Quantum Mechanics

Algebraic Methods

Harmonic Oscillator

1D Harmonic Oscillator

Hamiltonian

H=(1/(2m))(p^2)+(1/2)m (ω^2)(x^2).

Coordinate operator x,

Momentum operator p,

[x,p]=i ℏ.

p->p(Sqrt[ℏ m ω]), x->x(Sqrt[(ℏ/(m ω))]),

Energy scale set by ℏ ω.

New operators x and p are dimensionless.

Commutation relation for the rescaled operators

The Idea of Boson

Levels are equally spaced: oscillator can only absorb/emit energy in integer multiples of ℏ ω.

Each unit (quantum) of energy is a boson:

For mechanical oscillation (sound), the boson is also called a phonon.

For electromagnetic oscillation (light), the boson is also called a photon.

Each boson carries energy ℏ ω => can be considered as a particle.

({{oscillator, , boson}, {state (Ket[n]), (Subscript[E, n])/ℏ ω, }, {(Ket[0]) (ground), 1/2, vacuum}, {(Ket[1]), 3/2, 1 boson}, {(Ket[2]), 5/2, 2bosons}, {⋮, ⋮, ⋮}})

created by the operator (SuperDagger[a]): (Ket[0])->(Ket[1])->(Ket[2])->…,

annihilated by the operator a: …->(Ket[2])->(Ket[1])->(Ket[0]).

Annihilation and Creation Operators

a and (SuperDagger[a]) are Hermitian conjugate to each other.

Analogy: complex numbers z=x+i y, (SuperStar[z])=x-i y => coordinate ∼ real part, momentum ∼ imaginary part.

Boson Number Basis

(SuperDagger[a])a(Ket[n])=n(Ket[n]), a (SuperDagger[a])(Ket[n])=(n+1)(Ket[n]),

therefore a (SuperDagger[a])=(SuperDagger[a])a+1 indeed holds for all states, as an operator identity.

(SuperDagger[a])a is the boson number operator, which counts the number of bosons in the Fock state.

The Hilbert space spanned by Fock states is called the Fock space (or the occupation number space).

(OverHat[n]) is Hermitian ((SuperDagger[(OverHat[n])])=(OverHat[n])),

(OverHat[n]) is positive (semi)definite: ∀(Ket[ψ]):(Bra[ψ])(OverHat[n])(Ket[ψ])=(Bra[ψ])(SuperDagger[a])a(Ket[ψ])≥0,

(OverHat[n])(Ket[n])=(Subscript[λ, n])(Ket[n]).

(OverHat[n])a=a ((OverHat[n])-1), (OverHat[n])(SuperDagger[a])=(SuperDagger[a]) ((OverHat[n])+1),

(OverHat[n])(a(Ket[n]))=((Subscript[λ, n])-1)(a(Ket[n])), (OverHat[n])((SuperDagger[a])(Ket[n]))=((Subscript[λ, n])+1)((SuperDagger[a])(Ket[n])).

Matrix Representations

(Ket[0])≏(({{1}, {0}, {0}, {⋮}})), (Ket[1])≏(({{0}, {1}, {0}, {⋮}})), (Ket[2])≏(({{0}, {0}, {1}, {⋮}})),…

(Subscript[a, m n])=(Bra[m])a(Ket[n])=(Sqrt[n])(BraKet[m, n-1])=(Sqrt[n])(Subscript[δ, m,n-1]),(Subscript[((SuperDagger[a])), m n])=(Bra[m])(SuperDagger[a])(Ket[n])=(Sqrt[n+1])(BraKet[m, n+1])=(Sqrt[n+1])(Subscript[δ, m,n+1]),

a≏(({{0, (Sqrt[1]),  ,  }, { , 0, (Sqrt[2]),  }, { ,  , 0, ⋱}, { ,  ,  , ⋱}})),(SuperDagger[a])≏(({{0,  ,  ,  }, {(Sqrt[1]), 0,  ,  }, { , (Sqrt[2]), 0,  }, { ,  , ⋱, ⋱}})).

(OverHat[n])=(SuperDagger[a])a≏(({{0,  ,  ,  }, { , 1,  ,  }, { ,  , 2,  }, { ,  ,  , ⋱}})).

x=((a+(SuperDagger[a]))/(Sqrt[2]))≏(1/(Sqrt[2]))(({{0, (Sqrt[1]),  ,  }, {(Sqrt[1]), 0, (Sqrt[2]),  }, { , (Sqrt[2]), 0, ⋱}, { ,  , ⋱, ⋱}})),
p=((a-(SuperDagger[a]))/((Sqrt[2])i))≏(1/((Sqrt[2])i))(({{0, (Sqrt[1]),  ,  }, {-(Sqrt[1]), 0, (Sqrt[2]),  }, { , -(Sqrt[2]), 0, ⋱}, { ,  , ⋱, ⋱}})).

x p≏(i/2)(({{1, 0, -(Sqrt[2]),  ,  }, {0, 1, 0, -(Sqrt[6]),  }, {(Sqrt[2]), 0, 1, 0, ⋱}, { , (Sqrt[6]), 0, 1, ⋱}, { ,  , ⋱, ⋱, ⋱}})),
p x≏(i/2)(({{-1, 0, -(Sqrt[2]),  ,  }, {0, -1, 0, -(Sqrt[6]),  }, {(Sqrt[2]), 0, -1, 0, ⋱}, { , (Sqrt[6]), 0, -1, ⋱}, { ,  , ⋱, ⋱, ⋱}})),

Block[{n=5,a,ad,x,p},
a=SparseArray[{Band[{1,2}]->Sqrt[Range[n]]},{n+1,n+1}];
ad=Transpose[a];
x=(a+ad)/Sqrt[2];
p=(a-ad)/(Sqrt[2]I);
MatrixForm/@{x.p/(I/2),p.x/(I/2)}]

Matrix multiplication.

[x,p]≏(({{i,  ,  ,  }, { , i,  ,  }, { ,  , i,  }, { ,  ,  , ⋱}}))=i 𝟙.

Energy Spectrum

(OverHat[n])(Ket[n])=(SuperDagger[a])a(Ket[n])
=(1/(Sqrt[n!]))(SuperDagger[a])a ((((SuperDagger[a])))^n)(Ket[0])
=…
=(1/(Sqrt[n!]))(SuperDagger[a])(((((SuperDagger[a])))^n)a+((n((SuperDagger[a])))^(n-1)))(Ket[0])
=(1/(Sqrt[n!]))(SuperDagger[a])(((((SuperDagger[a])))^n)a(Ket[0])+((n((SuperDagger[a])))^(n-1))(Ket[0]))
=(1/(Sqrt[n!]))(SuperDagger[a])(0+n ((((SuperDagger[a])))^(n-1))(Ket[0]))
=(1/(Sqrt[n!]))n ((((SuperDagger[a])))^n)(Ket[0])
=n(Ket[n]).

Wave Functions

(x+(∂_x))(Subscript[ψ, 0])(x)=0,

(Subscript[ψ, 0])(x)=(1/(π^(1/4)))(e^(-(1/2)(x^2))).

DSolve[x ψ0[x]+ψ0'[x]==0,ψ0[x],x]

Solve differential equation.

Integrate[(Exp[-x^2/2]/π^(1/4))^2,{x,-∞,∞}]

Check normalization.

(Subscript[ψ, 1])(x)=(1/(Sqrt[2]))(x-(∂_x))(Subscript[ψ, 0])(x)=(1/(π^(1/4)))(Sqrt[2])x (e^(-(1/2)(x^2))).

Nest[(x#-D[#,x])/Sqrt[2]&,Exp[-x^2/2]/π^(1/4),1]

Apply (SuperDagger[a]) operator.

(Subscript[ψ, n])(x)=(1/((π^(1/4))(Sqrt[(2^n)n!])))(Subscript[H, n])(x)(e^(-(1/2)(x^2))),

TableForm[Table[Factor@{n,Nest[(x#-D[#,x])/Sqrt[2]&,Exp[-x^2/2]/π^(1/4),n]/Sqrt[n!],HermiteH[n,x],Plot[HermiteH[n,x]Exp[-x^2/2]/π^(1/4)/Sqrt[2^n n!],{x,-6,6},Ticks->None,Frame->None,PlotStyle->Black,PlotRange->{-1,1},AspectRatio->1/3,ImageSize->80]},{n,0,5}],TableHeadings->{None,{"n","\!\(ψ\_n(x)\)","\!\(H\_n(x)\)","plot of \!\(ψ\_n(x)\)"}}]

(Subscript[(OverTilde[ψ]), n])(p)=(1/((π^(1/4))(Sqrt[(2^n)n!])))(Subscript[H, n])(p)(e^(-(1/2)(p^2))).

Comparing Eq. () and Eq. (), there is an SO(2) rotational symmetry in the 2D phase space of (x,p), which is evident in Eq. ().

This also corresponds to the U(1) symmetry:

a->(e^(i θ))a, (SuperDagger[a])->(e^(-i θ))(SuperDagger[a]).

U(1) symmetry => conservation of the boson number.

Uncertainty Relation

(std x) (std p)≥(1/2)|[x,p]|=(1/2),

x=((a+(SuperDagger[a]))/(Sqrt[2])), p=((a-(SuperDagger[a]))/((Sqrt[2])i)).

(Bra[n])x(Ket[n])=(1/(Sqrt[2]))(Bra[n])a+(SuperDagger[a])(Ket[n])
=(1/(Sqrt[2]))((Sqrt[n])(BraKet[n, n-1])+(Sqrt[n+1])(BraKet[n, n+1]))
=0,

(Bra[n])(x^2)(Ket[n])=(1/2)(((Bra[n])(a+(SuperDagger[a])))^2)(Ket[n])
=(1/2)((Bra[n])a a(Ket[n])+(Bra[n])a (SuperDagger[a])(Ket[n])+(Bra[n])(SuperDagger[a]) a(Ket[n])+(Bra[n])(SuperDagger[a]) (SuperDagger[a])(Ket[n]))
=(1/2)(0+(n+1)+n+0)
=n+1/2.

(((std x))^2)=(Bra[n])(x^2)(Ket[n])-(Bra[n])x((Ket[n])^2)=n+1/2.

(((std p))^2)=(Bra[n])(p^2)(Ket[n])-(Bra[n])p((Ket[n])^2)=n+1/2.

(std x) (std p)=n+1/2≥1/2.

⟨(x^2)⟩⟨(p^2)⟩≥(1/4)((|[x,p]|)^2)=(1/4),

E=⟨H⟩=((ℏ ω)/2)(⟨(p^2)⟩+⟨(x^2)⟩)≥((ℏ ω)/2)(⟨(p^2)⟩+(1/(4⟨(p^2)⟩)))≥((ℏ ω)/2).

Coherent States

Definition

Note that a is non-Hermitian,

its eigenvalues α can be complex,

its eigenstates with different eigenvalues may not be orthogonal, i.e. (BraKet[(Subscript[α, 1]), (Subscript[α, 2])])≠δ((Subscript[α, 1])-(Subscript[α, 2])).

Nevertheless, we do assume that (Ket[α]) is normalized, i.e. (BraKet[α, α])=1.

Eq. () also implies

(Bra[α])x(Ket[α])=(1/(Sqrt[2]))(Bra[α])a+(SuperDagger[a])(Ket[α])
=(1/(Sqrt[2]))(α+(SuperStar[α]))(BraKet[α, α])
=(Sqrt[2])Re α,

(Bra[α])(x^2)(Ket[α])=(1/2)(((Bra[α])(a+(SuperDagger[a])))^2)(Ket[α])
=(1/2)((Bra[α])a a(Ket[α])+(Bra[α])a (SuperDagger[a])(Ket[α])+(Bra[α])(SuperDagger[a]) a(Ket[α])+(Bra[α])(SuperDagger[a]) (SuperDagger[a])(Ket[α]))
=(1/2)((Bra[α])a a(Ket[α])+(Bra[α])((SuperDagger[a])a+1)(Ket[α])+(Bra[α])(SuperDagger[a]) a(Ket[α])+(Bra[α])(SuperDagger[a]) (SuperDagger[a])(Ket[α]))
=(1/2)((α^2)+2(SuperStar[α])α+(α^( *2))+1)
=2(((Re α))^2)+1/2,

(((std x))^2)=(Bra[α])(x^2)(Ket[α])-(Bra[α])x((Ket[α])^2)
=2(((Re α))^2)+1/2-2(((Re α))^2)
=1/2.

