利用者:Nova replet laetitia/sandbox/宇宙・物理・自然科学/スピン角運動量

en:Spin (physics) (00:02, 13 July 2014‎)の版

量子力学quantum mechanicsと素粒子物理学particle physicsにおいて、素粒子elementary particle、ハドロンhadronの複合粒子、原子核atomic nucleusによって伝えられるスピンは角運動量angular momentumの本質となるものである。スピンは量子力学にのみの現象で古典力学classical mechanicsには（スピンという言葉から自転軸を回転する惑星のような現象を思い起こすが）対応する概念がない。

スピンは量子力学における2種類の角運動量の1つで、もう1つは軌道角運動量という。軌道角運動量は角運動量の古典的解釈に正対する量子力学的解釈であり、これは粒子が回転したりひねるような曲線を描くときに生じるときに現れる（核を回る電子軌道など）。スピン角運動量の存在は実験から推論されているものである。例えばシュテルン=ゲルラッハの実験Stern–Gerlach experiment。これについて、粒子が角運動量を持っていることを観測することからは軌道角運動量だけを説明することができない。

ある意味では、スピンはベクトル量Euclidean vectorのようなものである。それは大きさが定義され、方向を持つ（しかし量子化は通常のベクトル方向とは異なる「方向」を定義する）。すべての素粒子ははある同じ大きさのスピン各運度量をもち、それをその粒子のもつスピン量子数quantum numberとする。

スピンはSI単位系[SI unitで[$Js$]] であるが、これはちょうど古典的な角運動量と同じである。実際にはSI単位系でスピンを記述することはないが、そのかわり換算したプランク定数reduced Planck constant 構文解析に失敗 (SVG（ブラウザのプラグインで MathML を有効にすることができます）: サーバー「http://localhost:6011/ja-two.iwiki.icu/v1/」から無効な応答 ("Math extension cannot connect to Restbase."):): {\displaystyle ħ} を単位として表す。自然単位系では 構文解析に失敗 (構文エラー): {\displaystyle ħ} が省略され単位のない数として書かれる。この定義によりスピン量子数は常に単位が書かれない。

スピン統計定理（英: spin-statistics theorem）と組み合わせることで、電子のスピンはパウリの排他律Pauli exclusion principleに従う。これは化学物質の周期表の根幹に関わっている。

Wolfgang Pauliは最初にスピンの考え方を提案したが、それにはまだ名前を付けていなかった。1925年にライデン大学Leiden UniversityのRalph Kronig、George Uhlenbeck、Samuel Goudsmitが軸の周りを回転している粒子という物理的解釈を与えた。数学的な深い考察はPauliの1927年の業績である。さらに1928年のPaul Diracの相対論的量子力学で発展させ、電子スピンはその重要な部分となった。

量子数 Quantum Number

→詳細は「スピン量子数」を参照

Spin quantum number

名前の通り、スピンはもともと軸のまわりの粒子の回転として考えだされた。この描像は量子化された角運動量と同様にスピンが数学的に従っている限りは正しい。一方で軌道角運動量と区別されるスピンは独特の性質をもつ。：

• スピンの量子数は半整数値をとる。

• スピンの方向は変化することができるが、速いもしくは遅いスピンをもつ素粒子は存在できない。

• 荷電粒子のスピンは、磁気双極子モーメントmagnetic dipole momentと1 ではない g因子g-factorと関連している。これは、粒子の内部電荷が質量分布とは異なる場合に起こりうる古典的な磁気回転比によるためである。

The conventional definition of the spin quantum number, s, 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:

スピン量子数の従来の従来からの定義は、sで、s = n/2、ここで n は負でない任意の整数。つまり s は 0, 1/2, 1, 3/2, 2,などを取る。sの値は素粒子の種類によって決まり、（下記にのべるスピンの方向とは対照的な方法で）どのような方法でも変更できない。どの物理系でもスピン角運動量Sは量子化された角運動量angular momentum quantizationである。Sは、

$S={\frac {h}{2\pi }}\,{\sqrt {s(s+1)}}={\frac {h}{4\pi }}\,{\sqrt {n(n+2)}},$

ここで h がプランク定数である。対照的に、軌道角運動量（角運動量演算子）angular momentum operatorは整数値sだけをとる。すなわちnは偶数である。

