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# Spin angular momentum

**Spin angular momentum**(Spin motion,British: *spin angular momentum*) IsQuantum mechanicsWith the above conceptparticleUniqueAngular momentumIs. simply**spin**Also 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 momentum**Exists, and the sum of spin and orbital angular momentum**Full angle momentum**Called.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 like**Such 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^{[1]}^{: 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 of**Spin quantum number**That.

Spin quantum number is a half-integer 1/2, 3/2,… Particles that become**Fermion**,integer 0, 1, 2, ... Particles that become**Boson**The 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.

## ready

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.

### Math preparation

The mathematical knowledge required to define the spin angular momentum operator is briefly described.**R**The set of all real numbers,**C**Is the set of all complex numbers. 3D space**R**^{3}InRotation matrixThe whole set

Notated as.here The n Row n 列のExecution columnIt ’s a whole set,I TheIdentity matrixAnd^{t}*R* The R OfTranspose matrix.SO (3) With respect to matrix multiplicationgroupBecause it makesSO (3) 3D**Rotation group**That.

SO (3) like,"SmoothA group with a structure**Lie group**(See the Lie group section for a strict definition).In particularSO (3) Lie group consisting of a matrix like**Matrix Lie group**Or simply**Matrix group**That 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,*I*Is an identity matrix*A*^{*} The*A* OfHermitian conjugateIs:

- 3D rotation group … (G1)
- Unitary groupIn the linear map above, … (G2)
- Special unitary groupIn the linear map above, … (G3)

Vector spaceV **C**^{n}IfU(*V*), SU (*V*)Each thingU(*n*), SU (*n*)It is written as.

*G*TheSO (3), U (*V*), SU (*V*)When any of

- The
*G*In the above differentiable curve,*t*=0 When is the identity matrix … (G4)

The*G* Of**Lie algebra**Called,The origin of*G*upper**Infinitesimal transformation**Called.Lee "ringThe name isIs in the matrix**Commutator product**

Because it forms a ring with respect to.SO (3), U (*V*), SU (*V*)Each of the Lie algebras

- … (G5)
- The
*V*In the linear map above, The*V*upperSkew-Hermit operator … (G6) - The
*V*In 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*) ∈

**R**

^{3}On the other hand, the matrix belonging to so (3)

*F*

_{x}The

- … (G11)

When defined as^{[2]}^{: p344}^{[3]}^{: p36}The following holds^{[3]}^{: p36} :

- exp (exp (
*F*_{x}) The**x**The rotation matrix is about the axis, and the rotation angle is clockwise with respect to the axis. |||| RadianIs. ... (G12)**x** - … (G13)

here"×IsCross product.*G*,*H*TheSO (3), U (*V*), SU (*V*)And either,The*G*,*H*Lie algebra. (That is,, Theso (3), u (*V*), Su (*V*)Is either).

The*G*から*H*Differentiable toHomomorphismIt is a map.At this time*π* **Map to guide***π*_{*}The

- … (G14)

Defined by, this mapwell-definedbecome.Moreover, this map is known to be a homomorphic map as a Lie algebra.That is,

- … (G15)

.

*π*Guided by*π*_{*}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 **R**^{3} upper*L*^{2} 空间Is equal to, i.e.

.

The orbital angular momentum operator is derived as symmetry with respect to the rotation of space.^{[1]}^{: 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*

^{−1}

*) Move to.That is, each rotation matrix*

**x***R*∈ SO (3) On the other hand, the space of the wave function Unitary operator on

Is defined^{[3]}^{: p37}^{[2]}^{: p396 Def 17.1}.

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 *λ*_{R} Correspondence (continuousHomomorphism) Map

The SO (3) Ofupper**Unitary representation**That.

on the other hand,SO (3) The entire set of "infinitesimal transformations" corresponding to so (3) (**G1**), And (**G14**)λ **Map to guide**λ_{*}The

- upperSkew-Hermit operator

So the unit vector * n* = (

*x*,

*y*,

*z*) ∈

**R**

^{3}Against

*F*

_{n}(

**G11**), AndImaginary unit i とConverted Planck's constant

*ħ*Using,

- … (J1)

When defined as The*L*^{2}(**R**^{3})Become the Hermitian operator above.This operator is "infinitesimal rotation*F*_{n}Operator corresponding to "^{[1]}^{: p73}And this operator is the axis * n* = (

*x*,

*y*,

*z*) ∈

**R**

^{3}Around

**Orbital angular momentum operator**Call.

For example, z Orbital angular momentum around the axis Spherical coordinate system (*r*, *θ*, *φ*) Using

It can be confirmed as follows that it can be written as.ψLet be an arbitrary wavefunction, then (**G10**), (**G12**)Than

さ ら に x 軸、y Each orbital angular momentum around the axis,age,*F*_{x}=*F*_{(1,0,0)},*F*_{y}=*F*_{(0,1,0)},*F*_{z}=*F*_{(0,0,1)}Then (**G15**), (**G13**)ThanExchange relation

Follows.

