Photo Lala Anai was playing with a sticker for ballistic measurement on the driver's face on the first day (Photo: Wataru Murakami)
Lala Anai is disqualified for violating "Compatible Club" Play with a sticker on the driver's face
If you write the contents roughly
However, if it hits, the effect on the amount of spin and repulsion will not be zero.
<Daikin Orchid Ladies Day 2 ◇ 5th ◇ Ryukyu Golf Club (Okinawa Prefecture) ◇ 6561 yards par 72>… → Continue reading
Golf information ALBA.Net
Wikipedia related words
If there is no explanation, there is no corresponding item on Wikipedia.
Spin angular momentum
Spin angular momentum(Spin motion,British: spin angular momentum) IsQuantum mechanicsWith the above conceptparticleUniqueAngular momentumIs. simplyspinAlso called.The angular momentum of a particle is the angular momentum derived from the rotational motion of the particle in addition to spin.Orbital angular momentumExists, and the sum of spin and orbital angular momentumFull angle momentumCalled.The "particle" here isElectronic,quarkSuch asElementary particlesEvenHadron,Atomic nucleus,atomConsists of multiple elementary particlesComposite particlesIt may be.
The name "spin" is based on this concept of particles.rotationFor historical reasons, it was perceived as something likeSuch an interpretation is no longer considered correct..Because spin is the classical limit ħ→ 0Since it disappears in, the concept of spin, including "rotation"classicBecause it is completely meaningless to add an interpretation: p196.
Like other physical quantities in quantum mechanics, the spin angular momentum isoperatorIs defined using.This operator (Spin angular momentum operator) Is defined corresponding to the direction of the spin axis of rotation,x 軸、y 軸、z Axial spin operatorsIt is written as.Of these operatorseigenvalue(= Corresponds to these operatorsObservableThe value obtained when observingIntegerOrHalf-integerValue that is s 0 ≥ Using,
Can be written as.value s Is known to be determined only by the particles and independent of the direction of the axis of the spin operator.this s The particles ofSpin quantum numberThat.
Spin quantum number is a half-integer 1/2, 3/2,… Particles that becomeFermion,integer 0, 1, 2, ... Particles that becomeBosonThe physical properties of the two are very different (see each item for details). To the extent known as of 2016
- All spin quantum numbers of elementary particles that are fermions 1/2 .
- Elementary particles that are bosonsHiggs bosonOnly spin quantum number 0 And the spin quantum numbers of other boson particles are 1 .
- The spin quantum number of the composite particle can take other values, but it is not simply the total value of the spin quantum numbers of the elementary particles constituting the composite particle.For example, the spin quantum number of helium atom 0 However, the electrons and quarks that make up this are all fermions, and therefore their spin quantum numbers are half-integers.
In non-relativistic quantum mechanics, the spin angular momentum behaves very differently from other observables, so the theory must be revised only to describe the spin angular momentum.For itRelativistic quantum mechanicsThen, for example,Dirac equationThe definition of spin is formulated in a more natural way, such as the concept of spin being woven into itself.
In the following, unless otherwise specified, the concept of spin for non-relativistic quantum mechanics will be described.
This section first introduces the rotation group and the unitary group, and then uses these concepts to formulate the concept of orbital angular momentum from the perspective of rotational symmetry.The reason for reviewing the concept of orbital angular momentum in this section is to formulate the concept of spin angular momentum with reference to the definition of orbital angular momentum in the following sections.
The mathematical knowledge required to define the spin angular momentum operator is briefly described.RThe set of all real numbers,CIs the set of all complex numbers. 3D spaceR3InRotation matrixThe whole set
Notated as.here The n Row n 列のExecution columnIt ’s a whole set,I TheIdentity matrixAndtR The R OfTranspose matrix.SO (3) With respect to matrix multiplicationgroupBecause it makesSO (3) 3DRotation groupThat.
SO (3) like,"SmoothA group with a structureLie group(See the Lie group section for a strict definition).In particularSO (3) Lie group consisting of a matrix likeMatrix Lie groupOr simplyMatrix groupThat is.The Lie groups appearing in this section are limited to the following linear groups.Therefore, in this section, we avoid developing the general theory of Lie groups, and limit our discussion to the following linear groups.BelowVIs a complex metric vector space,IIs an identity matrixA* TheA OfHermitian conjugateIs:
- 3D rotation group … (G1)
- Unitary groupIn the linear map above, … (G2)
- Special unitary groupIn the linear map above, … (G3)
Vector spaceV CnIfU(V), SU (V)Each thingU(n), SU (n)It is written as.