(Bra[α])p(Ket[α])=(Sqrt[2])Im α,
(Bra[α])(p^2)(Ket[α])=2(((Im α))^2)+1/2,
(((std p))^2)=(Bra[α])(p^2)(Ket[α])-(Bra[α])p((Ket[α])^2)=1/2.

Fock State Representation

a(Ket[α])=(e^(-(1/2)((|α|)^2)))((∑_(n=1)^∞)((α^n)/(Sqrt[n!]))a(Ket[n])+a(Ket[0]))
=(e^(-(1/2)((|α|)^2)))((∑_(n=1)^∞)((α^n)/(Sqrt[n!]))(Sqrt[n])(Ket[n-1])+0)
=(e^(-(1/2)((|α|)^2)))(∑_(n=1)^∞)((α^n)/(Sqrt[(n-1)!]))(Ket[n-1]),

a(Ket[α])=(e^(-(1/2)((|α|)^2)))(∑_(m=0)^∞)((α^(m+1))/(Sqrt[m!]))(Ket[m])
=α(Ket[α]).

In particular, the ground state (vacuum state) (Ket[0]) is a coherent state with α=0.

Note: (Ket[α=0])=(Ket[n=0]) is the only case when the coherent state index α∈ℂ and the Fock state index n∈ℤ coincide. In general, α=n does not imply (Ket[α]) and (Ket[n]) to be the same state. For example,

(Ket[α=1])=(e^(-1/2))(∑_(n=0)^∞)(1/(Sqrt[n!]))(Ket[n])
=(e^(-1/2))((Ket[n=0])+(Ket[n=1])+(1/(Sqrt[2]))(Ket[n=2])+(1/(Sqrt[6]))(Ket[n=3])+…)
≠(Ket[n=1]).

Based on Eq. (), the probability to observe n bosons in the coherent state (Ket[α]) is given by

DynamicModule[{n0=5},Deploy@EventHandler[Dynamic[DiscretePlot[If[n0>0,n0^n/n!Exp[-n0],KroneckerDelta[n]],{n,0,20},GridLines->{{n0},{}},FrameLabel->{"\!\( n\)","\!\(p\_α(n)\)"},PlotRange->{0,1},PlotStyle->Black,ImageSize->200,PlotLabel->"\!\(\[LeftBracketingBar]α\[RightBracketingBar]\^2="<>ToString[n0]<>"\)"]],{"MouseDragged":>(n0=First@MousePosition["Graphics"])}]]

The mean boson number is determined by the expectation value of the boson number operator (OverHat[n])=(SuperDagger[a])a,

because by Eq. () and Eq. ()

(Bra[α])(OverHat[n])(Ket[α])=(Bra[α])(SuperDagger[a])a(Ket[α])
=(Bra[α])(SuperStar[α])α(Ket[α])
=((|α|)^2)(BraKet[α, α])
=((|α|)^2).

(Subscript[p, α])(n)=((Subscript[⟨n⟩, α]^n)/(n!))(e^(-(Subscript[⟨n⟩, α]))),

Time Evolution

U(t)(Ket[α])=((e^(-((i ω t)/2) ))(e^(-i ω t (OverHat[n]))))((e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((α^n)/(Sqrt[n!]))(Ket[n]))
=(e^(-((i ω t)/2) ))(e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((α^n)/(Sqrt[n!]))(e^(-i ω t (OverHat[n])))(Ket[n])
=(e^(-((i ω t)/2) ))(e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((((α (e^(-i ω t))))^n)/(Sqrt[n!]))(Ket[n])
=(e^(-((i ω t)/2) ))(e^(-(1/2)((|α (e^(-i ω t))|)^2)))(∑_(n=0)^∞)((((α (e^(-i ω t))))^n)/(Sqrt[n!]))(Ket[n]).

(Subscript[⟨x⟩, α(t)])=(Bra[α(t)])x(Ket[α(t)])=(Sqrt[2])Re α(t),
(Subscript[⟨p⟩, α(t)])=(Bra[α(t)])p(Ket[α(t)])=(Sqrt[2])Im α(t),
(Subscript[⟨n⟩, α(t)])=(Bra[α(t)])(OverHat[n])(Ket[α(t)])=((|α(t)|)^2).

(Subscript[⟨x⟩, α(t)])=A cos ω t,
(Subscript[⟨p⟩, α(t)])=-A sin ω t,
(Subscript[⟨n⟩, α(t)])=(1/2)(A^2).

The expectation values of coordinate and momentum do oscillate following the classical behavior of a harmonic oscillator.

The expectation value of the boson number remains constant. As a consequence, the energy of the oscillator is conserved.

(Subscript[⟨H⟩, α(t)])=ℏ ω((Subscript[⟨n⟩, α(t)])+(1/2))=((ℏ ω)/2)((A^2)+1).

U(1) Symmetry

Real Space Representation

α=(((Subscript[⟨x⟩, α])+i (Subscript[⟨p⟩, α]))/(Sqrt[2])).

Be aware, we are switching among three sets of basis! Don’t get confused.

coordinate basis: (OverHat[x])(Ket[x])=x(Ket[x]),
Fock state basis:(OverHat[n])(Ket[n])=n(Ket[n]),
coherent state "basis":a(Ket[α])=α(Ket[α]).

a =((x+i p)/(Sqrt[2]))=((x+(∂_x))/(Sqrt[2])),

a (Subscript[ψ, α])(x)=((x+(∂_x))/(Sqrt[2]))(1/(π^(1/4)))exp(i(Subscript[⟨p⟩, α])x-(1/2)(((x-(Subscript[⟨x⟩, α])))^2))
=((x+i(Subscript[⟨p⟩, α])-(x-(Subscript[⟨x⟩, α])))/(Sqrt[2]))(1/(π^(1/4)))exp(i(Subscript[⟨p⟩, α])x-(1/2)(((x-(Subscript[⟨x⟩, α])))^2))
=(((Subscript[⟨x⟩, α])+i(Subscript[⟨p⟩, α]))/(Sqrt[2]))(Subscript[ψ, α])(x)
=α (Subscript[ψ, α])(x).

DynamicModule[{a=2,xs,t=0,flag=False,pts},
xs=Range[-4,4,0.1];
EventHandler[Dynamic[
If[flag,t=t+0.1];
pts=Table[{x,Exp[-(x-a Cos[t])^2]/Sqrt[π]},{x,xs}];Graphics[{{AbsoluteThickness[1],Arrow[{#,0}&/@{First@xs,Last@xs}],Arrow[{0,#}&/@{0,(First@xs)^2/2}],Text["\!\( x\)",{Last@xs,0},{1,-1}],Text["\!\(V(x)\)",{0,(First@xs)^2/2},{1.2,1}]},{Line[Table[{x,x^2/2},{x,xs}]]},Translate[{Green,{Opacity[0.2],Polygon[Join[pts,{{Last@xs,0},{First@xs,0}}]]},Line[pts],Text["\!\(p\_α(x)\)",{0,1/Sqrt[π]},{0,-1}]},{0,(a^2+1)/2}]},ImageSize->120]],{"MouseClicked":>(flag=Not[flag];If[flag,Block[{x,y},{x,y}=CurrentValue[{"MousePosition","Graphics",{0,0}}];
a=If[y>1/2,Sqrt[2y-1],0]]])}]]

Baker-Campbell-Hausdorff Formula*

B(λ)=(e^(λ A))B (e^(-λ A)).

(∂_λ)B(λ)=(∂_λ)((e^(λ A))B (e^(-λ A)))
=((∂_λ)(e^(λ A)))B (e^(-λ A))+(e^(λ A))B ((∂_λ)(e^(-λ A)))
=A(e^(λ A))B (e^(-λ A))+(e^(λ A))B (e^(-λ A))(-A)
=A B(λ)-B(λ)A
=[A,B(λ)],

B(λ)=B+λ[A,B]+((λ^2)/(2!))[A,[A,B]]+((λ^3)/(3!))[A,[A,[A,B]]]+….

(∂_λ)B(λ)=[A,B]+λ [A,[A,B]]+((λ^2)/(2!))[A,[A,[A,B]]]+…=[A,B(λ)],
B(λ=0)=B+0+0+0+…=B,

(e^A)B (e^(-A))=(e^([A,•]))B=B+[A,B],

(∂_λ)g(λ)=(∂_λ)((e^(λ A))(e^(λ B)))
=((∂_λ)(e^(λ A)))(e^(λ B))+(e^(λ A))((∂_λ)(e^(λ B)))
=A (e^(λ A))(e^(λ B))+(e^(λ A))B (e^(λ B))
=A (e^(λ A))(e^(λ B))+(e^(λ A))B (e^(-λ A))(e^(λ A))(e^(λ B))
=(A+(e^(λ A))B (e^(-λ A)))g(λ),

(∂_λ)g(λ)=(A+B+λ[A,B])g(λ),

g(λ)=(e^(λ(A+B)+((λ^2)/2)[A,B])).

(e^A)(e^B)=(e^(A+B+(1/2)[A,B]))=(e^(A+B))(e^([A,B]/2)),

Displacement Operator*

D(α) is unitary,

which can be verified directly

(SuperDagger[D(α)])=(SuperDagger[((e^(α (SuperDagger[a])-(SuperStar[α])a)))])=(e^((SuperStar[α])a-α (SuperDagger[a])))=((((e^(α (SuperDagger[a])-(SuperStar[α])a))))^(-1))=((D(α))^(-1)).

D(α) implements phase space displacement,

To see this, let A=(SuperStar[α])a-α (SuperDagger[a]). We can see

[A,a]=[(SuperStar[α])a-α (SuperDagger[a]),a]=-α[(SuperDagger[a]),a]=α,

which is a constant (operator). Then by the operator expansion in Eq. ()

(SuperDagger[D(α)])a D(α)=(e^A)a (e^(-A))=a+[A,a]=a+α.

The case for (SuperDagger[a]) follows by Hermitian conjugating both sides.

Composition of displacements

Let A=α (SuperDagger[a])-(SuperStar[α])a, B=β (SuperDagger[a])-(SuperStar[β])a, we can see

[A,B]=[α (SuperDagger[a])-(SuperStar[α])a,β (SuperDagger[a])-(SuperStar[β])a]
=-α (SuperStar[β])[(SuperDagger[a]),a]-(SuperStar[α])β[a,(SuperDagger[a])]
=α (SuperStar[β])-(SuperStar[α])β
=2i Im α (SuperStar[β]),

which is a constant (operator). Then by the BCH formula in Eq. ()

D(α+β)=(e^(A+B))
=(e^A)(e^B)(e^(-[A,B]/2))
=D(α)D(β)(e^(-i Im α (SuperStar[β]))).

To prove this, we want to show that (Ket[α]) obtain in this way is indeed an eigenstate of a with eigenvalue α.

a(Ket[α])=a D(α)(Ket[0])
=D(α)(SuperDagger[D(α)])a D(α)(Ket[0])
=D(α)(a+α)(Ket[0])
=D(α)(a(Ket[0])+α(Ket[0]))
=D(α)α(Ket[0])
=α D(α)(Ket[0])
=α(Ket[α]).

Moreover, (Ket[α]) obtained in this way is automatically normalized (given that (Ket[0]) is normalized).

(BraKet[α, α])=(Bra[0])(SuperDagger[D(α)])D(α)(Ket[0])=(BraKet[0, 0])=1.

Another more explicit way to show Eq. () is to recall Eq. () and Eq. (),

(Ket[n])=(1/(Sqrt[n!]))((((SuperDagger[a])))^n)(Ket[0]),
(Ket[α])=(e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((α^n)/(Sqrt[n!]))(Ket[n]),

then we have

(Ket[α])=(e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((((α (SuperDagger[a])))^n)/(n!))(Ket[0])=(e^(-(1/2)((|α|)^2)))(e^(α (SuperDagger[a])))(Ket[0]).

Since a(Ket[0])=0, (e^(-(SuperStar[α]) a))(Ket[0])=(Ket[0]), so Eq. () can be written as

(Ket[α])=(e^(-(1/2)((|α|)^2)))(e^(α (SuperDagger[a])))(e^(-(SuperStar[α]) a))(Ket[0]).

Let A=α (SuperDagger[a]) and B=-(SuperStar[α]) a, then [A,B]=((|α|)^2), by the BCH formula in Eq. (),

D(α)=(e^(α (SuperDagger[a])-(SuperStar[α])a))=(e^(A+B))=(e^A)(e^B)(e^(-[A,B]/2))=(e^(α (SuperDagger[a])))(e^(-(SuperStar[α]) a))(e^(-(1/2)((|α|)^2))).

