 2 , + 1 2 ⟩ x = 1 2 2 ( − 3 − 1 1 3 ) , 4️⃣ | 3 2 , − 1 2 ⟩ x = 1 2 2 ( 3 − 1 − 1 3 4️⃣ ) , | 3 2 , − 3 2 ⟩ x = 1 2 2 ( − 1 3 − 4️⃣ 3 1 ) S y = ℏ 2 ( 0 − i 3 0 0 i 3 0 − 2 4️⃣ i 0 0 2 i 0 − i 3 0 0 i 3 0 ) , | 3 2 , 4️⃣ + 3 2 ⟩ y = 1 2 2 ( i − 3 − i 3 1 ) , | 4️⃣ 3 2 , + 1 2 ⟩ y = 1 2 2 ( − i 3 1 − i 3 4️⃣ ) , | 3 2 , − 1 2 ⟩ y = 1 2 2 ( i 3 1 i 4️⃣ 3 ) , | 3 2 , − 3 2 ⟩ y = 1 2 2 ( − i − 4️⃣ 3 i 3 1 ) S z = ℏ 2 ( 3 0 0 0 0 1 0 0 0 4️⃣ 0 − 1 0 0 0 0 − 3 ) , | 3 2 , + 3 2 ⟩ z 4️⃣ = ( 1 0 0 0 ) , | 3 2 , + 1 2 ⟩ z = ( 0 4️⃣ 1 0 0 ) , | 3 2 , − 1 2 ⟩ z = ( 0 0 1 0 4️⃣ ) , | 3 2 , − 3 2 ⟩ z = ( 0 0 0 1 ) {\displaystyle {\begin{array}{lclc}S_{x}={\frac 4️⃣ {\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 4️⃣ {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 4️⃣ {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 4️⃣ {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 4️⃣ {-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 4️⃣ 0 5 0 2 2 0 0 0 0 2 2 0 3 0 0 0 0 3 0 2 4️⃣ 2 0 0 0 0 2 2 0 5 0 0 0 0 5 0 ) , S y = 4️⃣ ℏ 2 ( 0 − i 5 0 0 0 0 i 5 0 − 2 i 2 0 0 4️⃣ 0 0 2 i 2 0 − 3 i 0 0 0 0 3 i 0 − 2 i 2 4️⃣ 0 0 0 0 2 i 2 0 − i 5 0 0 0 0 i 5 0 ) , 4️⃣ S z = ℏ 2 ( 5 0 0 0 0 0 0 3 0 0 0 0 0 0 4️⃣ 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 3 0 0 0 4️⃣ 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 4️⃣ {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 4️⃣ arbitrary spin s is ( S x ) a b = ℏ 2 ( δ a , b + 1 4️⃣ + δ a + 1 , b ) ( s + 1 ) ( a + b − 1 ) 4️⃣ − a b , ( S y ) a b = i ℏ 2 ( δ a , b + 4️⃣ 1 − δ a + 1 , b ) ( s + 1 ) ( a + b − 1 4️⃣ ) − a b , ( S z ) a b = ℏ ( s + 1 − a ) 4️⃣ δ a , b = ℏ ( s + 1 − b ) δ a , b , {\displaystyle {\begin{aligned}\left(S_{x}\right)_{ab}&={\frac 4️⃣ {\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}}} a , b {\displaystyle a,b} 1 4️⃣ ≤ a ≤ 2 s + 1 , 1 ≤ b ≤ 2 s + 1. {\displaystyle 1\leq a\leq 2s+1,\quad 4️⃣ 1\leq b\leq 2s+1.}
Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G n is defined to 4️⃣ consist of all n-fold tensor products of Pauli matrices.
The analog formula of Euler's formula in terms of the Pauli matrices
R 4️⃣ ^ ( θ , n ^ ) = e i θ 2 n ^ ⋅ σ = I cos 4️⃣ θ 2 + i ( n ^ ⋅ σ ) sin θ 2 {\displaystyle {\hat {R}}(\theta ,{\hat {\mathbf {n} 4️⃣ }})=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 4️⃣ }{2}}}
Parity [ edit ]
for higher spins is tractable, but less simple.