GTheSO (3), U (V), SU (V)When any of
- The G In the above differentiable curve,t=0 When is the identity matrix … (G4)
Because it forms a ring with respect to.SO (3), U (V), SU (V)Each of the Lie algebras
- … (G5)
- TheVIn the linear map above, TheVupperSkew-Hermit operator … (G6)
- TheVIn the linear map above, … (G7)
.so (3)Is in the above-mentioned form for the following reasons.R(t)TheSO (3) In the above differentiable curve,t=0 If it becomes an identity matrix atSO (3) From the definition of
So that t = 0 Differentiation at
This is to satisfy.u(V), Su (V)Can be proved in the same way thatIn addition, although the case where V is finite dimension is assumed here, it is infinite dimension.Hilbert spaceThe same thing holds true in the case of.
Theso (3), u (V), Su (V)And the matrixAgainstexp (exp (A) The
- … (G8)
When defined as, the following holds:
- A∈ so (3), u (V), Su (V)If,exp (exp (A) Are eachSO (3), U (V), SU (V)Is the source of. ... (G9)
- … (G10)
SO (3) The above-mentioned property can be described more concretely. 3D vector x = (x, y, z) ∈ R3On the other hand, the matrix belonging to so (3)FxThe
- … (G11)
- exp (exp (Fx) The x The rotation matrix is about the axis, and the rotation angle is clockwise with respect to the axis. ||x|| RadianIs. ... (G12)
- … (G13)
here"×IsCross product.G,HTheSO (3), U (V), SU (V)And either,TheG,HLie algebra. (That is,, Theso (3), u (V), Su (V)Is either).
TheGからHDifferentiable toHomomorphismIt is a map.At this timeπ Map to guideπ*The
- … (G14)
Defined by, this mapwell-definedbecome.Moreover, this map is known to be a homomorphic map as a Lie algebra.That is,
- … (G15)
πGuided byπ*And matrix exponentialexpSatisfies the following relationship:
- anyAgainst … (G16)
Orbital angular momentum operator from the viewpoint of rotational symmetry of space
In (non-relativistic) quantum mechanicsWave functionThe whole set isHilbert space In the case of a system consisting of one particle (without considering spin), it can be described as Is two-dimensionalEuclidean space R3 upperL2 空间Is equal to, i.e.
The orbital angular momentum operator is derived as symmetry with respect to the rotation of space.: p73..Therefore, in order to derive the orbital angular momentum operator, we investigate how the wave function changes depending on the rotation matrix. The Lie group of the entire three-dimensional rotation matrix SO (3) When writing, the rotation matrix R ∈ SO (3) When the coordinate system is rotated by, the wave function ϕ(x) The ϕ(R−1x) Move to.That is, each rotation matrix R ∈ SO (3) On the other hand, the space of the wave function Unitary operator on
Complex metric vector spaceVThe group of all the above unitary operatorsU(V)When, the rotation matrix R In contrast to the complex plane space Unitary operator above λR Correspondence (continuousHomomorphism) Map
The SO (3) OfupperUnitary representationThat.
- upperSkew-Hermit operator
- … (J1)
When defined as TheL2(R3)Become the Hermitian operator above.This operator is "infinitesimal rotationFnOperator corresponding to ": p73And this operator is the axis n = (x, y, z) ∈ R3AroundOrbital angular momentum operatorCall.
For example, z Orbital angular momentum around the axis Spherical coordinate system (r, θ, φ) Using
The orbital angular momentum operator for the two axes isSO (3) Unitary representation of λ Tie by.That is,R With a rotation matrix z Axis w If you move it to the axis,w Orbital angular momentum around the axis Is a composite map
Wave function space when spin is taken into consideration Mathematical formulation of
As mentioned in the previous section, the orbital angular momentum operator represents the position of a particle.(x,y,z)It can be defined as rotational symmetry in three-dimensional space.Spin, on the other hand, cannot be formulated in such a way.From various physics experimentsSpin(x,y,z)Fourth internal degree of freedom of particles independent ofThis is because it is known to be.Because of this, when considering spin,Wave functionThe whole Hilbert space Is a single particle system TheL2(R3) Is not equal to.