Plugging Eq. () into Eq. (), we obtain Eq. ().

Overcomplete Basis*

(BraKet[β, α])=(Bra[0])(SuperDagger[D(β)])D(α)(Ket[0])
=(Bra[0])(e^(-β (SuperDagger[a])))(e^((SuperStar[β]) a))(e^(α (SuperDagger[a])))(e^(-(SuperStar[α]) a))(Ket[0])(e^(-(1/2)(((|α|)^2)+((|β|)^2))))
=(Bra[0])(e^((SuperStar[β]) a))(e^(α (SuperDagger[a])))(Ket[0])(e^(-(1/2)(((|α|)^2)+((|β|)^2)))).

(e^((SuperStar[β]) a))(e^(α (SuperDagger[a])))=(e^(α (SuperDagger[a])))(e^((SuperStar[β]) a))(e^([(SuperStar[β]) a,α (SuperDagger[a])]))=(e^(α (SuperDagger[a])))(e^((SuperStar[β]) a))(e^(α (SuperStar[β]))),

(BraKet[β, α])=(Bra[0])(e^(α (SuperDagger[a])))(e^((SuperStar[β]) a))(Ket[0])(e^(-(1/2)(((|α|)^2)+((|β|)^2))+α (SuperStar[β])))
=(BraKet[0, 0])(e^(-(1/2)(((|α|)^2)+((|β|)^2))+α (SuperStar[β])))
=(e^(-(1/2)(((|α|)^2)+((|β|)^2))+α (SuperStar[β]))).

The non-vanishing scalar product (or transition probability) indicates that the coherent states are not orthogonal.

The transition probability only vanish in the limit of |α-β|≫1.

(Ket[α])=(1/π)(∫_ℂ)dβ(Ket[β])(BraKet[β, α])=(1/π)(∫_ℂ)dβ(Ket[β])(e^(-(1/2)(((|α|)^2)+((|β|)^2))+α (SuperStar[β]))).

(Ket[α])=(e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((α^n)/(Sqrt[n!]))(Ket[n]),

(1/π)(∫_ℂ)dα(Ket[α])(Bra[α])=(1/π)(∑_(m=0)^∞)(∑_(n=0)^∞)(1/(Sqrt[m!n!]))(∫_ℂ)dα (e^(-((|α|)^2)))(α^m)(α^( *n))(Ket[m])(Bra[n]).

(∫_ℂ)dα (e^(-((|α|)^2)))(α^m)(α^( *n))=(∫_0^∞)r dr (e^(-(r^2)))(r^(m+n))(∫_0^(2π))dφ (e^(i (m-n)φ))
=2π (Subscript[δ, m n])(∫_0^∞)rdr (e^(-(r^2)))(r^(m+n))
=π (Subscript[δ, m n])(∫_0^∞)d(r^2) (e^(-(r^2)))(r^(2n))
=π (Subscript[δ, m n])(∫_0^∞)dt (e^(-t))(t^n)
=π (Subscript[δ, m n])Γ(n+1)
=π (Subscript[δ, m n]) n!.

Integrate[Exp[-t]t^n,{t,0,∞},Assumptions->n>0]

(1/π)(∫_ℂ)dα(Ket[α])(Bra[α])=(∑_(n=0)^∞)(Ket[n])(Bra[n])=𝟙.

3D Harmonic Oscillator

Hamiltonian

H=((ℏ ω)/2)((p^2)+(x^2)).

Coordinate x=((Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])),

Momentum p=((Subscript[p, 1]),(Subscript[p, 2]),(Subscript[p, 3])).

[(Subscript[x, a]),(Subscript[p, b])]=i (Subscript[δ, a b]),
[(Subscript[x, a]),(Subscript[x, b])]=[(Subscript[p, a]),(Subscript[p, b])]=0.

boson annihilation  a=((Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3])) and creation  (SuperDagger[a])=((Subscript[a, 1]^†),(Subscript[a, 2]^†),(Subscript[a, 3]^†)) operators

a=(1/(Sqrt[2]))(x+i p), (SuperDagger[a])=(1/(Sqrt[2]))(x-i p).

They satisfy the commutation relation

[(Subscript[a, a]),(Subscript[a, b]^†)]=(Subscript[δ, a b]),
[(Subscript[a, a]),(Subscript[a, b])]=[(Subscript[a, a]^†),(Subscript[a, b]^†)]=0.

Block[{x,p,a,ad},
x={(Subscript[𝓍, 1]),(Subscript[𝓍, 2]),(Subscript[𝓍, 3])};
p={(Subscript[𝓅, 1]),(Subscript[𝓅, 2]),(Subscript[𝓅, 3])};
a=(x+I p)/Sqrt[2];
ad=(x-I p)/Sqrt[2];
MatrixForm/@OpReduce[𝓍,𝓅]@{Com[a,ad],Com[a,a],Com[ad,ad]}]

Verification.

Block[{x,p,a,ad,H},
x={(Subscript[𝓍, 1]),(Subscript[𝓍, 2]),(Subscript[𝓍, 3])};
p={(Subscript[𝓅, 1]),(Subscript[𝓅, 2]),(Subscript[𝓅, 3])};
a=(x+I p)/Sqrt[2];
ad=(x-I p)/Sqrt[2];
H=(ℏ ω/2)(Tr[x**x]+Tr[p**p]);
OpReduce[𝓍,𝓅][H-ℏ ω(Tr[ad**a]+3ℐ/2)]]

Verification.

Energy Level and Degeneracy

E=ℏ ω(N+(3/2)),
N=(Subscript[n, 1])+(Subscript[n, 2])+(Subscript[n, 3]).

Eigen state: (Ket[(Subscript[n, 1])(Subscript[n, 2])(Subscript[n, 3])])

Degeneracy

(∑_((Subscript[n, 1]),(Subscript[n, 2]),(Subscript[n, 3])))(Subscript[δ, N=(Subscript[n, 1])+(Subscript[n, 2])+(Subscript[n, 3])])=(1/2)(N+1)(N+2).

Assuming[Ν∈Integers&&Ν≥0,Factor@Sum[KroneckerDelta[n1+n2+n3,Ν],{n1,0,∞},{n2,0,∞},{n3,0,∞}]]

Style[TableForm[Thread@Table[{N+3/2,(N+1)(N+2)/2},{N,0,5}],TableHeadings->{{"\!\(E \)","deg."},Range[0,5]}],SingleLetterItalics->True,FontFamily->"CMU"]

Angular Momentum

L=x×p.

Block[{x,p,a,ad,L},
x={(Subscript[𝓍, 1]),(Subscript[𝓍, 2]),(Subscript[𝓍, 3])};
p={(Subscript[𝓅, 1]),(Subscript[𝓅, 2]),(Subscript[𝓅, 3])};
a=(x+I p)/Sqrt[2];
ad=(x-I p)/Sqrt[2];
L=x⋀p;
OpReduce[𝓍,𝓅][L+I ad⋀a]]

Verification.

The angular momentum operator generates the SO(3) rotation among different polarization modes.

The (quadratic) Casimir operator (the square of angular momentum) (L^2).

(L^2)=(Subscript[L, 1]^2)+(Subscript[L, 2]^2)+(Subscript[L, 3]^2)=(∑_(a≠b))((Subscript[(OverHat[n]), a])((Subscript[(OverHat[n]), b])+1)-(Subscript[a, a]^†)(Subscript[a, a]^†)(Subscript[a, b])(Subscript[a, b])).

To verify Eq. (), (the following summations are implicit)

(L^2)=(Subscript[L, a])(Subscript[L, a])=-(Subscript[ϵ, a b c])(Subscript[ϵ, a d e])(Subscript[a, b]^†)(Subscript[a, c])(Subscript[a, d]^†)(Subscript[a, e])
=-((Subscript[δ, b d])(Subscript[δ, c e])-(Subscript[δ, b e])(Subscript[δ, c d]))(Subscript[a, b]^†)(Subscript[a, c])(Subscript[a, d]^†)(Subscript[a, e])
=-(Subscript[a, b]^†)(Subscript[a, c])(Subscript[a, b]^†)(Subscript[a, c])+(Subscript[a, b]^†)(Subscript[a, c])(Subscript[a, c]^†)(Subscript[a, b]).

In the summation of b,c, the terms with b=c vanish, so we can restrict the summation to b≠c, (the following summation is explicit)

(L^2)=(∑_(b≠c))((Subscript[a, b]^†)(Subscript[a, c])(Subscript[a, c]^†)(Subscript[a, b])-(Subscript[a, b]^†)(Subscript[a, c])(Subscript[a, b]^†)(Subscript[a, c]))
=(∑_(b≠c))((Subscript[a, b]^†)(Subscript[a, b])((Subscript[a, c]^†)(Subscript[a, c])+1)-(Subscript[a, b]^†)(Subscript[a, b]^†)(Subscript[a, c])(Subscript[a, c])).

Fock State Basis

(Subscript[L, 1])=-i(Sqrt[((Subscript[n, 2])+1)(Subscript[n, 3])])(Ket[(Subscript[n, 1]),(Subscript[n, 2])+1,(Subscript[n, 3])-1])(Bra[(Subscript[n, 1])(Subscript[n, 2])(Subscript[n, 3])])+h.c.,
(Subscript[L, 2])=-i(Sqrt[((Subscript[n, 3])+1)(Subscript[n, 1])])(Ket[(Subscript[n, 1])-1,(Subscript[n, 2]),(Subscript[n, 3])+1])(Bra[(Subscript[n, 1])(Subscript[n, 2])(Subscript[n, 3])])+h.c.,
(Subscript[L, 3])=-i(Sqrt[((Subscript[n, 1])+1)(Subscript[n, 2])])(Ket[(Subscript[n, 1])+1,(Subscript[n, 2])-1,(Subscript[n, 3])])(Bra[(Subscript[n, 1])(Subscript[n, 2])(Subscript[n, 3])])+h.c. .

{(Ket[100]),(Ket[010]),(Ket[001])}.

(Subscript[L, 1])≏(({{0, 0, 0}, {0, 0, -i}, {0, i, 0}})),(Subscript[L, 2])≏(({{0, 0, i}, {0, 0, 0}, {-i, 0, 0}})),(Subscript[L, 3])≏(({{0, -i, 0}, {i, 0, 0}, {0, 0, 0}})).

(L^2)≏(({{2, 0, 0}, {0, 2, 0}, {0, 0, 2}})).

Block[{Ν=1,basis,ind,Ls,LL,vs},
basis=SortBy[Flatten[Table[{n1,n2,Ν-n1-n2},{n1,0,Ν},{n2,0,Ν-n1}],1],{Total[#^2],Reverse@#}&];
ind=AssociationThread[basis->Range@Length@basis];
Ls=SparseArray[Flatten@Block[{rep},
rep[{i_,j_,k_}->s_]:=Table[Block[{ms=ns,w},
ms[[{j,k}]]=ns[[{j,k}]]+{1,-1};
w=-I s Sqrt[(ns[[j]]+1)ns[[k]]];
If[w==0,Nothing,{i,ind@ms,ind@ns}->w]],{ns,basis}];
rep/@Most@ArrayRules@LeviCivitaTensor[3]],{3,#,#}&@Length@basis];
LL=TensorContract[Ls⊗Ls,{{1,4},{3,5}}];
MatrixForm/@Append[Ls,LL]
]

Matrix representations.

Angular Momentum Basis

(Ket[l=1,m=±1])=(1/(Sqrt[2]))((Ket[(RowDefault[1, 0, 0])])±i (Ket[(RowDefault[0, 1, 0])])),
(Ket[l=1,m=0])=(Ket[(RowDefault[0, 0, 1])]).

Classical picture:

(Ket[l=1,m=±1]) - uniform circular rotation in the x y-plane, ± sign corresponds to counterclockwise or clockwise.

(Ket[l=1,m=0]) - linear oscillation along the z-direction.

Quantum wave function:

Fock state basis

(Subscript[ψ, (Subscript[n, 1])(Subscript[n, 2])(Subscript[n, 3])])((Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3]))=(Subscript[ψ, (Subscript[n, 1])])((Subscript[x, 1]))(Subscript[ψ, (Subscript[n, 2])])((Subscript[x, 2]))(Subscript[ψ, (Subscript[n, 3])])((Subscript[x, 3])),

where (Subscript[ψ, n])(x) is the wave function of a 1D harmonic oscillator, given by Eq. ().