フェルミオンとボソン

半整数のスピンをもつ粒子（1/2、3/2、5/2など）はフェルミ粒子として知られる。一方で整数のスピンをもつ粒子（0、1、2など）はボーズ粒子として知られる。2つの粒子の系統は異なったルールに従い、この世界で「広範な」異なった役割を示す。2つの系統を区別する鍵はフェルミ粒子はパウリの排他律に従うということである。これは、2つのフェルミ粒子は同時同じ量子数を持つことができないということである（これはおおまかには同じ運動量で同じ場所にはいられないということである）。対照的にボーズ粒子はボーズ・アインシュタイン統計に従い、このような制約がない。つまり、同じ状態であっても共に群れていることができる。また、その複合粒子はそれを構成するそれぞれの粒子とはことなるスピンを持つことができる。例えば、ヘリウム原子はスピン0をもち、ボーズ粒子のように振る舞うが、それを構成するクオークや電子はすべてフェルミ粒子でできている。

このことが以下のような実態を示している。：

• クオークとレプトン（電子とニュートリノも含む）は、古典的な物質を形作るものであり、すべて 1/2 のスピンをもつ。「物質が空間を占める」という考え方は、同じ量子状態をとることから物質を形成するフェルミ粒子を妨げるような粒子を排除するPauliの排他律と共通の考え方となっている。

さらに電子が同じエネルギー状態を占めるように閉じ込めようとするため、ある種の圧力（時に電子の縮退圧として知られる）として振る舞いフェルミ粒子を近づけないようにする。これは星の内部での崩壊を妨げるが、重い星が巨大な重力圧で崩壊してしまうときには、最後には内部崩壊を起こして劇的な超新星爆発を起こす要因にもなる。

物質を構成するクオークやレプトンといった基本的な素粒子には、1/2 以外のスピンをもつものは 2014年現在見つかっていない。

• Elementary particles which are thought of as carrying forces are all bosons with spin 1. They include the photon which carries the electromagnetic force, the gluon (strong force), and the W and Z bosons (weak force). The ability of bosons to occupy the same quantum state is used in the laser, which aligns many photons having the same quantum number (the same direction and frequency), superfluid liquid helium resulting from helium-4 atoms being bosons, and superconductivity where pairs of electrons (which individually are fermions) act as single composite bosons.

Elementary bosons with other spins (0, 2, 3 etc.) were not historically known to exist, although they have received considerable theoretical treatment and are well established within their respective mainstream theories. In particular theoreticians have proposed the graviton (predicted to exist by some quantum gravity theories) with spin 2, and the Higgs boson (explaining electroweak symmetry breaking) with spin 0. Since 2013 the Higgs boson with spin 0 has been considered proven to exist. It is the first scalar particle (spin 0) known to exist in nature.

Theoretical and experimental studies have shown that the spin possessed by elementary particles cannot be explained by postulating that they are made up of even smaller particles rotating about a common center of mass analogous to a classical electron radius; as far as can be presently determined, these elementary particles have no inner structure. The spin of an elementary particle is therefore seen as a truly intrinsic physical property, akin to the particle's electric charge and rest mass.

=== スピン統計定理 === Spin-statistics theorem The proof that particles with half-integer spin (fermions) obey Fermi–Dirac statistics and the Pauli Exclusion Principle, and particles with integer spin (bosons) obey Bose–Einstein statistics, occupy "symmetric states", and thus can share quantum states, is known as the spin-statistics theorem. 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".

磁気モーメント

→詳細は「スピン磁気モーメント」を参照

Spin magnetic moment

Magnetic field lines around a magnetostatic dipole; the magnetic dipole itself is in the center and is seen from the side.

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

${\boldsymbol {\mu }}={\frac {g_{s}q}{2m}}\mathbf {S}$

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 −2.0023193043622(15), with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation. The value of 2 arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties, and the correction of 0.002319304... arises from the electron's interaction with the surrounding electromagnetic field, including its own field. 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:

$\mu _{\nu }\approx 3\times 10^{-19}\mu _{\mathrm {B} }{\frac {m_{\nu }}{\text{eV}}}$

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 cannot exceed about 1 eV, these radiative corrections must then be assumed to be fine tuned to cancel out to a large degree.

The measurement of neutrino magnetic moments is an active area of research. 2001年現在, the latest experimental results have put the neutrino magnetic moment at less than 1.2×10⁻¹⁰ times the electron's magnetic moment.

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. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, 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 spontaneously align with an externally applied magnetic field. In diamagnetic materials, on the other hand, the magnetic dipole moments of individual atoms spontaneously 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.