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 experiments**Spin( x,y,z)Fourth internal degree of freedom of particles independent of**This is because it is known to be.Because of this, when considering spin,Wave functionThe whole Hilbert space Is a single particle system The

*L*

^{2}(

**R**

^{3}) Is not equal to.

Therefore, to describe the spin, the space of the state vector of the spin*V*_{s} The*L*^{2}(**R**^{3})Prepared separately from

Need to think^{[4]}..Subscript here*s* 0 ≥Is an integer or half-integer,*V*_{s} The2*s*+1 It is a dimensional complex metric vector space.

One-particle system wave function space When can be written as aboves The particles of**Spin quantum number**To say^{[2]}^{: p384}.*V*_{s} The**Spinor space**^{[3]}^{: p50},*V*_{s} The origin of**Spinor**That.s When is a half-integer that is not an integer**Fermion**Good,s When is an integer**Boson**That.

### 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.

#### Ingredient labeling

From the definition of the tensor product, the wave function The

- … (B1)

Ingredients can be displayed in the form of.here TheL^{2}(**R**^{3})Is the source of*σ*_{j} The*V*_{s}Is the source of.Therefore,

If you define

Is.When written in this way, spin (representing spinor)*σ*_{j} (*x*,*y*,*z*)It is easy to understand that it is the fourth internal degree of freedom independent of.

#### Spinor display

Wave function considering spin*ψ*Ingredient display (**B1**) Is interpreted from another angle.Wave function considering spin*ψ*Against*ψ'*(*x*,*y*,*z*)The

Can be defined as.In the above formula, "・" is a vector*σ*_{j} OfScalar times by.A normal wave function that does not consider spin is a one-dimensional complex metric vector space**C**Whereas the value is taken to*ψ'*(*x*,*y*,*z*) The2*s*+1Dimension complex metric vector space** V_{s}Wavefunction that takes a value to**Can be regarded as.Wave function considering spin

*ψ*,

*V*

_{s}What is regarded as a wave function that takes a value in

*ψ*Of

**Spinor display**That.

In many physics textbooks*V*_{s}The source of is introduced in the form of component display.*e*_{-s}, *e*_{−(s − 1)},…, *e*_{s - 1}, *e*_{s} The*V*_{s}When it is the basis of*ψ'*(*x*,*y*,*z*)Is always

Because it can be written in the form of*ψ'*(*x*,*y*,*z*)Is a vector

Can be displayed as an ingredient.

Basis *e*_{-s}, *e*_{−(s − 1)},…, *e*_{s - 1}, *e*_{s} 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,*L*^{2}(**R**^{3}) 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 is*V*_{s}It 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.

*V*_{s} 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. *V*_{s} It can be defined as an operator for infinitesimal rotation.However, in the definition of the orbital angular momentum operatorSimply *V*_{s} 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 *V*_{s} When I try to define the spin angular momentum operator by replacing it withSO (3) Of *V*_{s} The above unitary representation is needed.However, it is known that such expressions do not always exist.^{[2]}^{: p375 Thm 17.10} :

**Theorem 1** - The following holds:

- sIf is an integerSO (3) Of
*V*_{s}The above irreducible unitary representation exists (uniquely except for the same type). - s If is a non-integer half-integerSO (3) Of
*V*_{s}The above irreducible unitary representation does not exist.

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 first solution is *V*_{s} Instead of thinking directly*V*_{s} The origin ofphaseIgnore the differenceEquivalence relation^{[2]}^{: p368}

The same equivalence relation for unitary operators

Identified byEquivalence class [*U*] Is to think about^{[2]}^{: p369}..The set of all equivalence classes of this unitary operator

It is written as.PU (*V*_{s}) The *V*_{s} upper**Projection unitary group**,PU (*V*_{s}) Equivalence class belonging to *V*_{s} upper**Projection unitary operator**Call.

Projection unitary operator [*U*] The *V*_{s} / ~ 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) Of**Projection unitary representation**

To use.

Unlike normal unitary representation, projective unitary representation is known to satisfy the following:^{[2]}^{: p383-384}

**Theorem 2** - s Whether is an integer or a half-integerSO (3) Of *V*_{s} The above irreducible projective unitary representation exists (uniquely except for the same type).