Block[{Ν=1,basis,wave},
basis=SortBy[Flatten[Table[{n1,n2,Ν-n1-n2},{n1,0,Ν},{n2,0,Ν-n1}],1],{Total[#^2],Reverse@#}&];
wave[ns_]:=Block[{x,y,z,Ψ,Ψ0,R=2.5Sqrt[Ν]},
Ψ0=0.5/Sqrt[2^# #!]&@Total@ns;Ψ=Exp[-(x^2+y^2+z^2)/2]Times@@MapThread[HermiteH[#1,#2]/Sqrt[2^#1 #1!]&,{ns,{x,y,z}}];
Show[ContourPlot3D[Abs[Ψ]==Ψ0,{x,-R,R},{y,-R,R},{z,-R,R},ColorFunction->Function[{x,y,z,f},ComplexColor[Ψ/Ψ0]],ColorFunctionScaling->False,Boxed->False,Axes->False,SphericalRegion->True,Mesh->None,PlotPoints->24,MaxRecursion->0,Lighting->"Neutral",ImageSize->80],Graphics3D[{Line[{-#,#}]&/@{{R,0,0},{0,R,0},{0,0,R}}}]]];
Style[Grid[{Ket/@Row/@basis,wave/@basis}],FontFamily->"CMU"]
]

Angular momentum basis

Block[{Ν=1,basis,wave,ind,Ls,LL,vs},
basis=SortBy[Flatten[Table[{n1,n2,Ν-n1-n2},{n1,0,Ν},{n2,0,Ν-n1}],1],Total[#^2]&];
wave[v_,l_]:=Block[{Ψ0=0.5/Sqrt[2^Ν Ν!],R=2Sqrt[Ν+1],x,y,z,Ψ},
Ψ=Exp[-(x^2+y^2+z^2)/2]v.Table[Times@@MapThread[HermiteH[#1,#2]/Sqrt[2^#1 #1!]&,{ns,{x,y,z}}],{ns,basis}];
Show[ContourPlot3D[Abs[Ψ]==Ψ0,{x,-R,R},{y,-R,R},{z,-R,R},RegionFunction->If[l==0,Function[{x,y,z},x<0||y>0],True&],ColorFunction->Function[{x,y,z,f},ComplexColor[Ψ/Ψ0]],ColorFunctionScaling->False,Boxed->False,Axes->False,SphericalRegion->True,Mesh->None,PlotPoints->24,MaxRecursion->0,Lighting->"Neutral",ImageSize->60],Graphics3D[{Line[{-#,#}]&/@{{R,0,0},{0,R,0},{0,0,R}}}]]];
ind=AssociationThread[basis->Range@Length@basis];
Ls=SparseArray[Flatten@Block[{rep},
rep[{i_,j_,k_}->s_]:=Table[Block[{ms=ns,w},
ms[[{j,k}]]=ns[[{j,k}]]+{1,-1};
w=-I s Sqrt[(ns[[j]]+1)ns[[k]]];
If[w==0,Nothing,{i,ind@ms,ind@ns}->w]],{ns,basis}];
rep/@Most@ArrayRules@LeviCivitaTensor[3]],{3,#,#}&@Length@basis];
LL=TensorContract[Ls⊗Ls,{{1,4},{3,5}}];
vs=Normalize/@Eigenvectors[LL+Ls[[3]]/(10(Ν+1))];
Style[Labeled[TableForm[Block[{l},{l=Simplify[Sqrt[Conjugate[#].LL.#+1/4]-1/2],Simplify[Conjugate[#].Ls[[3]].#],Simplify[#.(Ket/@Row/@basis)],wave[#,l]}]&/@vs,TableHeadings->{None,{"\!\(l \)","\!\(m \)","state"}}],"\!\(N="<>ToString[Ν]<>"\) sector",Top],SingleLetterItalics->True,FontFamily->"CMU"]
]

More Examples

Block[{Ν=0,basis,wave,ind,Ls,LL,vs},
basis=SortBy[Flatten[Table[{n1,n2,Ν-n1-n2},{n1,0,Ν},{n2,0,Ν-n1}],1],Total[#^2]&];
wave[v_,l_]:=Block[{Ψ0=0.5/Sqrt[2^Ν Ν!],R=2Sqrt[Ν+1],x,y,z,Ψ},
Ψ=Exp[-(x^2+y^2+z^2)/2]v.Table[Times@@MapThread[HermiteH[#1,#2]/Sqrt[2^#1 #1!]&,{ns,{x,y,z}}],{ns,basis}];
Show[ContourPlot3D[Abs[Ψ]==Ψ0,{x,-R,R},{y,-R,R},{z,-R,R},RegionFunction->If[l==0,Function[{x,y,z},x<0||y>0],True&],ColorFunction->Function[{x,y,z,f},ComplexColor[Ψ/Ψ0]],ColorFunctionScaling->False,Boxed->False,Axes->False,SphericalRegion->True,Mesh->None,PlotPoints->24,MaxRecursion->0,Lighting->"Neutral",ImageSize->60],Graphics3D[{Line[{-#,#}]&/@{{R,0,0},{0,R,0},{0,0,R}}}]]];
ind=AssociationThread[basis->Range@Length@basis];
Ls=SparseArray[Flatten@Block[{rep},
rep[{i_,j_,k_}->s_]:=Table[Block[{ms=ns,w},
ms[[{j,k}]]=ns[[{j,k}]]+{1,-1};
w=-I s Sqrt[(ns[[j]]+1)ns[[k]]];
If[w==0,Nothing,{i,ind@ms,ind@ns}->w]],{ns,basis}];
rep/@Most@ArrayRules@LeviCivitaTensor[3]],{3,#,#}&@Length@basis];
LL=TensorContract[Ls⊗Ls,{{1,4},{3,5}}];
vs=Normalize/@Eigenvectors[LL+Ls[[3]]/(10(Ν+1))];
Style[Labeled[TableForm[Block[{l},{l=Simplify[Sqrt[Conjugate[#].LL.#+1/4]-1/2],Simplify[Conjugate[#].Ls[[3]].#],Simplify[#.(Ket/@Row/@basis)],wave[#,l]}]&/@vs,TableHeadings->{None,{"\!\(l \)","\!\(m \)","state"}}],"\!\(N="<>ToString[Ν]<>"\) sector",Top],SingleLetterItalics->True,FontFamily->"CMU"]
]

Block[{Ν=1,basis,wave,ind,Ls,LL,vs},
basis=SortBy[Flatten[Table[{n1,n2,Ν-n1-n2},{n1,0,Ν},{n2,0,Ν-n1}],1],Total[#^2]&];
wave[v_,l_]:=Block[{Ψ0=0.5/Sqrt[2^Ν Ν!],R=2Sqrt[Ν+1],x,y,z,Ψ},
Ψ=Exp[-(x^2+y^2+z^2)/2]v.Table[Times@@MapThread[HermiteH[#1,#2]/Sqrt[2^#1 #1!]&,{ns,{x,y,z}}],{ns,basis}];
Show[ContourPlot3D[Abs[Ψ]==Ψ0,{x,-R,R},{y,-R,R},{z,-R,R},RegionFunction->If[l==0,Function[{x,y,z},x<0||y>0],True&],ColorFunction->Function[{x,y,z,f},ComplexColor[Ψ/Ψ0]],ColorFunctionScaling->False,Boxed->False,Axes->False,SphericalRegion->True,Mesh->None,PlotPoints->24,MaxRecursion->0,Lighting->"Neutral",ImageSize->60],Graphics3D[{Line[{-#,#}]&/@{{R,0,0},{0,R,0},{0,0,R}}}]]];
ind=AssociationThread[basis->Range@Length@basis];
Ls=SparseArray[Flatten@Block[{rep},
rep[{i_,j_,k_}->s_]:=Table[Block[{ms=ns,w},
ms[[{j,k}]]=ns[[{j,k}]]+{1,-1};
w=-I s Sqrt[(ns[[j]]+1)ns[[k]]];
If[w==0,Nothing,{i,ind@ms,ind@ns}->w]],{ns,basis}];
rep/@Most@ArrayRules@LeviCivitaTensor[3]],{3,#,#}&@Length@basis];
LL=TensorContract[Ls⊗Ls,{{1,4},{3,5}}];
vs=Normalize/@Eigenvectors[LL+Ls[[3]]/(10(Ν+1))];
Style[Labeled[TableForm[Block[{l},{l=Simplify[Sqrt[Conjugate[#].LL.#+1/4]-1/2],Simplify[Conjugate[#].Ls[[3]].#],Simplify[#.(Ket/@Row/@basis)],wave[#,l]}]&/@vs,TableHeadings->{None,{"\!\(l \)","\!\(m \)","state"}}],"\!\(N="<>ToString[Ν]<>"\) sector",Top],SingleLetterItalics->True,FontFamily->"CMU"]
]

Block[{Ν=2,basis,wave,ind,Ls,LL,vs},
basis=SortBy[Flatten[Table[{n1,n2,Ν-n1-n2},{n1,0,Ν},{n2,0,Ν-n1}],1],Total[#^2]&];
wave[v_,l_]:=Block[{Ψ0=0.5/Sqrt[2^Ν Ν!],R=2Sqrt[Ν+1],x,y,z,Ψ},
Ψ=Exp[-(x^2+y^2+z^2)/2]v.Table[Times@@MapThread[HermiteH[#1,#2]/Sqrt[2^#1 #1!]&,{ns,{x,y,z}}],{ns,basis}];
Show[ContourPlot3D[Abs[Ψ]==Ψ0,{x,-R,R},{y,-R,R},{z,-R,R},RegionFunction->If[l==0,Function[{x,y,z},x<0||y>0],True&],ColorFunction->Function[{x,y,z,f},ComplexColor[Ψ/Ψ0]],ColorFunctionScaling->False,Boxed->False,Axes->False,SphericalRegion->True,Mesh->None,PlotPoints->24,MaxRecursion->0,Lighting->"Neutral",ImageSize->60],Graphics3D[{Line[{-#,#}]&/@{{R,0,0},{0,R,0},{0,0,R}}}]]];
ind=AssociationThread[basis->Range@Length@basis];
Ls=SparseArray[Flatten@Block[{rep},
rep[{i_,j_,k_}->s_]:=Table[Block[{ms=ns,w},
ms[[{j,k}]]=ns[[{j,k}]]+{1,-1};
w=-I s Sqrt[(ns[[j]]+1)ns[[k]]];
If[w==0,Nothing,{i,ind@ms,ind@ns}->w]],{ns,basis}];
rep/@Most@ArrayRules@LeviCivitaTensor[3]],{3,#,#}&@Length@basis];
LL=TensorContract[Ls⊗Ls,{{1,4},{3,5}}];
vs=Normalize/@Eigenvectors[LL+Ls[[3]]/(10(Ν+1))];
Style[Labeled[TableForm[Block[{l},{l=Simplify[Sqrt[Conjugate[#].LL.#+1/4]-1/2],Simplify[Conjugate[#].Ls[[3]].#],Simplify[#.(Ket/@Row/@basis)],wave[#,l]}]&/@vs,TableHeadings->{None,{"\!\(l \)","\!\(m \)","state"}}],"\!\(N="<>ToString[Ν]<>"\) sector",Top],SingleLetterItalics->True,FontFamily->"CMU"]
]

Block[{Ν=3,basis,wave,ind,Ls,LL,vs},
basis=SortBy[Flatten[Table[{n1,n2,Ν-n1-n2},{n1,0,Ν},{n2,0,Ν-n1}],1],Total[#^2]&];
wave[v_,l_]:=Block[{Ψ0=0.5/Sqrt[2^Ν Ν!],R=2Sqrt[Ν+1],x,y,z,Ψ},
Ψ=Exp[-(x^2+y^2+z^2)/2]v.Table[Times@@MapThread[HermiteH[#1,#2]/Sqrt[2^#1 #1!]&,{ns,{x,y,z}}],{ns,basis}];
Show[ContourPlot3D[Abs[Ψ]==Ψ0,{x,-R,R},{y,-R,R},{z,-R,R},RegionFunction->If[l<Ν,Function[{x,y,z},x<0||y>0],True&],ColorFunction->Function[{x,y,z,f},ComplexColor[Ψ/Ψ0]],ColorFunctionScaling->False,Boxed->False,Axes->False,SphericalRegion->True,Mesh->None,PlotPoints->32,MaxRecursion->0,Lighting->"Neutral",ImageSize->60],Graphics3D[{Line[{-#,#}]&/@{{R,0,0},{0,R,0},{0,0,R}}}]]];
ind=AssociationThread[basis->Range@Length@basis];
Ls=SparseArray[Flatten@Block[{rep},
rep[{i_,j_,k_}->s_]:=Table[Block[{ms=ns,w},
ms[[{j,k}]]=ns[[{j,k}]]+{1,-1};
w=-I s Sqrt[(ns[[j]]+1)ns[[k]]];
If[w==0,Nothing,{i,ind@ms,ind@ns}->w]],{ns,basis}];
rep/@Most@ArrayRules@LeviCivitaTensor[3]],{3,#,#}&@Length@basis];
LL=TensorContract[Ls⊗Ls,{{1,4},{3,5}}];
vs=Normalize/@Eigenvectors[LL+Ls[[3]]/(10(Ν+1))];
Style[Labeled[TableForm[Block[{l},{l=Simplify[Sqrt[Conjugate[#].LL.#+1/4]-1/2],Simplify[Conjugate[#].Ls[[3]].#],Simplify[#.(Ket/@Row/@basis)],wave[#,l]}]&/@vs,TableHeadings->{None,{"\!\(l \)","\!\(m \)","state"}}],"\!\(N="<>ToString[Ν]<>"\) sector",Top],SingleLetterItalics->True,FontFamily->"CMU"]
]

U(3) Symmetry*

a->U a.