== 方向 == Direction

→詳細は「角運動量演算子」を参照

Angular momentum operator

=== スピン射影量子数と多重度 === Spin projection quantum number and multiplicity 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 measured along any direction can only take on the values

$S_{i}=\hbar s_{i},\quad s_{i}\in \{-s,-(s-1),\dots ,s-1,s\}\,\!$

where S_i is the spin component along the i-axis (either x, y, or z), s_i is the spin projection quantum number along the i-axis, and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z-axis:

$S_{z}=\hbar s_{z},\quad s_{z}\in \{-s,-(s-1),\dots ,s-1,s\}\,\!$

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 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.

=== ベクトル === Vector For a given quantum state, one could think of a spin vector $\langle S\rangle$ whose components are the expectation values of the spin components along each axis, i.e., $\langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]$. 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 maximum probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180 degrees—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 degrees 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 degree rotation. A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180 degrees can bring it back to the same quantum state and a spin-4 particle should be rotated 90 degrees 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 degrees and a spin 0 particle can be imagined as sphere which looks the same after whatever angle it is turned through.

== 数学的な公式 == Mathematical formulation

=== 演算子 === Operator Spin obeys commutation relations analogous to those of the orbital angular momentum:

$[S_{i},S_{j}]=i\hbar \epsilon _{ijk}S_{k}$

where $\epsilon _{ijk}$ is the Levi-Civita symbol. It follows (as with angular momentum) that the eigenvectors of S² and S_z (expressed as kets in the total S basis) are:

${\begin{aligned}S^{2}|s,m\rangle &=\hbar ^{2}s(s+1)|s,m\rangle \\S_{z}|s,m\rangle &=\hbar m|s,m\rangle .\end{aligned}}$

The spin raising and lowering operators acting on these eigenvectors give:

$S_{\pm }|s,m\rangle =\hbar {\sqrt {s(s+1)-m(m\pm 1)}}|s,m\pm 1\rangle$, where $S_{\pm }=S_{x}\pm iS_{y}.$

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.

In addition to their other properties, all quantum mechanical particles possess an intrinsic spin (though it may have the intrinsic spin 0, too). The spin is quantized in units of the reduced Planck constant, such that the state function of the particle is, say, not $\psi =\psi (\mathbf {r} )$, but $\psi =\psi (\mathbf {r} ,\sigma )\,,$ where $\sigma$ is out of the following discrete set of values:

$\sigma \in \{-s\hbar ,-(s-1)\hbar ,\cdots ,+(s-1)\hbar ,+s\hbar \}.$

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.

=== パウリ行列 === Pauli matrices

The quantum mechanical operators associated with spin observables are:

$S_{x}={\hbar \over 2}\sigma _{x},\quad S_{y}={\hbar \over 2}\sigma _{y},\quad S_{z}={\hbar \over 2}\sigma _{z}.$

In the special case of spin-1/2 particles, σ_x, σ_y and σ_z are the three Pauli matrices, given by:

$\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.$

=== パウリの排他律 === Pauli exclusion principle For systems of N identical particles this is related to the Pauli exclusion principle, which states that by interchanges of any two of the N particles one must have

$\psi (\cdots \mathbf {r} _{i},\sigma _{i}\cdots \mathbf {r} _{j},\sigma _{j}\cdots )=(-1)^{2s}\psi (\cdots \mathbf {r} _{j},\sigma _{j}\cdots \mathbf {r} _{i},\sigma _{i}\cdots ).$

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 chemists or biologists.

=== 回転 === Rotations

→「量子力学の対称性」も参照

symmetries in quantum mechanics

As described above, quantum mechanics states that components of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum mechanical description of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis. For instance, for a spin 1/2 particle, we would need two numbers a_±1/2, giving amplitudes of finding it with projection of angular momentum equal to ħ/2 and −ħ/2, satisfying the requirement

$\left|a_{\frac {1}{2}}\right|^{2}+\left|a_{-{\frac {1}{2}}}\right|^{2}\,=1.$

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's 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:

$\sum _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}\left(\sum _{n=-j}^{j}U_{nm}a_{n}\right)^{*}\left(\sum _{k=-j}^{j}U_{km}b_{k}\right)$

$\sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.$

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). 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 ${\hat {\theta }}$, U can be written

$U=e^{-{\frac {i}{\hbar }}{\vec {\theta }}\cdot {\vec {S}}},$

where ${\vec {\theta }}=\theta {\hat {\theta }}$ and ${\vec {S}}$ is the vector of spin operators.