In tables of the spin quantum number s for nuclei 4️⃣ or particles, the spin is often followed by a "+" or "−". This refers to the parity with "+" for 4️⃣ even parity (wave function unchanged by spatial inversion) and "−" for odd parity (wave function negated by spatial inversion). For 4️⃣ example, see the isotopes of bismuth, in which the list of isotopes includes the column nuclear spin and parity. For 4️⃣ Bi-209, the longest-lived isotope, the entry 9/2– means that the nuclear spin is 9/2 and the parity is odd.
Applications [ 4️⃣ edit ]
Spin has important theoretical implications and practical applications. Well-established direct applications of spin include:
Electron spin plays an important role 4️⃣ in magnetism, with applications for instance in computer memories. The manipulation of nuclear spin by radio-frequency waves (nuclear magnetic resonance) 4️⃣ is important in chemical spectroscopy and medical imaging.
Spin–orbit coupling leads to the fine structure of atomic spectra, which is used 4️⃣ in atomic clocks and in the modern definition of the second. Precise measurements of the g-factor of the electron have 4️⃣ played an important role in the development and verification of quantum electrodynamics. Photon spin is associated with the polarization of 4️⃣ light (photon polarization).
An emerging application of spin is as a binary information carrier in spin transistors. The original concept, proposed 4️⃣ in 1990, is known as Datta–Das spin transistor.[24] Electronics based on spin transistors are referred to as spintronics. The manipulation 4️⃣ of spin in dilute magnetic semiconductor materials, such as metal-doped ZnO or TiO 2 imparts a further degree of freedom 4️⃣ and has the potential to facilitate the fabrication of more efficient electronics.[25]
There are many indirect applications and manifestations of spin 4️⃣ and the associated Pauli exclusion principle, starting with the periodic table of chemistry.
History [ edit ]
Spin was first discovered in 4️⃣ the context of the emission spectrum of alkali metals. In 1924, Wolfgang Pauli introduced what he called a "two-valuedness not 4️⃣ describable classically"[26] associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating 4️⃣ that no two electrons can have the same quantum state in the same quantum system.
The physical interpretation of Pauli's "degree 4️⃣ of freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in early 1925 that it was produced by 4️⃣ the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical 4️⃣ surface would have to be moving faster than the speed of light in order for it to rotate quickly enough 4️⃣ to produce the necessary angular momentum. This would violate the theory of relativity. Largely due to Pauli's criticism, Kronig decided 4️⃣ not to publish his idea.[27]
In the autumn of 1925, the same thought came to Dutch physicists George Uhlenbeck and Samuel 4️⃣ Goudsmit at Leiden University. Under the advice of Paul Ehrenfest, they published their results.[28] It met a favorable response, especially 4️⃣ after Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished 4️⃣ results). This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.
Mathematically speaking, a 4️⃣ fiber bundle description is needed. The tangent bundle effect is additive and relativistic; that is, it vanishes if c goes 4️⃣ to infinity. It is one half of the value obtained without regard for the tangent-space orientation, but with opposite sign. 4️⃣ Thus the combined effect differs from the latter by a factor two (Thomas precession, known to Ludwik Silberstein in 1914).
Despite 4️⃣ his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics invented by 4️⃣ Schrödinger and Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators and introduced a 4️⃣ two-component spinor wave-function. Uhlenbeck and Goudsmit treated spin as arising from classical rotation, while Pauli emphasized, that spin is a 4️⃣ non-classical and intrinsic property.[29]
Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac published the Dirac equation, which described 4️⃣ the relativistic electron. In the Dirac equation, a four-component spinor (known as a "Dirac spinor") was used for the electron 4️⃣ wave-function. Relativistic spin explained gyromagnetic anomaly, which was (in retrospect) first observed by Samuel Jackson Barnett in 1914 (see Einstein–de 4️⃣ Haas effect). In 1940, Pauli proved the spin–statistics theorem, which states that fermions have half-integer spin, and bosons have integer 4️⃣ spin.
In retrospect, the first direct experimental evidence of the electron spin was the Stern–Gerlach experiment of 1922. However, the correct 4️⃣ explanation of this experiment was only given in 1927.[30]
See also [ edit ]
References [ edit ]
Further reading [ edit ]