Therefore, to describe the spin, the space of the state vector of the spinVs TheL2(R3)Prepared separately from
Need to think..Subscript heres 0 ≥Is an integer or half-integer,Vs The2s+1 It is a dimensional complex metric vector space.
One-particle system wave function space When can be written as aboves The particles ofSpin quantum numberTo say: p384.Vs TheSpinor space: p50,Vs The origin ofSpinorThat.s When is a half-integer that is not an integerFermionGood,s When is an integerBosonThat.
How to display the wave function considering spin
In many physics textbooks, spin-aware wavefunctionsIs written in two ways.Therefore, I will introduce these two notation methods next.
From the definition of the tensor product, the wave function The
- … (B1)
Ingredients can be displayed in the form of.here TheL2(R3)Is the source ofσj TheVsIs the source of.Therefore,
If you define
Is.When written in this way, spin (representing spinor)σj (x,y,z)It is easy to understand that it is the fourth internal degree of freedom independent of.
Wave function considering spinψIngredient display (B1) Is interpreted from another angle.Wave function considering spinψAgainstψ'(x,y,z)The
Can be defined as.In the above formula, "・" is a vectorσj OfScalar times by.A normal wave function that does not consider spin is a one-dimensional complex metric vector spaceCWhereas the value is taken toψ'(x,y,z) The2s+1Dimension complex metric vector spaceVsWavefunction that takes a value toCan be regarded as.Wave function considering spinψ,VsWhat is regarded as a wave function that takes a value inψ OfSpinor displayThat.
In many physics textbooksVsThe source of is introduced in the form of component display.e-s, e−(s − 1),…, es - 1, es TheVsWhen it is the basis ofψ'(x,y,z)Is always
Because it can be written in the form ofψ'(x,y,z)Is a vector
Can be displayed as an ingredient.
Basis e-s, e−(s − 1),…, es - 1, es Is usually an eigenvector corresponding to the spin operator (on some axis).
Observable when considering spin
Observables when spin is not considered in quantum mechanicsIt is,L2(R3) It is formulated as the Hermitian operator above.This operator when considering spinThe
By equating with, the space of the wave function considering the spinConsider it as the above observable. (hereid TheConformal mapIs).
As will be described later, the spin angular momentum operator isVsIt can be formulated as the Hermitian operator above, but this is also due to the same kind of identification.Consider it as the above observable.That is,When is the spin angular momentum (on some axis) The
To equate with.
Vs Problems with the above unitary representation
The orbital angular momentum operatorThe spin angular momentum operator is similar to what could be defined as the "operator for infinitesimal rotation" above. Vs It can be defined as an operator for infinitesimal rotation.However, in the definition of the orbital angular momentum operatorSimply Vs The spin angular momentum operator cannot be defined simply by replacing it with.This is due to the following reasons.
In the case of the orbital angular momentum operator, a group of three-dimensional rotation matrices SO (3) OfUnitary representation above
The t The orbital angular momentum operator was defined by differentiating with respect to.
Therefore, in the definition of the orbital angular momentum operator, simplyThe Vs When I try to define the spin angular momentum operator by replacing it withSO (3) Of Vs The above unitary representation is needed.However, it is known that such expressions do not always exist.: p375 Thm 17.10 :
That is, in the methodology described above,s The spin angular momentum operator cannot be defined for the case where is a half-integer.There are two solutions to this problem, and the two are essentially equivalent, as described below.
Solution using projected unitary representation
The same equivalence relation for unitary operators
It is written as.PU (Vs) The Vs upperProjection unitary group,PU (Vs) Equivalence class belonging to Vs upperProjection unitary operatorCall.
Projection unitary operator [U] The Vs / ~ It is known to be the map above:
So to describe the behavior of the spin operatorSO (3) Instead of the unitary representation of SO (3) OfProjection unitary representation
Unlike normal unitary representation, projective unitary representation is known to satisfy the following:: p383-384
Therefore, the spin angular momentum operator can be defined by using the projected unitary representation instead of the unitary representation.