The Abelian U(1) subgroup does not lead to degeneracy.

The non-Abelian SU(3) subgroup => irreducible representations (Subscript[D, p q]) are labeled by two quantum numbers p,q=0,1,2,…

Block[{box},
box[col_,{x_,y_}]:=Translate[{Lighter[col,0.8],Rectangle[{0,0},{1,1}]},{x,y}];
Graphics[{{EdgeForm[Directive[Black,AbsoluteThickness[1]]],
Table[If[y==2||x<4,box[If[x<4,Red,Blue],{x,y}]],{x,7},{y,2}]},
{Arrowheads[{-0.1,0.1}],{Red,Arrow[{#,3.7}&/@{1,4}],Text[Style["\!\( q\)",Background->White],{2.5,3},{0,-1}]},
{Blue,Arrow[{#,3.7}&/@{4,8}],Text[Style["\!\( p\)",Background->White],{6,3},{0,-1}]}}},ImageSize->100]
]

and their dimensions are

dim (Subscript[D, p q])=(1/2)(p+1)(q+1)(p+q+2).

Angular Momentum

Operator Algebra

Definition

[(Subscript[x, a]),(Subscript[p, b])]=i (Subscript[δ, a b]),

[(Subscript[L, a]),(Subscript[L, b])]=i (Subscript[ϵ, a b c])(Subscript[L, c]).

Block[{ϵ,x,p,L},
ϵ=LeviCivitaTensor[3];
x={(Subscript[𝓍, 1]),(Subscript[𝓍, 2]),(Subscript[𝓍, 3])};
p={(Subscript[𝓅, 1]),(Subscript[𝓅, 2]),(Subscript[𝓅, 3])};
L=x⋀p;
MatrixForm@OpReduce[𝓍,𝓅][Com[L,L]-I ϵ.L]]

Verification.

Equivalently, in vector form, J×J=i J.

This general definition applies to orbital and spin angular momenta. The spin angular momentum goes beyond the classical definition of x×p.

The Hermitian operators J generate a unitary group - the SU(2) group.

Casimir Operator

(J^2) is Hermitian.

By Eq. (), one can verify that (for a=1,2,3)

[(J^2),(Subscript[J, a])]=0.

Common Eigenstates

(J^2)(Ket[j,m])=(Subscript[λ, j])(Ket[j,m]),
(Subscript[J, 3])(Ket[j,m])=m(Ket[j,m]),

(Subscript[λ, j]) is the the eigenvalue of (J^2) on (Ket[j,m]),

m is the the eigenvalue of (Subscript[J, 3]) on (Ket[j,m]).

Raising and Lowering Operators

In analogy to (e^(±i θ))=cos θ±i sin θ.

(Subscript[J, ±]) are not Hermitian. Under Hermitian conjugate: (Subscript[J, ±]^†)=(Subscript[J, ∓]).

(Subscript[J, 3])(Subscript[J, ±])=(Subscript[J, 3])((Subscript[J, 1])±i (Subscript[J, 2]))
=((Subscript[J, 1])(Subscript[J, 3])+i (Subscript[J, 2]))±i ((Subscript[J, 2])(Subscript[J, 3])-i (Subscript[J, 1]))
=((Subscript[J, 1])±i (Subscript[J, 2]))((Subscript[J, 3])±1)
=(Subscript[J, ±])((Subscript[J, 3])±1).

(Subscript[J, ±])(Ket[j,m])=(Subscript[c, m]^±)(Ket[j,m±1]).

Therefore (Subscript[J, ±]) is called the raising/lowering operator.

Also note that, under the action of (Subscript[J, ±]), the quantum number m can only change by 1 (not change continuously) => angular momentum quantization.

Bounds on Quantum Numbers

(SubPlus[J])(SubMinus[J])=((Subscript[J, 1])+i (Subscript[J, 2]))((Subscript[J, 1])-i (Subscript[J, 2]))
=(Subscript[J, 1]^2)+i (Subscript[J, 2])(Subscript[J, 1])-i (Subscript[J, 1])(Subscript[J, 2])+(Subscript[J, 2]^2)
=((Subscript[J, 1]^2)+(Subscript[J, 2]^2))-i((Subscript[J, 1])(Subscript[J, 2])-(Subscript[J, 2])(Subscript[J, 1]))
=((J^2)-(Subscript[J, 3]^2))+(Subscript[J, 3]).

(Bra[j,m])(SubPlus[J])(SubMinus[J])(Ket[j,m])=(Subscript[λ, j])-(m^2)+m,
(Bra[j,m])(SubMinus[J])(SubPlus[J])(Ket[j,m])=(Subscript[λ, j])-(m^2)-m.

(Bra[j,m])(SubPlus[J])(SubMinus[J])(Ket[j,m])=(Bra[j,m])(Subscript[J, -]^†)(SubMinus[J])(Ket[j,m])=((|(Subscript[c, m]^(-))|)^2)≥0,
(Bra[j,m])(SubMinus[J])(SubPlus[J])(Ket[j,m])=(Bra[j,m])(Subscript[J, +]^†)(SubPlus[J])(Ket[j,m])=((|(Subscript[c, m]^(+))|)^2)≥0.

(Subscript[λ, j])-m(m±1)≥0 =>
-(1/2)((Sqrt[1+4(Subscript[λ, j])])-1)≤m≤(1/2)((Sqrt[1+4(Subscript[λ, j])])-1).

Reduce[λ-m(m+1)≥0&&λ-m(m-1)≥0&&λ≥0,m]

Reduce the inequality.

Block[{j=1/2},Plot[{Max[m(m+1),m(m-1)],m(m+1),m(m-1)},{m,-1.2,1.2},
PlotStyle->{None,Red,Blue},Filling->{1->{10,LightGray}},
Epilog->{GraphicsComplex[{{-j,j(j+1)},{j,j(j+1)},{-j,0},{j,0}},{Line[{1,2}],AbsoluteThickness[1],Dotted,Line[{{1,3},{2,4}}]}],Text["\!\(m\_max\)",{j,j(j+1)},{-0.5,-1}],
Text["\!\(m\_min\)",{-j,j(j+1)},{0.5,-1}]},Frame->None,AxesLabel->{"\!\( m\)","\!\( λ\_j\)"}]]

Representation Theory

Highest and Lowest Weight State

Highest weight state: (Ket[j,(Subscript[m, max])]), corresponds to m=(Subscript[m, max]). It can not be further raised by (SubPlus[J]) => (SubPlus[J])(Ket[j,(Subscript[m, max])])=0.

Lowest weight state: (Ket[j,(Subscript[m, min])]), corresponds to m=(Subscript[m, min]). It can not be further lowered by (SubMinus[J]) => (SubMinus[J])(Ket[j,(Subscript[m, min])])=0.

0=(Bra[j,(Subscript[m, min])])(SubPlus[J])(SubMinus[J])(Ket[j,(Subscript[m, min])])=(Subscript[λ, j])-(Subscript[m, min]^2)+(Subscript[m, min]),
0=(Bra[j,(Subscript[m, max])])(SubMinus[J])(SubPlus[J])(Ket[j,(Subscript[m, max])])=(Subscript[λ, j])-(Subscript[m, max]^2)-(Subscript[m, max]).

((Subscript[m, max])+(Subscript[m, min]))((Subscript[m, max])-(Subscript[m, min])+1)=0,

(Subscript[m, max])=j, (Subscript[m, min])=-j => -j≤m≤j.

(Subscript[λ, j])=j(j+1).

Angular Momentum Quantization

j can only be integer or half-integer: j=0,1/2,1,3/2,2,…

m takes values in m=-j,-j+1,…,j-1,j.

For orbital angular momentum j takes integer values. For spin angular momentum j can also be half-integer.

Plot[{Max[m(m+1),m(m-1)],m(m+1),m(m-1)},{m,-2,2},
PlotStyle->{None,Red,Blue},Filling->{1->{10,LightGray}},
Epilog->Table[{GraphicsComplex[{{-j,j(j+1)},{j,j(j+1)},{-j,0},{j,0}},{AbsoluteThickness[1],Dashed,Line[{1,2}],Dotted,Line[{{1,3},{2,4}}]}],Point[{#,j(j+1)}&/@Range[-j,j]]},{j,0,2,1/2}],Frame->None,AxesLabel->{"\!\( m\)","\!\( λ\_j=j(j+1)\)"}]

Representation

(Subscript[c, m]^±)=(Sqrt[j(j+1)-m(m±1)]).

either raised from the lowest weight state,

(Ket[j,m])=((((((j-m)!)/((2j)!(j+m)!))))^(1/2))(Subscript[J, +]^(j+m))(Ket[j,-j]),

or lowered from the highest weight state,

(Ket[j,m])=((((((j+m)!)/((2j)!(j-m)!))))^(1/2))(Subscript[J, -]^(j-m))(Ket[j,j]).

((SubPlus[J])/(Sqrt[2j]))(Ket[j,-j+n])=(Sqrt[n+1])(Ket[j,-j+n+1])+𝒪((j^(-1/2))),
((SubMinus[J])/(Sqrt[2j]))(Ket[j,-j+n])=(Sqrt[n])(Ket[j,-j+n-1])+𝒪((j^(-1/2))).

(Ket[j,-j+n])→(Ket[n]),
(((2j))^(-1/2))(SubMinus[J])->a, (((2j))^(-1/2))(SubPlus[J])→(SuperDagger[a]),

Spin-1/2

MatrixForm/@PauliMatrix/@Range[3]

Pauli matrices.

Block[{j=1/2,JJ,J3,Jp,Jm,J1,J2},
{JJ,J3,Jp}=SparseArray[#,{#,#}&@2j+1]&/@{Table[j+1-{m,m}->j(j+1),{m,-j,j}],Table[j+1-{m,m}->m,{m,-j,j}],
Table[j+1-{m+1,m}->Sqrt[j(j+1)-m(m+1)],{m,-j,j-1}]};
Jm=Transpose@Jp;
{J1,J2}={Jp+Jm,-I Jp+I Jm}/2;
MatrixForm/@{J1,J2,J3}
]

Spin-1/2 operators.

Basis: {(Ket[↑])=(Ket[1/2,+1/2]),(Ket[↓])=(Ket[1/2,-1/2])}.

Matrix representations:

(Subscript[S, 1])≏(1/2)(({{0, 1}, {1, 0}})),(Subscript[S, 2])≏(1/2)(({{0, -i}, {i, 0}})),(Subscript[S, 3])≏(1/2)(({{1, 0}, {0, -1}})).

Eigenstates and eigenvalues

(Subscript[S, 1]):(1/(Sqrt[2]))(({{1}, {1}}))->+(1/2),(1/(Sqrt[2]))(({{1}, {-1}}))->-(1/2),
(Subscript[S, 2]):(1/(Sqrt[2]))(({{1}, {i}}))->+(1/2),(1/(Sqrt[2]))(({{1}, {-i}}))->-(1/2),
(Subscript[S, 1]):(({{1}, {0}}))->+(1/2),(({{0}, {1}}))->-(1/2).

Spin-1

Basis: {(Ket[1,+1]),(Ket[1,0]),(Ket[1,-1])}.

