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Os pilotos ucranianos começaram a pilotar F-16, disse Volodymyro Zelenskiy afirmando que os caça de fabricação norte-americana chegaram há mais 💸 do 29 meses desde o início da invasão russa.
O líder ucraniano anunciou o uso de F-16, que Kyiv há muito 💸 tempo faz lobby para s.a quando ele conheceu pilotos militares bacana play bonus uma base aérea ladeada por dois dos jatos com 💸 mais duas sobrevoando a aeronave
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A chegada dos jatos é um marco para a Ucrânia após muitos meses 💸 de espera, embora ainda não esteja claro quantos estão disponíveis e quanto impacto terão no aprimoramento das defesas aéreas.
A Rússia 💸 tem alvejado bases que podem abrigá-las e prometeu derrubálos.
Os F-16 estavam na lista de desejos da Ucrânia há muito tempo, 💸 equipados com um canhão 20mm e podem transportar bombas.
Dois F-16 voaram sobre a base aérea enquanto Zelenskiy falava.
: Valentyn Ogirenko/Reuters
Falando 💸 com repórteres na pista de um aeródromo, Zelenskiy disse que ainda não há pilotos suficientes treinados para usar os F-16 💸 ou o suficiente dos próprios jatos.
"O positivo é que estamos esperando mais F-16s... muitos caras estão treinando agora", disse ele.
A 💸 Ucrânia já contou anteriormente com uma frota envelhecida de aviões da era soviética que são superados pela mais avançada e 💸 muito maior esquadra russa.
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A Rússia usou essa vantagem para realizar ataques regulares de 💸 mísseis bacana play bonus alvos na Ucrânia e bater posições ucranianas da linha dianteira com milhares das bombas guiada, apoiando suas forças 💸 que estão avançando lentamente no leste.
"Este é o novo estágio de desenvolvimento da força aérea das forças armadas ucranianas", disse 💸 Zelenskiy.
"Fizemos muito para que as forças ucranianas fizessem a transição de um novo padrão da aviação, o combate ocidental", acrescentou 💸 ele citando centenas das reuniões e diplomacia implacável na obtenção dos F-16.
Ainda não está claro com quais mísseis os jatos 💸 estão equipados. Um alcance mais longo de míssil permitiria que eles tivessem um impacto maior no campo do batalha, dizem 💸 analistas militares
Zelenskiy disse que também espera fazer lobby entre os países vizinhos aliados para ajudar a interceptar mísseis russos sendo 💸 lançados na Ucrânia por meio de conversas no Conselho OTAN-Ucrânia.
"Esta é outra ferramenta, e eu quero tentar isso para que 💸 os países da Otan possam conversar com a Ucrânia sobre uma pequena coalizão de vizinhos derrubando mísseis inimigos", disse ele.
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No Brasil a educação em Educação Física (EiF) surgiu na década de 1950, período em que a educação física se 😆 tornou mais centralizado nos centros de educação de nível superior.
Foi uma forma de educação do conhecimento superior, focada nas necessidades 😆 educativas e no desenvolvimento das competências cognitivas do indivíduo.
A partir da década de 1980, surgiu a ideia de formar profissionais 😆 de gestão em educação física que explondessem o conhecimento das ciências e do comportamento social
por meio da educação física.
É nesse 😆 contexto, que surgiu, em 1980, o Fórum Educativo, que foi criado por representantes das entidades de educação para a educação 😆 física.
jogos de matemática online
Intrinsic quantum property of particles
This article is about the concept in quantum
mechanics. For the concept in classical mechanics, see 🧲 Rotation
Spin is an intrinsic
form of angular momentum carried by elementary particles, and thus by composite
particles such as hadrons, 🧲 atomic nuclei, and atoms.[1][2]: 183–184 Spin should not be
conceptualized as involving the "rotation" of a particle's "internal mass", as 🧲 ordinary
use of the word may suggest: spin is a quantized property of waves.[3]
The existence of
electron spin angular momentum 🧲 is inferred from experiments, such as the Stern–Gerlach
experiment, in which silver atoms were observed to possess two possible discrete
🧲 angular momenta despite having no orbital angular momentum.[4] The existence of the
electron spin can also be inferred theoretically from 🧲 the spin–statistics theorem and
from the Pauli exclusion principle—and vice versa, given the particular spin of the
electron, one may 🧲 derive the Pauli exclusion principle.