This paper does not describe the details of the definition of the spin angular momentum operator using the projected unitary representation.This is because there are few physics textbooks that describe spin operators using projected unitary representations.However, as already mentioned, the solution by the projected unitary representation is essentially equivalent to the other solution described later, so the projected unitary representation was used from the definition of the spin angular momentum operator using the other solution. The definition of the spin angular momentum operator can be derived.
The projective unitary representation solution can describe the spin angular momentum operator in a format similar to other observables, except that it is identified in a phase that has no physical meaning, so another solution described later. Compared to, the advantage is that its physical meaning is easy to understand.
Solution using spin groups
Another solution isSO (3) 3D instead ofSpin group Spin (3) Is to be used.Therefore, we first introduce the definition and properties of spin groups.n The dimensional spin group satisfies the following propertiesLinkIMatrix groupIt is a thing. (It is known that there is only one connected matrix group satisfying such a property except for the same type):
- Differentiable homomorphism Φn: Spin (n) → SO (n) And there is something that is a 2: 1 surjective function. ...C1
hereSO (n) ThenIt is a group of dimensional rotation matrices.What is needed to define the spin angular momentum is the spin group when the dimension is 3.Spin (3)AndSpin (3)Is a 2D special unitary transformation group SU (2) Known to be isomorphic to:
Therefore, unless otherwise specified, Spin (3) と SU (2) To equate.
From the definition of spin group, rotation matrix R Is the source of some spin group U Using
Can be written as.This is the rotation matrix R Instead of dealing directly with, the source of the spin group U Means that rotation can be described by.Therefore SO (3) Instead of the unitary representation of Spin (3) Consider the unitary representation of.SO (3) Unlike the unitary representation ofSpin (3) Unitary representation of: p383-384:
Therefore SO (3) Instead of the unitary representation of Spin (3) The spin angular momentum operator can be defined by using the unitary representation of.Details will be described later.
Equivalence of the two solutions
The two solutions mentioned above are essentially equivalent.this is Spin (3) With unitary representation of SO (3) This is because the projected unitary representation of is naturally one-to-one.In particular,πs(S) The origin of the spin group S Of Vs As the unitary representation above,γ(R) The rotation matrix R Of Vs With the above projected unitary representation, the following scheme is commutative (if properly replaced with an isomorphic one).here design Is a map that takes equivalence classes.
Specific notation of space and function used to define spin
Based on the above discussionSpin (3) = SU (2)It was found that the spin angular momentum can be defined by using.Therefore, in this section, it is necessary to define the spin angular momentum.
- Spinor spaceVs
- Theorem 3Said inSpin (3) = SU (2)Irreducible unitary representation of
- Spin (3) = SU (2)からSO (3)Map to
Write down concretely.However, in this sectionVsとπsMost important abouts= 1 / 2I will only describe the case of.Excluding thatsSee later chapters for more information.
spin1/2In the case ofVsとπsSpecific notation of
M2, 2(C) To complex quadraticSquare matrixAs a whole setI When is an identity matrixSpin (3) = SU (2)Is the set of all 2D unitary transformations
Is a subset of.Therefore
- … (H1)
Defined as an inclusion map
The Spin (3) = SU (2) Original V1/2 It is the above unitary representation.This unitary expressionTheorem 3Of the irreducible unitary representation mentioned in s= 1 / 2 Corresponds to the case of.That is,
- … (H2)
Set of infinitesimal transformationsspin (3) = su (2)Specific notation of
When defining orbital angular momentumSO (3)Set of infinitesimal transformations ofso (3)For the same reason that was needed, the definition of spin angular momentumSpin (3) = SU (2) The whole set of "infinitesimal transformations"spin (3) = su (2)In this section, we will investigate its concrete form and basic properties. (G4), (G7)Than,
- The Spin (3) = SU (2) In the above differentiable curve,t=0 When is the identity matrix. ... (L1)
.su (2) aboveinner product
If you define,su (2) Can be regarded as a metric vector space with three degrees of freedom.
nextsu (2) Describes the basis of.Pauli matrices σ1, σ2, σ3 The
- … (L3)
Defined bysu (2)Source ofX1,X2,X3The
- .... (L4)
So 3D vectorx=(x, y, z) ∈R3Against
- … (L6)
Defined as a map
By と spin (3) = su (2) Can be equated as a metric vector space.Moreover, in this identification, the following holds.: p65 :
here"×IsCross productAnd[A,B] = AB-BAIs the commutator product.