Matrix representations (the same matrices also represent spin-1)

(Subscript[L, 1])≏(1/(Sqrt[2]))(({{0, 1, 0}, {1, 0, 1}, {0, 1, 0}})),(Subscript[L, 2])≏(1/((Sqrt[2])i))(({{0, 1, 0}, {-1, 0, 1}, {0, -1, 0}})),(Subscript[L, 3])≏(({{1, 0, 0}, {0, 0, 0}, {0, 0, -1}})).

Block[{j=1,JJ,J3,Jp,Jm,J1,J2},
{JJ,J3,Jp}=SparseArray[#,{#,#}&@2j+1]&/@{Table[j+1-{m,m}->j(j+1),{m,-j,j}],Table[j+1-{m,m}->m,{m,-j,j}],
Table[j+1-{m+1,m}->Sqrt[j(j+1)-m(m+1)],{m,-j,j-1}]};
Jm=Transpose@Jp;
{J1,J2}={Jp+Jm,-I Jp+I Jm}/2;
MatrixForm/@{J1,J2,J3}
]

Spin-1 operators.

Basis transformation

(({{(Ket[1,+1])}, {(Ket[1,0])}, {(Ket[1,-1])}}))=(({{(i/(Sqrt[2])), (1/(Sqrt[2])), 0}, {0, 0, -i}, {((-i)/(Sqrt[2])), (1/(Sqrt[2])), 0}}))(({{(Ket[x])}, {(Ket[y])}, {(Ket[z])}})).

Denote the transformation matrix by U, the operators transform as (Subscript[L, a])->(SuperDagger[U])(Subscript[L, a])U,

(Subscript[L, 1])≏(({{0, 0, 0}, {0, 0, -i}, {0, i, 0}})),(Subscript[L, 2])≏(({{0, 0, i}, {0, 0, 0}, {-i, 0, 0}})),(Subscript[L, 3])≏(({{0, -i, 0}, {i, 0, 0}, {0, 0, 0}})).

Block[{j=1,JJ,J3,Jp,Jm,J1,J2,U},
{JJ,J3,Jp}=SparseArray[#,{#,#}&@2j+1]&/@{Table[j+1-{m,m}->j(j+1),{m,-j,j}],Table[j+1-{m,m}->m,{m,-j,j}],
Table[j+1-{m+1,m}->Sqrt[j(j+1)-m(m+1)],{m,-j,j-1}]};
Jm=Transpose@Jp;
{J1,J2}={Jp+Jm,-I Jp+I Jm}/2;
U={{I,1,0}/Sqrt[2],{0,0,-I},{-I,1,0}/Sqrt[2]};
MatrixForm[ConjugateTranspose[U].#.U]&/@{J1,J2,J3}
]

Basis transformation.

Rotation Operators

direction of θ: the direction of the rotational axis (following the right-handed rule),

magnitude of θ: the amount of angle to rotate.

Examples: rotate along (0,0,1) axis by θ => θ=(0,0,θ).

Spin-1/2 representation, in the {(Ket[↑]),(Ket[↓])} basis,

(Subscript[R, 1/2])(θ)≏(({{(e^(i θ/2)), 0}, {0, (e^(-i θ/2))}})).

Block[{j=1/2,JJ,J3,Jp,Jm,J1,J2,U},
{JJ,J3,Jp}=SparseArray[#,{#,#}&@2j+1]&/@{Table[j+1-{m,m}->j(j+1),{m,-j,j}],Table[j+1-{m,m}->m,{m,-j,j}],
Table[j+1-{m+1,m}->Sqrt[j(j+1)-m(m+1)],{m,-j,j-1}]};
Jm=Transpose@Jp;
{J1,J2}={Jp+Jm,-I Jp+I Jm}/2;
MatrixForm[MatrixExp[I θ #]]&/@{J1,J2,J3}
]

Matrix Exp.

Spin-1 representation, in the {(Ket[x]),(Ket[y]),(Ket[z])} basis,

(Subscript[R, 1])(θ)≏(({{cos θ, sin θ, 0}, {-sin θ, cos θ, 0}, {0, 0, 1}})).

Block[{j=1,JJ,J3,Jp,Jm,J1,J2,U},
{JJ,J3,Jp}=SparseArray[#,{#,#}&@2j+1]&/@{Table[j+1-{m,m}->j(j+1),{m,-j,j}],Table[j+1-{m,m}->m,{m,-j,j}],
Table[j+1-{m+1,m}->Sqrt[j(j+1)-m(m+1)],{m,-j,j-1}]};
Jm=Transpose@Jp;
{J1,J2}={Jp+Jm,-I Jp+I Jm}/2;
U={{I,1,0}/Sqrt[2],{0,0,-I},{-I,1,0}/Sqrt[2]};
MatrixForm[MatrixExp[I θ ConjugateTranspose[U].#.U]]&/@{J1,J2,J3}
]

Matrix Exp.

This is indeed the case for spin-1 representation

(Subscript[R, 1])(0,0,2π)≏(({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}))≏𝟙.

But not for spin-1/2,

(Subscript[R, 1/2])(0,0,2π)≏(({{-1, 0}, {0, -1}}))≏-𝟙,

The extra minus sign means that spin-1/2 is a projective representation of SO(3). Nevertheless, it is a linear representation of SU(2).

Mathematical Interlude: Group and Group Representation

(Subscript[D, 3])={e,(Subscript[C, 3]),(Subscript[C, 3]^2),σ,σ (Subscript[C, 3]),σ (Subscript[C, 3]^2)}.

Each group element is a symmetry operation

e: identity operation (do nothing)

(Subscript[C, 3]): three-fold rotation (clockwise)

σ: reflection with respect to the vertical axis (x->-x)

The group multiplication can be defined by multiplication table

({{∘, e, (Subscript[C, 3]), (Subscript[C, 3]^2), σ, σ (Subscript[C, 3]), σ (Subscript[C, 3]^2)}, {e, e, (Subscript[C, 3]), (Subscript[C, 3]^2), σ, σ (Subscript[C, 3]), σ (Subscript[C, 3]^2)}, {(Subscript[C, 3]), (Subscript[C, 3]), (Subscript[C, 3]^2), e, σ (Subscript[C, 3]^2), σ, σ (Subscript[C, 3])}, {(Subscript[C, 3]^2), (Subscript[C, 3]^2), e, (Subscript[C, 3]), σ (Subscript[C, 3]), σ (Subscript[C, 3]^2), σ}, {σ, σ, σ (Subscript[C, 3]), σ (Subscript[C, 3]^2), e, (Subscript[C, 3]), (Subscript[C, 3]^2)}, {σ (Subscript[C, 3]), σ (Subscript[C, 3]), σ (Subscript[C, 3]^2), σ, (Subscript[C, 3]^2), e, (Subscript[C, 3])}, {σ (Subscript[C, 3]^2), σ (Subscript[C, 3]^2), σ, σ (Subscript[C, 3]), (Subscript[C, 3]), (Subscript[C, 3]^2), e}})

Block[{D3},
D3=AssociationThread[GroupElements[DihedralGroup[3]][[{1,4,5,6,2,3}]]->{e,(Subscript[C, 3]),(Subscript[C, 3]^2),σ,σ (Subscript[C, 3]),σ (Subscript[C, 3]^2)}];
Grid@Table[D3[g\[PermutationProduct]h],{g,Keys@D3},{h,Keys@D3}]
]

Multiplication Table.

In fact, the following definition relations are sufficient to specify all the multiplications in (Subscript[D, 3]) group

(Subscript[C, 3]^3)=e, (σ^2)=e, σ (Subscript[C, 3])σ (Subscript[C, 3])=e.

g∘h=g h=>R(g)R(h)=R(g h).

Underlying each representation is a set of basis (on which group element can act as symmetry transformations). For example, the following vertex-labeled triangles form a set of basis

{(Ket),(Ket),(Ket),(Ket),(Ket),(Ket)}

Block[{D3,st},
D3=GroupElements[DihedralGroup[3]][[{1,4,5,6,2,3}]];
st[g_]:=Deploy[Graphics[GraphicsComplex[Permute[CirclePoints[3],g],{{Yellow,Triangle[{1,2,3}]},
Text[Style[#1,10],#2]&@@@{{"A",2},{"B",1},{"C",3}}}],ImageSize->30]];
Ket/@st/@D3
]

Triangle basis.

They are related to each other by symmetry transformations (as unitary operators)

(Subscript[C, 3])=(RowWithSeparators[+, "+", (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)])]),
σ=(RowWithSeparators[+, "+", (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)]), (RowDefault[(Ket), (Bra)])]),

Block[{D3,st,rep},
D3=GroupElements[DihedralGroup[3]][[{1,4,5,6,2,3}]];
st[g_]:=Deploy[Graphics[GraphicsComplex[Permute[CirclePoints[3],g],{{Yellow,Triangle[{1,2,3}]},
Text[Style[#1,10],#2]&@@@{{"A",2},{"B",1},{"C",3}}}],ImageSize->30]];
rep[g_]:=Row[Table[Row[{Ket[st[g\[PermutationProduct]a]],Bra[st[a]]}],{a,D3}],"+"];
rep[D3[[#]]]&/@{2,4}
]

Operator representation.

which can be written in the matrix form

(Subscript[C, 3])≏(({{0, 0, 1, 0, 0, 0}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 1, 0, 0}})),σ≏(({{0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}})).

This is called the canonical representation of (Subscript[D, 3]).

The canonical representation happens to be reducible, because we can find a new set of basis, related to the original ones by

(({{(Ket[0])}, {(Ket[s])}, {(Ket[x])}, {(Ket[y])}, {(Ket[x'])}, {(Ket[y'])}}))=(({{(1/(Sqrt[6])), (1/(Sqrt[6])), (1/(Sqrt[6])), (1/(Sqrt[6])), (1/(Sqrt[6])), (1/(Sqrt[6]))}, {(1/(Sqrt[6])), (1/(Sqrt[6])), (1/(Sqrt[6])), -(1/(Sqrt[6])), -(1/(Sqrt[6])), -(1/(Sqrt[6]))}, {(1/(Sqrt[3])), -(1/(2 (Sqrt[3]))), -(1/(2 (Sqrt[3]))), (1/(Sqrt[3])), -(1/(2 (Sqrt[3]))), -(1/(2 (Sqrt[3])))}, {0, (1/2), -(1/2), 0, -(1/2), (1/2)}, {0, -(1/2), (1/2), 0, -(1/2), (1/2)}, {(1/(Sqrt[3])), -(1/(2 (Sqrt[3]))), -(1/(2 (Sqrt[3]))), -(1/(Sqrt[3])), (1/(2 (Sqrt[3]))), (1/(2 (Sqrt[3])))}}))(({{(Ket)}, {(Ket)}, {(Ket)}, {(Ket)}, {(Ket)}, {(Ket)}})).

Let U be the matrix in Eq. (). Under the basis transformation,

(Subscript[C, 3])→U (Subscript[C, 3])(SuperDagger[U])≏(({{1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, -(1/2), -((Sqrt[3])/2), 0, 0}, {0, 0, ((Sqrt[3])/2), -(1/2), 0, 0}, {0, 0, 0, 0, -(1/2), -((Sqrt[3])/2)}, {0, 0, 0, 0, ((Sqrt[3])/2), -(1/2)}})),
σ→U σ (SuperDagger[U])≏(({{1, 0, 0, 0, 0, 0}, {0, -1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, -1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, -1}})),

Block[{D3,st,rep,U},
D3=AssociationThread[GroupElements[DihedralGroup[3]][[{1,4,5,6,2,3}]]->Range[6]];
st[g_]:=Deploy[Graphics[GraphicsComplex[Permute[CirclePoints[3],g],{{Yellow,Triangle[{1,2,3}]},Text[Style[#1,10],#2]&@@@{{"A",2},{"B",1},{"C",3}}}],ImageSize->30]];
rep[g_]:=SparseArray[Table[{D3[g\[PermutationProduct]a],D3[a]}->1,{a,Keys@D3}],{6,6}];
U=Orthogonalize[{{1,1,1,1,1,1},{1,1,1,-1,-1,-1},{2,-1,-1,2,-1,-1},{0,1,-1,0,-1,1},{0,-1,1,0,-1,1},{2,-1,-1,-2,1,1}}];
MatrixForm[U.rep[Keys[D3][[#]]].ConjugateTranspose[U]]&/@{2,4}
]

Irreducible representation.

The group elements can actually be represented as smaller matrices => irreducible representations. The canonical representation is reducible because it can be split into a direct sum of irreducible representations,

→(Subscript[A, 1])⊕(Subscript[A, 2])⊕B⊕B

(Subscript[A, 1])=span{(Ket[0])}: trivial representation, one-dimensional.