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 of**Spin 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 Φ
_{n}: 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^{[2]}^{: p383-384}:

**Theorem 3** - sWhether is an integer or a half-integerSpin (3) Of *V*_{s} The above irreducible unitary representation exists (uniquely except for the same type).

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,*π*_{s}(*S*) The origin of the spin group S Of *V*_{s} As the unitary representation above,γ(*R*) The rotation matrix R Of *V*_{s} 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 space
*V*_{s} **Theorem 3**Said inSpin (3) = SU (2)Irreducible unitary representation of- Spin (3) = SU (2)からSO (3)Map to

Write down concretely.However, in this section*V*_{s}と*π*_{s}Most important about*s*= 1 / 2I will only describe the case of.Excluding thatsSee later chapters for more information.

### spin1/2In the case of*V*_{s}と*π*_{s}Specific notation of

*M*_{2, 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 *V*_{1/2} It is the above unitary representation.This unitary expression**Theorem 3**Of 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

- ...L2

If you define^{[5]},su (2) Can be regarded as a metric vector space with three degrees of freedom.

nextsu (2) Describes the basis of.**Pauli matrices** *σ*_{1}, *σ*_{2}, *σ*_{3} The

- … (L3)

Defined bysu (2)Source of*X*_{1},*X*_{2},*X*_{3}The

- .... (L4)

Defined by^{[3]}^{: p39-40}^{[6]}^{: p31,73}, (**L1**), (**L2**) Shows that the following holds.

*X*_{1},*X*_{2},*X*_{3} Thespin (3) = su (2) Orthonormal basis above^{[3]}^{: p39-40}^{[6]}^{: p31,73}.. ... (L5)

So 3D vector* x*=(

*x*,

*y*,

*z*) ∈

**R**

^{3}Against

- … (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.^{[6]}^{: p65} :

here"×IsCross productAnd[*A*,*B*] = *AB*-*BA*Is the commutator product.

### Spin (3)Original concrete notation of

Spin (3) = SU (2) It is,*α*, *β* Using the real number of

- … (X1)

Follow from a simple calculation that can be written as^{[3]}^{: p38}^{[6]}^{: p65}.

on the other hand,* n*=(

*x*,

*y*,

*z*) ∈

**R**

^{3}Is 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* ∈ [0,2*π*]Against

- ... (X3)

Follow^{[6]}^{: p28-29}.

Then (**L4**)spin (3) = su (2)The fact that the elements of the spin group can be written as follows using the basis of (**X1**), (**X2**), (**X3**):

- Spin (3) = SU (2)Any source of
*U*Is a unit vector=(**n***x*,*y*,*z*) ∈**R**^{3}と *θ*∈ [0,4*π*]- Using
- It can be written in the form of.Moreover,
*S*≠*I*, -*I*If so, it can be written like this,**n***θ*Is unique. ... (X4)

### Spin (3) から SO (3) Homomorphic mapping to *Φ*_{3}

As mentioned in the previous sectionsu (2) Is a three-dimensional metric vector space, so**R**^{3}Can be equated with.*U* ∈ Spin (3) = SU (2)とAgainst*SLEEP*^{−1}AlsoIt can be seen from a simple calculation that it is the source of.Moreover, a linear map*Φ*_{3}(U)The

When defined as*Φ*_{3}(U)But(**L2**It can be confirmed by a simple calculation that the inner product defined in) and the orientation of the space are maintained.That is,*Φ*_{3}(U)Is a rotational transformation, so*Φ*_{3}(U) ∈ SO (3).

From the above,Spin (3) から SO (3) Homomorphic mapping to

Was defined.this *Φ*_{3}The specific notation of is described in a later section.

*Φ*_{3}Guided by(*Φ*_{3})_{*}Definition and its concrete notation

(**G14**)in accordance with,*Φ*_{3} **Map to guide**(*Φ*_{3})_{*} ,

- … (D1)

Defined by.At this time(*Φ*_{3})_{*} The

- … (D2)

Meet^{[3]}^{: p43}^{[6]}^{: p73}..If you write in ingredients

Is. Especially

- … (D3)

Is an isomorphism.

**(**

**D2**) Proof(**X4**)Than,Spin (3)Curve above*U*(*t*)Is a three-dimensional space curve* x*(

*t*) ∈

**R**

^{3}Using

It can be written in the form of.* x*=(

*x*

_{1},

*x*

_{2},

*x*

_{3})=

*(0)Then any*

**x***Y*=

*v*

_{1}

*X*

_{1}+

*v*

_{2}

*X*

_{2}+

*v*

_{3}

*X*

_{3}∈ spin (3) = su (2)Against

Is. here[A, B] = AB-BA TheCommutator product.