Spin (3)Original concrete notation of
Spin (3) = SU (2) It is,α, β Using the real number of
- … (X1)
on the other hand,n=(x, y, z) ∈R3Is a unit vector, and Pauli matrices are used.
- … (X2)
By a simple calculation,
I understand.So the matrixAExponential function againstexp (exp (A)(A3) When defined as an expression,τ ∈ [0,2π]Against
- ... (X3)
- Spin (3) = SU (2)Any source ofUIs a unit vectorn=(x, y, z) ∈R3と
- θ∈ [0,4π]
- It can be written in the form of.Moreover,S≠I, -IIf so, it can be written like thisn,θIs unique. ... (X4)
Spin (3) から SO (3) Homomorphic mapping to Φ3
As mentioned in the previous sectionsu (2) Is a three-dimensional metric vector space, soR3Can be equated with.U ∈ Spin (3) = SU (2)とAgainstSLEEP−1AlsoIt can be seen from a simple calculation that it is the source of.Moreover, a linear mapΦ3(U)The
When defined asΦ3(U)But(L2It can be confirmed by a simple calculation that the inner product defined in) and the orientation of the space are maintained.That is,Φ3(U)Is a rotational transformation, soΦ3(U) ∈ SO (3).
From the above,Spin (3) から SO (3) Homomorphic mapping to
Was defined.this Φ3The specific notation of is described in a later section.
Φ3Guided by(Φ3)*Definition and its concrete notation
(G14)in accordance with,Φ3 Map to guide(Φ3)* ,
- … (D1)
Defined by.At this time(Φ3)* The
- … (D2)
- … (D3)
Is an isomorphism.
Φ3Specific notation of
- … (E1)
Is. (X4)Than,Spin (3)What is the origin ofθ∈ [0, 4π]Using,exp (exp (θXx)Since it can be written in the form ofΦ3It is possible to completely describe the behavior of.
Therefore, the definition of the spin group (C1)Φ3We can see the fact that is a 2: 1 mapping.
Definition and properties of the spin angular momentum operator
Definition of spin angular momentum operator
Based on the above preparations, the spin angular momentum is defined.
The Spin (3) = SU (2) Of VsThe existence and uniqueness (excluding isomorphisms) of such a unitary representation as the above irreducible unitary representationTheorem 3Guaranteed by).It should be noted that s= 1 / 2 AgainstVs,πs は(H1), (H2) Has already been described.For other sVs,πs Will be described later in the following sections.
さ ら に
(C1)Spin (3) から SO (3)To 2: 1 map (The specific form of this map is (E1) See equation).The figures of these maps are as follows.Here the symbol "Means that G is a set of matrices on the vector space V (ie G is in V)effectTo).
πs Guided by (πs)*Is defined as:
- Hermitian operator above … (F1)
Unit vector n = (x, y, z) ∈ R3Infinitesimal rotation Xn ∈ su (2) (L6) And a composite map
- Skew-Hermitian operator aboveHermitian operator above
Hermitian operator determined by
- … (F2)
Given that, (D2)Than,
Because it can be writtenIs an infinitesimal rotation in 3D spaceFnCan be regarded as the operator corresponding to.
Properties of spin angular momentum operator
xAxis (1,0,0),yAxis (0,1,0),zAxis (0,0,1) ∈Spin angular momentum operator withThen,
Change of rotation axis
Next, let us look at the relationship between spin angular momentums with different axes of rotation.n,m∈R3Is a two unit vectornとmIs a rotation matrixRBy
Let's say that they were moving.MapΦ3 : Spin (3) = SU (2) → SO (3)Is a XNUMX: XNUMX surjective function, so
Spin angular momentum operator,From its definitionVsThe above unitary operators, both
It is tied in the relationship.Here on the right sideとπs(U)Is the product as a matrix of.
spin1/2Specific notation in the case of
Spin quantum numbers 1/2If, the spinor space is (H1)Than
ThenAlways eigenvalue regardless
eachStandardizationThe resulting eigenvectors are as follows.
Spin (3)Unitary representation and angular momentum
In this section, 3D spin groupsSpin (3) = SU (2)The unitary representation of is described in detail, and the properties of the orbital angular momentum, the spin angular momentum, and the total angular momentum, which is the sum of them, are investigated based on this.