Spin is described
mathematically as a vector for some particles such as photons, and as spinors 🧲 and
bispinors for other particles such as electrons. Spinors and bispinors behave similarly
to vectors: they have definite magnitudes and 🧲 change under rotations; however, they use
an unconventional "direction". All elementary particles of a given kind have the same
magnitude 🧲 of spin angular momentum, though its direction may change. These are
indicated by assigning the particle a spin quantum number.[2]: 🧲 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 🧲 is usually given as a dimensionless spin quantum number by dividing the
spin angular momentum by the reduced Planck constant 🧲 ħ, which has the same dimensions
as angular momentum. Often, the "spin quantum number" is simply called "spin".
Relation
to classical 🧲 rotation [ edit ]
The very earliest models for electron spin imagined a
rotating charged mass, but this model fails when 🧲 examined in detail: the required space
distribution does not match limits on the electron radius: the required rotation speed
exceeds 🧲 the speed of light. In the Standard Model, the fundamental particles are all
considered "point-like": they have their effects through 🧲 the field that surrounds
them.[5] Any model for spin based on mass rotation would need to be consistent with
that 🧲 model.
The classical analog for quantum spin is a circulation of energy or
momentum-density in the particle wave field: "spin is 🧲 essentially a wave property".[3]
This same concept of spin can be applied to gravity waves in water: "spin is generated
🧲 by subwavelength circular motion of water particles".[6]
Photon spin is the
quantum-mechanical description of light polarization, where spin +1 and spin 🧲 −1
represent two opposite directions of circular polarization. Thus, light of a defined
circular polarization consists of photons with the 🧲 same spin, either all +1 or all −1.
Spin represents polarization for other vector bosons as well.
Relation to orbital
angular 🧲 momentum [ edit ]
As the name suggests, spin was originally conceived as the
rotation of a particle around some axis. 🧲 Historically orbital angular momentum related
to particle orbits.[7]: 131 While the names based on mechanical models have survived,
the physical 🧲 explanation has not. Quantization fundamentally alters the character of
both spin and orbital angular momentum.
Since elementary particles are point-like,
self-rotation 🧲 is not well-defined for them. However, spin implies that the phase of the
particle depends on the angle as e 🧲 i S θ {\displaystyle e^{iS\theta }} , for rotation
of angle θ around the axis parallel to the spin S. 🧲 This is equivalent to the
quantum-mechanical interpretation of momentum as phase dependence in the position, and
of orbital angular momentum 🧲 as phase dependence in the angular position.
For fermions,
the picture is less clear. Angular velocity is equal by Ehrenfest theorem 🧲 to the
derivative of the Hamiltonian to its conjugate momentum, which is the total angular
momentum operator J = L 🧲 + S. Therefore, if the Hamiltonian H is dependent upon the spin
S, dH/dS is non-zero, and the spin causes 🧲 angular velocity, and hence actual rotation,
i.e. a change in the phase-angle relation over time. However, whether this holds for
🧲 free electron is ambiguous, since for an electron, S2 is constant, and therefore it is
a matter of interpretation whether 🧲 the Hamiltonian includes such a term. Nevertheless,
spin appears in the Dirac equation, and thus the relativistic Hamiltonian of the
🧲 electron, treated as a Dirac field, can be interpreted as including a dependence in the
spin S.[8] Under this interpretation, 🧲 free electrons also self-rotate, with the
zitterbewegung effect understood as this rotation.
Quantum number [ edit ]
Spin obeys
the mathematical laws 🧲 of angular momentum quantization. The specific properties of spin
angular momenta include:
Spin quantum numbers may take half-integer values.
Although
the direction 🧲 of its spin can be changed, the magnitude of the spin of an elementary
particle cannot be changed.