(Subscript[A, 2])=span{(Ket[s])}: signed representation, one-dimensional.

B=span{(Ket[x]),(Ket[y])}: standard (isomorphic) representation, two-dimensional.

Addition of Angular Momentum

Example: Fusion of Spins

State tensor products:

(Subscript[{(Ket[↑]),(Ket[↓])}, A])×(Subscript[{(Ket[↑]),(Ket[↓])}, B])={(Ket[↑↑]),(Ket[↑↓]),(Ket[↓↑]),(Ket[↓↓])}.

Angular momentum adds:

(Subscript[S, A])=(1/2)(Subscript[σ, A])⊗(Subscript[𝟙, B])≏(1/2)((σ^10),(σ^20),(σ^30)),
(Subscript[S, B])=(1/2)(Subscript[𝟙, A])⊗(Subscript[σ, B])≏(1/2)((σ^01),(σ^02)(σ^03)).

(S^2)=((((Subscript[S, A])+(Subscript[S, B])))^2)=(Subscript[S, A]^2)+(Subscript[S, B]^2)+2(Subscript[S, A])·(Subscript[S, B])
=(3/2)𝟙+2(Subscript[S, A])·(Subscript[S, B]).

(Subscript[S, A])·(Subscript[S, B])≏(1/4)((σ^11)+(σ^22)+(σ^33))=(1/4)(({{1, 0, 0, 0}, {0, -1, 2, 0}, {0, 2, -1, 0}, {0, 0, 0, 1}})).

Spin singlet state, (Subscript[S, A])·(Subscript[S, B]) =-3/4 => (S^2)=0 (s=0):

Spin triplet states, (Subscript[S, A])·(Subscript[S, B]) =1/4 => (S^2)=2 (s=1):

Block[{S,H,U},
S=Represent@(σ[#,0]+σ[0,#])/2&/@Range[3];
H=Total[#.#&/@S]-Last@S/10;
U=Transpose[Normalize/@Last/@SortBy[First]@Thread@Eigensystem[H]];
Style[Grid[Join[{{"space:","\!\(\(1\/2\)⊗\(1\/2\)\)","\!\(0⊕1\)"},{"basis:","\!\((Ket[↑↑]),(Ket[↑↓]),(Ket[↓↑]),(Ket[↓↓])\)","\!\((Ket[0,0]);(Ket[1,+1]),(Ket[1,0]),(Ket[1,-1])\)"}},{"\!\("<>#1<>"≏\)",MatrixForm[#2],MatrixForm[{{Grid[ConjugateTranspose[U].#2.U,Dividers->{2->Dashed,2->Dashed}]}}]}&@@@Thread@{{"S\_1","S\_2","S\_3","S\^2"},Append[S,tTr[{S,S},{{1,4},{3,5}}]]}]],FontSize->14,SingleLetterItalics->True,FontFamily->"CMU"]]

The original basis {(Ket[↑↑]),(Ket[↑↓]),(Ket[↓↑]),(Ket[↓↓])} form a reducible representation of SU(2).

The new basis {(Ket[0,0])} and {(Ket[1,+1]),(Ket[1,0]),(Ket[1,-1])} respectively form irreducible representations of SU(2).

1/2⊗1/2→0 channel:

((Ket[0,0]))=(({{0, (1/(Sqrt[2])), -(1/(Sqrt[2])), 0}}))(({{(Ket[↑↑])}, {(Ket[↑↓])}, {(Ket[↓↑])}, {(Ket[↓↓])}})).

1/2⊗1/2→1 channel:

(({{(Ket[1,+1])}, {(Ket[1,0])}, {(Ket[1,-1])}}))=(({{1, 0, 0, 0}, {0, (1/(Sqrt[2])), (1/(Sqrt[2])), 0}, {0, 0, 0, 1}}))(({{(Ket[↑↑])}, {(Ket[↑↓])}, {(Ket[↓↑])}, {(Ket[↓↓])}})).

Fusion Rules

Irreducible representations are simple objects in the category, labeled by their spin quantum numbers j=0,1/2,1,…

Other reducible representations are not simple. They can be considered as spin systems.

Semisimple property: each reducible representation (spin system) can split in to direct sum of irreducible representations (spins).

Matrix representation of group element g∈ SU(2) in representation j can be depicted as

Graphics[{{Red,Line[{{-1,0},{1,0}}],Text["\!\( j\)",{-0.8,0},{0,1}],Text["\!\( j\)",{0.8,0},{0,1}]},Text[Style["\!\( g\)",Background->White],{0,0}]},PlotRange->{-0.8,0.5},ImageSize->60]

Example: rotation θ=(0,0,θ), c.f. Eq. () and Eq. ()

=(({{(e^(i θ/2)), 0}, {0, (e^(-i θ/2))}})),
=(({{cos θ, sin θ, 0}, {-sin θ, cos θ, 0}, {0, 0, 1}})).

Row[{Graphics[{{Red,Line[{{-1,0},{1,0}}],Text["\!\(j\_1\)",{0,0},{0,1}]}},ImageSize->30],Style[" ⊗ ",14,FontFamily->"CMU"],Graphics[{{Blue,Line[{{-1,0},{1,0}}],Text["\!\(j\_2\)",{0,0},{0,1}]}},ImageSize->30],Style[" = ",14,FontFamily->"CMU"],Graphics[{{Red,Line[{{-1,0.5},{1,0.5}}],Text["\!\(j\_1\)",{0,0.5},{0,-1}]},
{Blue,Line[{{-1,-0.5},{1,-0.5}}],Text["\!\(j\_2\)",{0,-0.5},{0,1}]}},ImageSize->30]}]

The tensor structure specifies the fusion rule:

Composition of a spin-(Subscript[j, 1]) and a spin-(Subscript[j, 2]) systems is a spin system that contains irreducible representations ranging from |(Subscript[j, 1])-(Subscript[j, 2])| to (Subscript[j, 1])+(Subscript[j, 2]) (each of them appear once).

Example: 1/2⊗1/2→0⊕1.

Notice that the representation dimensions add up correctly

dim (Subscript[ℋ, (Subscript[j, 1])⊗(Subscript[j, 2])])=(∑_((Subscript[j, 3])=|(Subscript[j, 1])-(Subscript[j, 2])|)^((Subscript[j, 1])+(Subscript[j, 2])))(2(Subscript[j, 3])+1)=(2(Subscript[j, 1])+1)(2(Subscript[j, 2])+1).

The decomposition is implemented by the fusion vertex (a projective morphism), which can be viewed as three-leg tensors preserving the SU(2) symmetry.

Graphics[{{Red,BSplineCurve[{{-1.5,0.5},{-1,0.5},{-0.5,0.5},{-0.2,0.2},{0,0}}],Text["\!\(j\_1\)",{-1.2,0.5},{0,1}]},
{Green,BSplineCurve[{{-1.5,-0.5},{-1,-0.5},{-0.5,-0.5},{-0.2,-0.2},{0,0}}],Text["\!\(j\_2\)",{-1.2,-0.5},{0,1}]},
{Blue,Line[{{0,0},{1.5,0}}],Text["\!\(j\_3\)",{1.2,0},{0,1}]},
{EdgeForm[Black],White,Disk[{0,0},0.35]},
Text["\!\( π\)",{0,0}]},BaselinePosition->{0,0},ImageSize->85]

such that ∀g∈ SU(2):

Graphics[{Translate[{{Red,BSplineCurve[{{-1.8,0.5},{-1,0.5},{-0.5,0.5},{-0.2,0.2},{0,0}}],Text["\!\(j\_1\)",{-1.5,0.5},{0,1}]},
Text[Style["\!\( g\)",Background->White],{-1,0.5}],
{Green,BSplineCurve[{{-1.8,-0.5},{-1,-0.5},{-0.5,-0.5},{-0.2,-0.2},{0,0}}],Text["\!\(j\_2\)",{-1.5,-0.5},{0,1}]},
Text[Style["\!\( g\)",Background->White],{-1,-0.5}],
{Blue,Line[{{0,0},{1,0}}],Text["\!\(j\_3\)",{0.8,0},{0,1}]},
{EdgeForm[Black],White,Disk[{0,0},0.35]},
Text["\!\( π\)",{0,0}]},{-1.5,0}],
Text["=",{0,0}],Translate[{{Red,BSplineCurve[{{-1.5,0.5},{-1,0.5},{-0.5,0.5},{-0.2,0.2},{0,0}}],Text["\!\(j\_1\)",{-1.2,0.5},{0,1}]},
{Green,BSplineCurve[{{-1.5,-0.5},{-1,-0.5},{-0.5,-0.5},{-0.2,-0.2},{0,0}}],Text["\!\(j\_2\)",{-1.2,-0.5},{0,1}]},
{Blue,Line[{{0,0},{1.8,0}}],Text["\!\(j\_3\)",{1.5,0},{0,1}]},
Text[Style["\!\( g\)",Background->White],{1,0}],
{EdgeForm[Black],White,Disk[{0,0},0.35]},
Text["\!\( π\)",{0,0}]},{2,0}]},BaselinePosition->{0,0},ImageSize->200]

Example: fusion of two spin-1/2s, c.f. Eq. () and Eq. ()

(Subscript[π, 0]):1/2⊗1/2->0,
(Subscript[π, 0])≏(({{0, (1/(Sqrt[2])), -(1/(Sqrt[2])), 0}})),
(Subscript[π, 1]):1/2⊗1/2->1,
(Subscript[π, 1])≏(({{1, 0, 0, 0}, {0, (1/(Sqrt[2])), (1/(Sqrt[2])), 0}, {0, 0, 0, 1}})).

(({{(Subscript[j, 1]), (Subscript[j, 2]), (Subscript[j, 3])}, {(Subscript[m, 1]), (Subscript[m, 2]), (Subscript[m, 3])}}))=(BraKet[(Subscript[j, 3]),(Subscript[m, 3]), (Subscript[j, 1]),(Subscript[m, 1]);(Subscript[j, 2]),(Subscript[m, 2])]).

π((Subscript[j, 1])⊗(Subscript[j, 2])→(Subscript[j, 3]))=(∑_((Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3])))(Ket[(Subscript[j, 3]),(Subscript[m, 3])])(({{(Subscript[j, 1]), (Subscript[j, 2]), (Subscript[j, 3])}, {(Subscript[m, 1]), (Subscript[m, 2]), (Subscript[m, 3])}}))(Bra[(Subscript[j, 1]),(Subscript[m, 1]);(Subscript[j, 2]),(Subscript[m, 2])]).

either look up in Mathematica:

ClebschGordan[{1/2,-1/2},{1/2,1/2},{1,0}]

or calculate as eigenbasis of the Casimir operator (J^2):

Graphics[{Translate[{{Red,Line[{{-1,0.4},{1,0.4}}],Text["\!\(j\_1\)",{-0.8,0.4},{0,-1}],
Text["\!\(j\_1\)",{0.8,0.4},{0,-1}]},
{Blue,Line[{{-1,-0.4},{1,-0.4}}],Text["\!\(j\_2\)",{-0.8,-0.4},{0,1}],
Text["\!\(j\_2\)",{0.8,-0.4},{0,1}]},
{{EdgeForm[Black],White,Rectangle[{-0.4,-0.6},{0.4,0.6},RoundingRadius->0.2]},Text["\!\(J\^2\)",{0,0}]}},{-1.8,0}],Text["\!\(=\*UnderscriptBox[\(∑\), \(j\_3\)]\)",{0,0},{0,0.4}],
Translate[{{Green,Line[{{-1,0},{1,0}}],Text["\!\(j\_3\)",{-0.5,0},{0,1}],
Text["\!\(j\_3\)",{0.5,0},{0,1}]},
{EdgeForm[Black],White,Polygon@CirclePoints[{0.35,0},4]},
Text["\!\(λ\)"],Translate[{Scale[{{Red,BSplineCurve[{{-1,0.4},{-0.8,0.4},{-0.4,0.4},{-0.2,0.2},{0,0}}]},
{Blue,BSplineCurve[{{-1,-0.4},{-0.8,-0.4},{-0.4,-0.4},{-0.2,-0.2},{0,0}}]}},{#1,1},{0,0}],
{Red,Text["\!\(j\_1\)",{-#1 0.8,0.4},{0,-1}],
Blue,Text["\!\(j\_2\)",{-#1 0.8,-0.5},{0,1}]},
{EdgeForm[Black],White,Disk[{0,0},0.35]},
Text["\!\( π"<>#2<>"\)",{0,0}]},{-#1,0}]&@@@{{1,"\^*"},{-1,""}}},{2.8,0}]},ImageSize->220]

Represent (J^2) in the (Subscript[j, 1])⊗(Subscript[j, 2]) space.