Notation of orbital momentum and total angular momentum by spin group
nWhen vectoring a unit in three-dimensional spacenThe orbital angular momentum of one particle with the axis of rotation isSO (3)Unitary representation ofλGuided byλ*IsomorphismUsing
- Unitary operator above
- Skew-Hermitian operator above
Was defined as.
nOf a single particle with a rotation axisFull-angle momentum operatorThe
- Skew-Hermitian operator above
When defined as
Can be written as. The
- Unitary operator above
Since it is a map induced by, all of the orbital angular momentum, spin angular momentum, and total angular momentum for one particle
- (Spin (3)Map guided by the unitary representation ofXn)… (K1)
You can see that it can be written in the form of.
Since the orbital angular momentum, spin angular momentum, and total angular momentum for multiple particles can be expressed as the sum of those of one particle, (K1You can see that it can be written in the form of).
ThereforeSpin (3)If the specific form of the unitary representation of can be specified, the orbital angular momentum, spin angular momentum, and total angular momentum (for one or more particles) can be specifically written down.Therefore, in the main installation,Spin (3)Write down the unitary representation ofSpin (3)Using the unitary representation of (K1Examine the properties of operators that can be expressed in the form of).
Spin (3)Unitary representation of
u≧ 0Is an integer or a half-integerWuThe2u+1Let it be a dimensional complex metric vector space.In particular
- When considering the spin angular momentum of one particle,u=sso,WuIs a spinor spaceVs
- When considering the orbital angular momentum of one particle,Wu TheL2(R3) Of2u+1Dimensional subspace
- When considering the total angular momentum of one particle,Wu The Of2u+1Dimensional subspace
Is assumed.The same applies to the case of a plurality of particles.Theorem 1Than,Spin (3) OfWsSince the irreducible unitary representation above exists uniquely except for the same type, this irreducible unitary representation is used.
generaluAgainstWuとDu TheW1/2とD1/2Can be configured from: p25-27.
Symmetric tensor product
WuIn preparation for constructing, we define a symmetric tensor product.W1/2 Of2uTensor product of copies
Think aboutSource ofAgainstψ OfSymmetrizationThe
- … (M1)
Defined by.here TheSubstitution groupIs.That is,Is eachjAgainstThe sum of all the subscripts of(2u)!It is divided by. (Even if defined in this way, it is well-defined).Symmetrized tensorSymmetric tensorThe subvector space formed by the entire symmetric tensor is called
It is written as.e0,e1TheW1/2=C2As the basis of
When defined asE0, ...,E2sObviouslyIs the basis of.Therefore The2u+1It is a dimension.
- … (M2)
Define as: p25-27.
U∈ Spin (3)Against
Defined by TheIt is a linear map that keeps the inner product above.clearlyMoves the symmetric tensor to the symmetric tensor, so OfRestricted mapping to,
- … (N1)
It is defined as
Keeps the dot product, so this is
Means.This map is the irreducible unitary representation that should be sought.: p25-27.
Observables and their nature
In this section, defined in the previous sectionSpin (3)Irreducible unitary representation ofDuTo define an observable and examine the nature of the observable.
And unit vectors in 3D spaceObservable with
Can be defined.hereiIs an imaginary unit,Xnは(L6).In particular
- WuIs a spinor spaceVsWhenu=sso,Is the spin angular momentum operator of one particle
- Wu L2(R3) Of2u+1In the case of a dimensional subspaceIs the one-particle orbital angular momentum operator
- Wu The Of2u+1In the case of a dimensional subspaceIs a one-particle full-angle momentum operator
(Du)*Is written concretely.U(t)The
Is. hereIIs always the identity matrixIIs a map that returns.
spin1/2Observable by the same discussion as atD1/2Is two eigenvalues
Since each has a unique state corresponding to these,age,k = −u, − (u − 1),…, (u − 1), uAgainst
- … (P1)
herec(k)Is a normalization constant: p25-27
- … (P2)
So,En,kIs an eigenvalueIt is a unique state corresponding to.
n x軸、y軸、zWhen it is an axisThe,,age,
Wu,WvEach2u+1dimension,2v+1A dimensional complex metric vector space
As an irreducible unitary representation
Is not always an irreducible unitary representation.However, it is known that the following facts hold true if the basis is replaced properly:
The basis of the left side of the above equation is
Can be described in the format of.hereIs an eigenvaluej1Corresponding toDuIs the eigenstate of.On the other hand, the base on the right side is
Can be described in the format of.here TheEigenvalue injCorresponding toDwIs the eigenstate of.Since both are connected by a basis transformation, some coefficientc(u,v,w,j1,j2,j)Using
Can be written.c(u,v,w,j1,j2,j)TheClebsch-Gordan coefficientTo say: p60-61.