(Click "show" at right to see a proof or "hide" to hide it.)

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Working in the coordinate system where ${\hat {\theta }}={\hat {z}}$, 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:

${\begin{aligned}S_{x}\rightarrow U^{\dagger }S_{x}U&{}=e^{i\theta S_{z}}S_{x}e^{-i\theta S_{z}}\\&{}=S_{x}+(i\theta )[S_{z},S_{x}]+\left({\frac {1}{2!}}\right)(i\theta )^{2}[S_{z},[S_{z},S_{x}]]+\left({\frac {1}{3!}}\right)(i\theta )^{3}[S_{z},[S_{z},[S_{z},S_{x}]]]+\cdots \\\end{aligned}}$

Using the spin operator commutation relations, we see that the commutators evaluate to iS_y for the odd terms in the series, and to S_x for all of the even terms. Thus:

${\begin{aligned}U^{\dagger }S_{x}U&{}=S_{x}\left[1-{\frac {\theta ^{2}}{2!}}+...\right]-S_{y}\left[\theta -{\frac {\theta ^{3}}{3!}}\cdots \right]\\&{}=S_{x}\cos \theta -S_{y}\sin \theta \\\end{aligned}}$

as expected. Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension (i.e., for any principal spin quantum number s).

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A generic rotation in 3-dimensional space can be built by compounding operators of this type using Euler angles:

${\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha S_{x}}e^{-i\beta S_{y}}e^{-i\gamma S_{z}}$

An irreducible representation of this group of operators is furnished by the Wigner D-matrix:

$D_{m'm}^{s}(\alpha ,\beta ,\gamma )\equiv \langle sm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|sm\rangle =e^{-im'\alpha }d_{m'm}^{s}(\beta )e^{-im\gamma },$

where

$d_{m'm}^{s}(\beta )=\langle sm'|e^{-i\beta s_{y}}|sm\rangle$

is Wigner's small d-matrix. Note that for γ = 2π and α = β = 0; i.e., a full rotation about the z-axis, the Wigner D-matrix elements become

$D_{m'm}^{s}(0,0,2\pi )=d_{m'm}^{s}(0)e^{-im2\pi }=\delta _{m'm}(-1)^{2m}.$

Recalling that a generic spin state can be written as a superposition of states with definite m, we see that if s is an integer, the values of m are all integers, and this matrix corresponds to the identity operator. However, if s is a half-integer, the values of m are also all half-integers, giving (−1)^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.

=== ローレンツ変換 === Lorentz transformations 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 $\psi$ with each particle. These spinors transform under Lorentz transformations according to the law

$\psi '=\exp {\left({\frac {1}{8}}\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi$

where $\gamma _{\mu }$ are gamma matrices and $\omega _{\mu \nu }$ is an antisymmetric 4×4 matrix parametrizing the transformation. It can be shown that the scalar product

$\langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi$

is preserved. It is not, however, positive definite, so the representation is not unitary.

=== x、y、z 軸に沿った度量衡 === Metrology along the x, y, and z axes Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:

パウリ行列のハミルトニアンはそれぞれ2つの固有値 +1 と −1 を持つ。関連する正規化波動関数の固有値は、

${\begin{array}{lclc}\psi _{x+}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{1}\end{pmatrix}},&\psi _{x-}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-1}\end{pmatrix}},\\\psi _{y+}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{i}\end{pmatrix}},&\psi _{y-}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-i}\end{pmatrix}},\\\psi _{z+}=&{\begin{pmatrix}{1}\\{0}\end{pmatrix}},&\psi _{z-}=&{\begin{pmatrix}{0}\\{1}\end{pmatrix}}.\end{array}}$

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:

量子力学の仮定から、

$\psi ={\begin{pmatrix}{a+bi}\\{c+di}\end{pmatrix}}.$

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 $\left\vert \langle \psi _{x+}\vert \psi \rangle \right\vert ^{2}$. Correspondingly, the probability that its spin will be measured as –ħ/2 is just $\left\vert \langle \psi _{x-}\vert \psi \rangle \right\vert ^{2}$. Following the measurement, the spin state of the particle will collapse 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 $\left\vert \langle \psi _{x+}\vert \psi _{x+}\rangle \right\vert ^{2}=1$, etc), provided that no measurements of the spin are made along other axes.