Therefore*Y*Arbitrariness and (**G11**) The following conclusions can be drawn:

*Φ*_{3}Specific notation of

- … (E1)

Is. (**X4**)Than,Spin (3)What is the origin of*θ*∈ [0*, 4π*]Using,exp (exp (*θX*_{x})Since it can be written in the form of*Φ*_{3}It is possible to completely describe the behavior of.

Moreover

Therefore, the definition of the spin group (**C1**)*Φ*_{3}We can see the fact that is a 2: 1 mapping.

Spin (3) = SU (2)Original ingredient display (**X1**)*Φ*_{3}Is also known to be able to be displayed as follows^{[6]}^{: p74} :

## 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 *V*_{s}The existence and uniqueness (excluding isomorphisms) of such a unitary representation as the above irreducible unitary representation**Theorem 3**Guaranteed by).It should be noted that *s*= 1 / 2 Against*V*_{s},π_{s} は(**H1**), (**H2**) Has already been described.For other s*V*_{s},π_{s} 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).

π_{s} Guided by (π_{s})_{*}Is defined as:

- Hermitian operator above … (F1)

As well *Φ*_{3} Induces(*Φ*_{3})_{*} (**D1**) When defined as an expression,(*Φ*_{3})_{*} は(**D2**) Write like an expression, (**D3**)Than

.

Unit vector * n* = (

*x*,

*y*,

*z*) ∈

**R**

^{3}Infinitesimal rotation

*X*

_{n}∈ 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 space*F*_{n}Can be regarded as the operator corresponding to.

This,**n****Spin angular momentum operator with**Call^{[3]}^{: p50-51,60}^{[7]}.

### Properties of spin angular momentum operator

#### Exchange relation

xAxis (1,0,0),yAxis (0,1,0),zAxis (0,0,1) ∈Spin angular momentum operator withThen,

Will be.Therefore (**G15**)Than,Orbital angular momentumSimilar toExchange relationHolds:

#### Change of rotation axis

Next, let us look at the relationship between spin angular momentums with different axes of rotation.**n**,**m**∈**R**^{3}Is a two unit vector**n**と**m**Is a rotation matrix*R*By

Let's say that they were moving.MapΦ_{3} : Spin (3) = SU (2) → SO (3)Is a XNUMX: XNUMX surjective function, so

Meet*U*Exists.

Spin angular momentum operator,From its definition*V*_{s}The above unitary operators, both

It is tied in the relationship.Here on the right sideと*π*_{s}(*U*)Is the product as a matrix of.

**Proof**

So,(*Φ*_{3})_{*}Is an isomorphism,

Therefore any*t*∈**R**Against

Therefore

### spin1/2Specific notation in the case of

Spin quantum number*s* 1/2If, the spinor space is (**H1**)Than

And the unit vector * n* = (

*x*,

*y*,

*z*) ∈

**R**

^{3}The spin angular momentum operator with is on the axis of rotation is

**H2**), (

**L6**), (

**F1**), (

**F2**)Than,

Is.So especially

The* n*Always eigenvalue regardless

have.

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

* n*When vectoring a unit in three-dimensional space

*The orbital angular momentum of one particle with the axis of rotation isSO (3)Unitary representation of*

**n***λ*Guided by

*λ*

_{*}IsomorphismUsing

- Unitary operator above

Can be written as (**J1**)When(**D2**).here""Is a composition of functions.The spin angular momentum of one particle is also (**F2**From)

- Skew-Hermitian operator above

Was defined as.

* n*Of a single particle with a rotation axis

**Full-angle momentum operator**The

- 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 of
*X*_{n})… (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, (**K1**You 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 (**K1**Examine the properties of operators that can be expressed in the form of).

### Spin (3)Unitary representation of

*u*≧ 0Is an integer or a half-integer*W*_{u}The2*u*+1Let it be a dimensional complex metric vector space.In particular

- When considering the spin angular momentum of one particle,
*u*=*s*so,*W*_{u}Is a spinor space*V*_{s} - When considering the orbital angular momentum of one particle,
*W*_{u}TheL^{2}(**R**^{3}) Of2*u*+1Dimensional subspace - When considering the total angular momentum of one particle,
*W*_{u}The Of2*u*+1Dimensional subspace

Is assumed.The same applies to the case of a plurality of particles.**Theorem 1**Than,Spin (3) Of*W*_{s}Since the irreducible unitary representation above exists uniquely except for the same type, this irreducible unitary representation is used.