In an experiment observing the sodium spectrum, it was discovered that the D-line in the magnetic field splits into two (Zeeman effect), This is because the electron has a binary quantum degree of freedom that is not yet known, and in 2.UhlenbeckとGoudsmitIn addition to the orbital angular momentum of the electron revolving around the nucleus, he hypothesized that the electron has a size rather than a mass point, and that the electron itself is rotating...In this assumption, its rotationAngular momentumThe size ofHowever, since the direction of rotation of the rotation is different, the result of the experiment could be explained well by considering that the energy level was split into two by the interaction with the angular momentum accompanying the revolution.And this degree of freedom was called the spin angular momentum of the electron.
However, considering that the spin angular momentum is actually derived from the rotation of the electron according to this assumption, it means that the electron must have a magnitude and rotate at a speed exceeding the speed of light, which isSpecial relativityIs inconsistent with.Therefore, in 1925Ralph KronigThough proposed byPauliWas denied by.Pauli abandoned the classic picture of having to think about the rotation itself, and the general angular momentum Paying attention to the fact that a half-integer value is allowed as the eigenvalue of.
Then developedStandard modelEven if the electron is treated as a mass point of size 0, there is no contradiction with high accuracy experimentally, and it is not known whether the electron has an internal structure (whether it originates from internal degrees of freedom such as spin angular momentum). ..
Spin and statistics
s Particles that have a half-integer valueFermionsAnds Particles that take an integer valueBose particleIs known to be.s Such a relationship between the value of and the statisticalityRelativityTypicalQuantum field theoryCan be explained by.
- ^ a b c Landau-Lifshitz lesson
- ^ a b c d e f g h H13
- ^ a b c d e f g h i j k l m A07
- ^ H13, P383.In addition, on this pagenotIs written, Since is a finite dimension, they are the same (the description immediately before Def 17.21 on the same page).
- ^ A07 pp. 39–40.It should be noted thatA07Then the inner productHowever, since this is for the space pasted by Pauli matrices, this is defined assu (2)When copied to, the inner product becomes the form defined in this section.
- ^ a b c d e f g W16
- ^ H13 pp. 383–384.H13Is defined using the projective representation, so it is necessary to read this as the unitary representation of spin.
- ^ a b c d e S 12
- ^ W16
- ^ GE Uhlenbeck, S. Goudsmit (1925). “Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons”. natural sciences 13 (47): 953–954. two:10.1007 / BF01558878.
- ^ GE Uhlenbeck, S. Goudsmit (1926). “Spinning Electrons and the Structure of Spectra”. Nature 117: 264–265. two:10.1038 / 117264a0.
- ^ Shigenobu Sunakawa "Quantum Mechanics"Iwanami Shoten, 1991.ISBN 4000061399.
- [Landau-Lifshitz lesson] LD Landau,EM LifshitsWritten by Shigehiro Yoshimura and translated by Takeo Inoue (June 2008, 6). Landau-Lifshitz Physics Lesson Quantum Mechanics. Chikuma art library
- [A07] Joṥe Alvarado (December 2007, 12). “Group Theoretical Aspects of Quantum Mechanics (pdf) ”. 2016/12/1Browse.
- [H13] Brian C. Hall (2013/7/1). Quantum Theory for Mathematicians. Graduate Texts in Mathematics 267. Springer
- [S93] JJ Sakurai (1993 / 9 / 10). Modern Quantum Mechanics, Revised Edition. Addison Wesley. ISBN 978-0201539295
- [S12] DE Soper (January 2012, 1). “The rotation group and quantum mechanics (pdf) ”. 2016/12/27Browse.
- [W16] Peter Woit (December 2016, 12). “Quantum Theory, Groups and Representations: An Introduction (pdf) ”. 2016/12/16Browse.