The spin of 🧲 a charged particle is associated with a
magnetic dipole moment with a g -factor that differs from 1. (In the 🧲 classical context,
this would imply the internal charge and mass distributions differing for a rotating
object.[9])
The conventional definition of the 🧲 spin quantum number is s = n/2, where n
can be any non-negative integer. Hence the allowed values of s 🧲 are 0, 1/2, 1, 3/2, 2,
etc. The value of s for an elementary particle depends only on the type 🧲 of particle and
cannot be altered in any known way (in contrast to the spin direction described below).
The spin 🧲 angular momentum S of any physical system is quantized. The allowed values of
S are
S = ℏ s ( s 🧲 + 1 ) = h 2 π n 2 ( n + 2 ) 2 = h 4 π n 🧲 ( 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 🧲 2 π {\textstyle \hbar
={\frac {h}{2\pi }}}
s
n
Fermions and bosons [ edit ]
whereis the Planck constant ,
andis the reduced Planck 🧲 constant. In contrast, orbital angular momentum can only take
on integer values of; i.e., even-numbered values of
Those particles with half-integer
🧲 spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer
spins, such as 0, 1, 🧲 2, are known as bosons. The two families of particles obey
different rules and broadly have different roles in the 🧲 world around us. A key
distinction between the two families is that fermions obey the Pauli exclusion
principle: that is, 🧲 there cannot be two identical fermions simultaneously having the
same quantum numbers (meaning, roughly, having the same position, velocity and 🧲 spin
direction). Fermions obey the rules of Fermi–Dirac statistics. In contrast, bosons obey
the rules of Bose–Einstein statistics and have 🧲 no such restriction, so they may "bunch
together" in identical states. Also, composite particles can have spins different from
their 🧲 component particles. For example, a helium-4 atom in the ground state has spin 0
and behaves like a boson, even 🧲 though the quarks and electrons which make it up are all
fermions.
This has some profound consequences:
Spin–statistics theorem [ edit ]
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 theory states that
particles with an integer spin are bosons, while all 🧲 other particles have half-integer
spins and are fermions. 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 relies on both quantum mechanics and 🧲 the theory of special
relativity, and this connection between spin and statistics has been called "one of the
most important 🧲 applications of the special relativity theory".[11]
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[12]
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
−2.00231930436256(35), with the 🧲 digits in parentheses denoting measurement uncertainty
in the last two digits at one standard deviation.[13] 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 electromagnetic field, including its own
field.[14]
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:[15][16][17]
μ ν ≈ 3 × 10 − 19 μ
B m ν c 2 eV , {\displaystyle 🧲 \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 🧲 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/c2,
fine-tuning would be necessary in order to prevent large contributions 🧲 to the neutrino
mass via radiative corrections.[18] 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 a photon or a Z boson, do not
have a magnetic 🧲 moment.
Curie temperature and loss of alignment [ edit ]
In ordinary
materials, the magnetic dipole moments of individual atoms produce magnetic 🧲 fields that
cancel one another, because each dipole points in a random direction, with the overall
average being very near 🧲 zero. Ferromagnetic materials below their Curie temperature,
however, exhibit magnetic domains in which the atomic dipole moments spontaneously
align locally, 🧲 producing a macroscopic, non-zero magnetic field from the domain. These
are the ordinary "magnets" with which we are all familiar.
In 🧲 paramagnetic materials,
the magnetic dipole moments of individual atoms will partially align with an externally
applied magnetic field. In diamagnetic 🧲 materials, on the other hand, the magnetic
dipole moments of individual atoms align oppositely to any externally applied magnetic
field, 🧲 even if it requires energy to do so.
The study of the behavior of such "spin
models" is a thriving area 🧲 of research in condensed matter physics. For instance, the
Ising model describes spins (dipoles) that have only two possible states, 🧲 up and down,
whereas in the Heisenberg model the spin vector is allowed to point in any direction.
These models 🧲 have many interesting properties, which have led to interesting results in
the theory of phase transitions.
Direction [ edit ]
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[19]
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 2s + 1 possible values of s z . The number "2s +
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 ]
A single point in space can rotate continuously without becoming
tangled. Notice that after a 360-degree rotation, the spiral 🧲 flips between clockwise
and counterclockwise orientations. It returns to its original configuration after
spinning a full 720°.