=== 任意の軸での度量衡 === Metrology along an arbitrary axis 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

$S_{u}={\frac {\hbar }{2}}(u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z})$.

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:

${\frac {1}{\sqrt {2+2u_{z}}}}{\begin{pmatrix}1+u_{z}\\u_{x}+iu_{y}\end{pmatrix}}.$

The above spinor is obtained in the usual way by diagonalizing the $\sigma _{u}$ 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.

=== 度量衡の可換性 === Compatibility of metrology 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:

$\mid \langle \psi _{x\pm }\mid \psi _{y\pm }\rangle \mid ^{2}=\mid \langle \psi _{x\pm }\mid \psi _{z\pm }\rangle \mid ^{2}=\mid \langle \psi _{y\pm }\mid \psi _{z\pm }\rangle \mid ^{2}={\frac {1}{2}}.$

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 $\mid \psi _{x+}\rangle$. When we then subsequently measure the particle's spin along the y-axis, the spin state will now collapse into either $\mid \psi _{y+}\rangle$ or $\mid \psi _{y-}\rangle$, 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 $\mid \langle \psi _{x+}\mid \psi _{y-}\rangle \mid ^{2}$ and $\mid \langle \psi _{x-}\mid \psi _{y-}\rangle \mid ^{2}$ 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.

== パリティ == Parity 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.

== 応用 == Applications Spin has important theoretical implications and practical applications. Well-established direct applications of spin include:

• Nuclear magnetic resonance spectroscopy in chemistry;
• Electron spin resonance spectroscopy in chemistry and physics;
• Magnetic resonance imaging (MRI) in medicine, which relies on proton spin density;
• Giant magnetoresistive (GMR) drive head technology in modern hard disks.

Electron spin plays an important role in magnetism, with applications for instance in computer memories. The manipulation of nuclear spin by radiofrequency 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.

A possible future direct application of spin is as a binary information carrier in spin transistors. Original concept proposed in 1990 is known as Datta-Das spin transistor. Electronics based on spin transistors is called spintronics, which includes the manipulation of spins in semiconductor devices.

There are many indirect applications and manifestations of spin and the associated Pauli exclusion principle, starting with the periodic table of chemistry.

歴史　History

Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can share the same quantum state at the same time.

Wolfgang Pauli

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.

In the autumn of 1925, the same thought came to two Dutch physicists, George Uhlenbeck and Samuel Goudsmit at Leiden University. Under the advice of Paul Ehrenfest, they published their results. 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 ones). 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).

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.

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. In 1940, Pauli proved the spin-statistics theorem, which states that fermions have half-integer spin and bosons 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.

See also

• Spinor
• Stern–Gerlach experiment
• Einstein–de Haas effect
• Spin-orbital
• Angular momentum
• Chirality (physics)
• Dynamic nuclear polarisation
• Helicity (particle physics)
• Pauli equation
• Pauli–Lubanski pseudovector
• Rarita–Schwinger equation
• Representation theory of SU(2)
• Spin-½
• Spin-flip
• Spin isomers of hydrogen
• Spin magnetic moment
• Spin quantum number
• Spin-statistics theorem
• Spin tensor
• Spin wave
• Spin Engineering
• Spintronics
• Yrast
• Zitterbewegung

References

• Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2006). Quantum Mechanics (2 volume set ed.). John Wiley & Sons. ISBN 978-0-471-56952-7 
• Condon, E. U.; Shortley, G. H. (1935). “Especially Chapter 3”. The Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4 
• Hipple, J. A.; Sommer, H.; Thomas, H.A. (1949). A precise method of determining the faraday by magnetic resonance. doi:10.1103/PhysRev.76.1877.2 https://www.academia.edu/6483539/John_A._Hipple_1911-1985_technology_as_knowledge
• Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton University Press. ISBN 0-691-07912-9 
• Jackson, John David (1998). Classical Electrodynamics (3rd ed.). John Wiley & Sons. ISBN 978-0-471-30932-1 
• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7 
• Thompson, William J. (1994). Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. Wiley. ISBN 0-471-55264-X 
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4 

External links

• "Spintronics. Feature Article" in Scientific American, June 2002.
• Goudsmit on the discovery of electron spin.
• Nature: "Milestones in 'spin' since 1896."
• ECE 495N Lecture 36: Spin Online lecture by S. Datta