Notated as. (**H1**), (**H2**) As already mentioned

.

general*u*Against*W*_{u}と*D*^{u} The*W*_{1/2}と*D*^{1/2}Can be configured from^{[8]}^{: p25-27}.

*W*_{u}Composition of

##### Symmetric tensor product

*W*_{u}In preparation for constructing, we define a symmetric tensor product.*W*_{1/2} Of2*u*Tensor product of copies

Think aboutSource ofAgainst*ψ* Of**Symmetrization**The

- … (M1)

Defined by.here TheSubstitution groupIs.That is,Is eachjAgainstThe sum of all the subscripts of(2*u*)!It is divided by. (Even if defined in this way, it is well-defined).Symmetrized tensor**Symmetric tensor**The subvector space formed by the entire symmetric tensor is called

It is written as.**e**_{0},**e**_{1}The*W*_{1/2}=**C**^{2}As the basis of

When defined as*E*_{0}, ...,*E*_{2s}ObviouslyIs the basis of.Therefore The2*u*+1It is a dimension.

##### Definition

*W*_{u}The

- … (M2)

Define as^{[8]}^{: p25-27}.

*D*^{u}Composition of

*U*∈ Spin (3)Against

The

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.^{[8]}^{: p25-27}.

### Observables and their nature

In this section, defined in the previous sectionSpin (3)Irreducible unitary representation ofD^{u}To define an observable and examine the nature of the observable.

#### Observable

Guided by

And unit vectors in 3D spaceObservable with

Can be defined.here*i*Is an imaginary unit,*X*_{n}は(**L6**).In particular

*W*_{u}Is a spinor space*V*_{s}When*u*=*s*so,Is the spin angular momentum operator of one particle*W*_{u}L^{2}(**R**^{3}) Of2*u*+1In the case of a dimensional subspaceIs the one-particle orbital angular momentum operator*W*_{u}The Of2*u*+1In the case of a dimensional subspaceIs a one-particle full-angle momentum operator

.

(D^{u})_{*}Is written concretely.*U*(*t*)The

If you take to meetLeibniz ruleWhen(**N1**)Than

Is. here*I*Is always the identity matrix*I*Is a map that returns.

#### Unique state

spin1/2Observable by the same discussion as atD^{1/2}Is two eigenvalues

Since each has a unique state corresponding to these,age,*k* = −*u*, − (*u* − 1),…, (*u* − 1), *u*Against

- … (P1)

To^{[8]}^{: p25-27}.

here*c*(*k*)Is a normalization constant^{[8]}^{: p25-27}

- … (P2)

Then,

So,*E*_{n,k}Is an eigenvalueIt is a unique state corresponding to.

#### Ladder operator

* n* x軸、y軸、zWhen it is an axisThe,,age,

If^{[3]}^{: p50},

Is^{[3]}^{: p50}.

**Proof**

#### Clebsch-Gordan coefficient

*W*_{u},*W*_{v}Each2*u*+1dimension,2*v*+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 above formula**Clebsch-Gordan decomposition**To say^{[3]}^{: p59}^{[9]}^{: p116}.

The basis of the left side of the above equation is

Can be described in the format of.hereIs an eigenvalue*j*_{1}Corresponding toD^{u}Is the eigenstate of.On the other hand, the base on the right side is

Can be described in the format of.here TheEigenvalue in*j*Corresponding toD^{w}Is the eigenstate of.Since both are connected by a basis transformation, some coefficient*c*(*u*,*v*,*w*,*j*_{1},*j*_{2},*j*)Using

Can be written.*c*(*u*,*v*,*w*,*j*_{1},*j*_{2},*j*)The**Clebsch-Gordan coefficient**To say^{[3]}^{: p60-61}.

## History

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.^{[10]}^{[11]}..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^{[12]}.

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.

## footnote

- ^
^{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.

## Related item

## References

- [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.

## 外部 リンク

- Spin, generalized angular momentum, angular momentum synthesis (PDF) (Japanese)
- Chapter 19 Spin Chapter 14 Orbital Angular Momentum Chapter 20 Angular Momentum Synthesis (PDF) (Japanese)
- 9th Lecture Material (PDF) (Japanese) Pauli Of
**spin**Derivation of matrix - Chapter 2 Spin (Part 1) Chapter 2 Spin (Part 2) (PDF) (Japanese) Spinor notation-Pauli Of
**spin**Derivation of matrix - 3.3 Rotational symmetry (PDF) (Japanese)
**spin**とSymmetry operation