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 ]
Operator [ edit ]
Spin obeys 🧲 commutation
relations[20] 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},}
ε jkl
S ^ 2 {\displaystyle 🧲 {\hat {S}}^{2}}
S ^ z
{\displaystyle {\hat {S}}_{z}}
S
: 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}}}
whereis the
Levi-Civita symbol . It follows (as with angular momentum ) that the eigenvectors
ofand(expressed as kets in 🧲 the total basis ) are
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 ,}
S ^ 🧲 ± = S ^ x ± i S ^ y
{\displaystyle {\hat {S}}_{\pm }={\hat {S}}_{x}\pm i{\hat {S}}_{y}}
: 166
where
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 ]
The quantum-mechanical operators
associated with spin- 1/2 🧲 observables are
S ^ = ℏ 2 σ , {\displaystyle {\hat {\mathbf
{S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }},}
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}.}
where in Cartesian
components
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 ]
For
systems 🧲 of N identical particles this is related to the Pauli exclusion principle,
which states that its wavefunction ψ ( r 🧲 1 , σ 1 , … , r N , σ N ) {\displaystyle \psi
(\mathbf {r} _{1},\sigma _{1},\dots ,\mathbf 🧲 {r} _{N},\sigma _{N})} 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)2s will reduce to +1, for fermions to −1. In
quantum mechanics 🧲 all particles are either bosons or fermions. In some speculative
relativistic quantum field theories "supersymmetric" particles also exist, where linear
🧲 combinations of bosonic and fermionic components appear. In two dimensions, the
prefactor (−1)2s can be replaced by any complex number 🧲 of magnitude 1 such as in the
anyon.
The above 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 ]
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 2s
+ 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).[21] 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 2n). 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} },}
θ = θ θ 🧲 ^
{\textstyle {\boldsymbol {\theta }}=\theta {\hat {\boldsymbol {\theta }}}}
S
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}}} s )[22] : 164 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
where, andis the vector of spin operators
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 },}
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
}
γ = 2π
α = β = 0
z
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}.}
whereis Wigner's small d-matrix 🧲 . Note that forand; i.e., a full
rotation about theaxis, the Wigner D-matrix elements become
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)2m = −1 for all m, and hence upon rotation by 🧲 2π the state
picks up a minus sign. This fact is a crucial element of the proof of the
spin–statistics 🧲 theorem.
Lorentz transformations [ edit ]
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
u }[\gamma _{\mu },\gamma
_{
u }]\right)}\psi ,}
γ ν
ω μν
⟨ ψ | ϕ ⟩ = ψ ¯ ϕ = ψ 🧲 † γ 0 ϕ {\displaystyle \langle
\psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi }
Measurement
of spin along 🧲 the x , y , or z axes [ edit ]
whereare gamma matrices , andis an
antisymmetric 4 × 4 🧲 matrix parametrizing the transformation. It can be shown that the
scalar productis preserved. It is not, however, positive-definite, so the
🧲 representation is not unitary.
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 ]
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 ]
Since the Pauli matrices do not
commute, measurements of spin along the different axes 🧲 are incompatible. This means
that if, for example, we know the spin along the x axis, and we then measure 🧲 the spin
along the y axis, we have invalidated our previous knowledge of the x axis spin. This
can be 🧲 seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli
matrices that
| ⟨ ψ x ± | ψ 🧲 y ± ⟩ | 2 = | ⟨ ψ x ± | ψ z ± ⟩ | 2 = | 🧲 ⟨ ψ y ± | ψ z ± ⟩ |
2 = 1 2 . {\displaystyle {\big |}\langle \psi _{x\pm 🧲 }|\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 🧲 |}^{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
🧲 spin state collapses into the eigenstate | ψ x + ⟩ {\displaystyle |\psi _{x+}\rangle }
. When we then subsequently 🧲 measure the particle's spin along the y axis, the spin
state will now collapse into either | ψ y + 🧲 ⟩ {\displaystyle |\psi _{y+}\rangle } or |
ψ y − ⟩ {\displaystyle |\psi _{y-}\rangle } , each with probability 1/2. 🧲 Let us say, in
our example, that we measure − ħ/2. When we now return to measure the particle's spin
🧲 along the x axis again, the probabilities that we will measure ħ/2 or − ħ/2 are each
1/2 (i.e. they 🧲 are | ⟨ ψ x + | ψ y − ⟩ | 2 {\displaystyle {\big |}\langle \psi
_{x+}|\psi _{y-}\rangle {\big 🧲 |}^{2}} and | ⟨ ψ x − | ψ y − ⟩ | 2 {\displaystyle {\big
|}\langle \psi _{x-}|\psi _{y-}\rangle 🧲 {\big |}^{2}} respectively). This implies that
the original measurement of the spin along the x axis is no longer valid, 🧲 since the
spin along the x axis will now be measured to have either eigenvalue with equal
probability.
Higher spins [ 🧲 edit ]
The spin- 1/2 operator S = ħ/2σ 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}}} a , b {\displaystyle a,b} 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}}}
Parity [ edit ]
for higher spins is tractable, but
less simple.
In tables of the spin quantum number s for nuclei 🧲 or particles, the spin
is often followed by a "+" or "−". 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.
Applications [ 🧲 edit ]
Spin has important theoretical implications and practical
applications. Well-established direct applications of spin include:
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.[24] 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.[25]
There are many indirect
applications and manifestations of spin 🧲 and the associated Pauli exclusion principle,
starting with the periodic table of chemistry.
History [ edit ]
Spin was first
discovered in 🧲 the context of the emission spectrum of alkali metals. In 1924, Wolfgang
Pauli introduced what he called a "two-valuedness not 🧲 describable classically"[26]
associated with the electron in the outermost shell. This allowed him to formulate the
Pauli exclusion principle, stating 🧲 that no two electrons can have the same quantum
state in the same quantum system.
The physical interpretation of Pauli's "degree 🧲 of
freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in
early 1925 that it was produced by 🧲 the self-rotation of the electron. When Pauli heard
about the idea, he criticized it severely, noting that the electron's hypothetical
🧲 surface would have to be moving faster than the speed of light in order for it to
rotate quickly enough 🧲 to produce the necessary angular momentum. This would violate the
theory of relativity. Largely due to Pauli's criticism, Kronig decided 🧲 not to publish
his idea.[27]
In the autumn of 1925, the same thought came to Dutch physicists George
Uhlenbeck and Samuel 🧲 Goudsmit at Leiden University. Under the advice of Paul Ehrenfest,
they published their results.[28] It met a favorable response, especially 🧲 after
Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental
results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished 🧲 results).
This discrepancy was due to the orientation of the electron's tangent frame, in
addition to its position.
Mathematically speaking, a 🧲 fiber bundle description is
needed. The tangent bundle effect is additive and relativistic; that is, it vanishes if
c goes 🧲 to infinity. It is one half of the value obtained without regard for the
tangent-space orientation, but with opposite sign. 🧲 Thus the combined effect differs
from the latter by a factor two (Thomas precession, known to Ludwik Silberstein in
1914).
Despite 🧲 his initial objections, Pauli formalized the theory of spin in 1927,
using the modern theory of quantum mechanics invented by 🧲 Schrödinger and Heisenberg. He
pioneered the use of Pauli matrices as a representation of the spin operators and
introduced a 🧲 two-component spinor wave-function. Uhlenbeck and Goudsmit treated spin as
arising from classical rotation, while Pauli emphasized, that spin is a 🧲 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 🧲 the relativistic
electron. In the Dirac equation, a four-component spinor (known as a "Dirac spinor")
was used for the electron 🧲 wave-function. Relativistic spin explained gyromagnetic
anomaly, which was (in retrospect) first observed by Samuel Jackson Barnett in 1914
(see Einstein–de 🧲 Haas effect). In 1940, Pauli proved the spin–statistics theorem, which
states that fermions have half-integer spin, and bosons have integer 🧲 spin.
In
retrospect, the first direct experimental evidence of the electron spin was the
Stern–Gerlach experiment of 1922. However, the correct 🧲 explanation of this experiment
was only given in 1927.[30]
See also [ edit ]
References [ edit ]
Further reading [
edit ]
artigos relacionados
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