Diagonalize (J^2), group eigenvectors by eigenvalues.

Within each degenerated subspace: the eigenvalue (Subscript[λ, (Subscript[j, 3])])=(Subscript[j, 3])((Subscript[j, 3])+1) or the degeneracy 2(Subscript[j, 3])+1 infers the spin quantum number (Subscript[j, 3]), the eigenvectors form the fusion vertex π((Subscript[j, 1])⊗(Subscript[j, 2])→(Subscript[j, 3])).

Code Example

Jvec[j_]:=Block[{JJ,J3,Jp,Jm,J1,J2},
{JJ,J3,Jp}=SparseArray[#,{#,#}&@2j+1]&/@{
Table[j+1-{m,m}->j(j+1),{m,-j,j}],
Table[j+1-{m,m}->m,{m,-j,j}],
Table[j+1-{m+1,m}->Sqrt[j(j+1)-m(m+1)],{m,-j,j-1}]};
Jm=Transpose@Jp;
{J1,J2}={Jp+Jm,-I Jp+I Jm}/2;
{J1,J2,J3}];
MatrixForm/@Jvec[1]

j1=1/2;j2=1;
JA=KroneckerProduct[#,IdentityMatrix[2j2+1]]&/@Jvec[j1];
JB=KroneckerProduct[IdentityMatrix[2j1+1],#]&/@Jvec[j2];
J=JA+JB;
Jsqr=Total[#.#&/@J];
MatrixForm@Jsqr

Eigenvalues@Jsqr

({{(Subscript[λ, j])=j(j+1), deg.=2j+1, j}, {3/4, 2, 1/2}, {15/4, 4, 3/2}})

pi=Orthogonalize/@GroupBy[Thread@Eigensystem@Jsqr,First->Last];
MatrixForm/@pi

MatrixForm[pi[3/4].J[[3]].Transpose[pi[3/4]]]

π(1/2⊗1->1/2)=(({{0, -(1/(Sqrt[3])), 0, (Sqrt[(2/3)]), 0, 0}, {0, 0, -(Sqrt[(2/3)]), 0, (1/(Sqrt[3])), 0}})).

π(1⊗1/2->1/2)=(({{0, (Sqrt[(2/3)]), -(1/(Sqrt[3])), 0, 0, 0}, {0, 0, 0, (1/(Sqrt[3])), -(Sqrt[(2/3)]), 0}})),

({{1/2⊗1 basis, 1⊗1/2 basis}, {(Ket[1/2,+1/2;1,+1]), (Ket[1,+1;1/2,+1/2])}, {(Ket[1/2,+1/2;1,0]), (Ket[1,+1;1/2,-1/2])}, {(Ket[1/2,+1/2;1,-1]), (Ket[1,0;1/2,+1/2])}, {(Ket[1/2,-1/2;1,+1]), (Ket[1,0;1/2,-1/2])}, {(Ket[1/2,-1/2;1,0]), (Ket[1,-1;1/2,+1/2])}, {(Ket[1/2,-1/2;1,+1]), (Ket[1,-1;1/2,-1/2])}})

F-Symbols

Graphics[GraphicsComplex[{{1,0},{0,1},{-1,2},{1,2},{-2,3},{0,3},{2,3}},{Line[{{1,5},{3,6},{2,7}}],{EdgeForm[Black],White,Disk[#,0.4]&/@{2,3}},
Text["\!\(π\)",#]&/@{2,3},
Text@@@{{"\!\(j\_1\)",5,{0,-1}},{"\!\(j\_2\)",6,{0,-1}},{"\!\(j\_3\)",7,{0,-1}},{"\!\(j\_4\)",3,{-0.5,1.5}},{"\!\(j\_6\)",2,{-0.5,1.5}}}}],ImageSize->80]

The composition follows from the tensor network contraction.

For (Subscript[j, 4])=0,

(Subscript[α, 0])=π(1/2⊗1/2->0)∘π(0⊗1/2->1/2)
≏(({{1, 0}, {0, 1}}))[(({{0, (1/(Sqrt[2])), -(1/(Sqrt[2])), 0}}))⊗(({{1, 0}, {0, 1}}))]
=(({{1, 0}, {0, 1}}))(({{0, 0, (1/(Sqrt[2])), 0, -(1/(Sqrt[2])), 0, 0, 0}, {0, 0, 0, (1/(Sqrt[2])), 0, -(1/(Sqrt[2])), 0, 0}}))
=(({{0, 0, (1/(Sqrt[2])), 0, -(1/(Sqrt[2])), 0, 0, 0}, {0, 0, 0, (1/(Sqrt[2])), 0, -(1/(Sqrt[2])), 0, 0}})),

For (Subscript[j, 4])=1,

(Subscript[α, 1])=π(1/2⊗1/2->1)∘π(1⊗1/2->1/2)
≏(({{0, (Sqrt[(2/3)]), -(1/(Sqrt[3])), 0, 0, 0}, {0, 0, 0, (1/(Sqrt[3])), -(Sqrt[(2/3)]), 0}}))[(({{1, 0, 0, 0}, {0, (1/(Sqrt[2])), (1/(Sqrt[2])), 0}, {0, 0, 0, 1}}))⊗(({{1, 0}, {0, 1}}))]
=(({{0, (Sqrt[(2/3)]), -(1/(Sqrt[3])), 0, 0, 0}, {0, 0, 0, (1/(Sqrt[3])), -(Sqrt[(2/3)]), 0}}))(({{1, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, (1/(Sqrt[2])), 0, (1/(Sqrt[2])), 0, 0, 0}, {0, 0, 0, (1/(Sqrt[2])), 0, (1/(Sqrt[2])), 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1}}))
=(({{0, (Sqrt[(2/3)]), -(1/(Sqrt[6])), 0, -(1/(Sqrt[6])), 0, 0, 0}, {0, 0, 0, (1/(Sqrt[6])), 0, (1/(Sqrt[6])), -(Sqrt[(2/3)]), 0}})).

Graphics[GraphicsComplex[{{1,0},{0,1},{-1,2},{1,2},{-2,3},{0,3},{2,3}},{Line[{{1,5},{4,6},{2,7}}],{EdgeForm[Black],White,Disk[#,0.4]&/@{2,4}},
Text["\!\(π\)",#]&/@{2,4},
Text@@@{{"\!\(j\_1\)",5,{0,-1}},{"\!\(j\_2\)",6,{0,-1}},{"\!\(j\_3\)",7,{0,-1}},{"\!\(j\_5\)",4,{0.5,1.5}},{"\!\(j\_6\)",2,{-0.5,1.5}}}}],ImageSize->80]

For (Subscript[j, 5])=0,

(Subscript[β, 0])=π(1/2⊗1/2->0)∘π(1/2⊗0->1/2)
≏(({{1, 0}, {0, 1}}))[(({{1, 0}, {0, 1}}))⊗(({{0, (1/(Sqrt[2])), -(1/(Sqrt[2])), 0}}))]
=(({{1, 0}, {0, 1}}))(({{0, (1/(Sqrt[2])), -(1/(Sqrt[2])), 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, (1/(Sqrt[2])), -(1/(Sqrt[2])), 0}}))
=(({{0, (1/(Sqrt[2])), -(1/(Sqrt[2])), 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, (1/(Sqrt[2])), -(1/(Sqrt[2])), 0}})),

For (Subscript[j, 5])=1,

(Subscript[β, 1])=π(1/2⊗1/2->1)∘π(1/2⊗1->1/2)
≏(({{0, -(1/(Sqrt[3])), 0, (Sqrt[(2/3)]), 0, 0}, {0, 0, -(Sqrt[(2/3)]), 0, (1/(Sqrt[3])), 0}}))[(({{1, 0}, {0, 1}}))⊗(({{1, 0, 0, 0}, {0, (1/(Sqrt[2])), (1/(Sqrt[2])), 0}, {0, 0, 0, 1}}))]
=(({{0, -(1/(Sqrt[3])), 0, (Sqrt[(2/3)]), 0, 0}, {0, 0, -(Sqrt[(2/3)]), 0, (1/(Sqrt[3])), 0}}))(({{1, 0, 0, 0, 0, 0, 0, 0}, {0, (1/(Sqrt[2])), (1/(Sqrt[2])), 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, (1/(Sqrt[2])), (1/(Sqrt[2])), 0}, {0, 0, 0, 0, 0, 0, 0, 1}}))
=(({{0, -(1/(Sqrt[6])), -(1/(Sqrt[6])), 0, (Sqrt[(2/3)]), 0, 0, 0}, {0, 0, 0, -(Sqrt[(2/3)]), 0, (1/(Sqrt[6])), (1/(Sqrt[6])), 0}})).

=(∑_(Subscript[j, 5]))(Subscript[((Subscript[F, (Subscript[j, 6])]^((Subscript[j, 1])(Subscript[j, 2])(Subscript[j, 3])))), (Subscript[j, 5])]^(Subscript[j, 4])).

(({{(Subscript[α, 0])}, {(Subscript[α, 1])}}))=(({{-(1/2), -((Sqrt[3])/2)}, {((Sqrt[3])/2), -(1/2)}}))(({{(Subscript[β, 0])}, {(Subscript[β, 1])}})).

The matrix (Subscript[F, (Subscript[j, 6])]^((Subscript[j, 1])(Subscript[j, 2])(Subscript[j, 3]))) is called F-matrix, whose elements are called F-symbols. The F-matrix is unitary.

F-symbol is related to Wigner 6j-symbol by a normalization factor

(Subscript[((Subscript[F, (Subscript[j, 6])]^((Subscript[j, 1])(Subscript[j, 2])(Subscript[j, 3])))), (Subscript[j, 5])]^(Subscript[j, 4]))=(((-))^((Subscript[j, 1])+(Subscript[j, 2])+(Subscript[j, 3])+(Subscript[j, 6])))(Sqrt[(2(Subscript[j, 4])+1)(2(Subscript[j, 5])+1)]){({{(Subscript[j, 1]), (Subscript[j, 2]), (Subscript[j, 4])}, {(Subscript[j, 3]), (Subscript[j, 6]), (Subscript[j, 5])}})}.

SixJSymbol[{1/2,1/2,0},{1/2,1/2,0}]

Graphics[{MapThread[Translate[#1,#2]&,{{GraphicsComplex[{{-3,0},{-1,0},{1,0},{3,0},{-2,-1},{0,-1},{2,-1},{-1,-2},{1,-2},{0,-3},{1,-4}},
{Line[{{1,11},{2,9},{3,6},{4,10}}],
Text@@@{{"\!\(j\_1\)",1,{0,-1}},{"\!\(j\_2\)",2,{0,-1}},{"\!\(j\_3\)",3,{0,-1}},{"\!\(j\_4\)",4,{0,-1}},{"\!\(j\_5\)",11,{2,0}},{"\!\(j\_10\)",9,{1.5,0}},{"\!\(j\_9\)",10,{-2,0}}}}],
GraphicsComplex[{{-3,0},{-1,0},{1,0},{3,0},{-2,-1},{0,-1},{2,-1},{-1,-2},{1,-2},{0,-3},{1,-4}},
{Line[{{1,11},{2,9},{3,7},{4,10}}],
Text@@@{{"\!\(j\_1\)",1,{0,-1}},{"\!\(j\_2\)",2,{0,-1}},{"\!\(j\_3\)",3,{0,-1}},{"\!\(j\_4\)",4,{0,-1}},{"\!\(j\_5\)",11,{2,0}},{"\!\(j\_9\)",10,{-2,0}},{"\!\(j\_8\)",9,{-2,0}}}}],
GraphicsComplex[{{-3,0},{-1,0},{1,0},{3,0},{-2,-1},{0,-1},{2,-1},{-1,-2},{1,-2},{0,-3},{1,-4}},
{Line[{{1,11},{2,5},{3,7},{4,10}}],
Text@@@{{"\!\(j\_1\)",1,{0,-1}},{"\!\(j\_2\)",2,{0,-1}},{"\!\(j\_3\)",3,{0,-1}},{"\!\(j\_4\)",4,{0,-1}},{"\!\(j\_5\)",11,{2,0}},{"\!\(j\_6\)",8,{2,0}},{"\!\(j\_8\)",9,{-2,0}}}}],
GraphicsComplex[{{-3,0},{-1,0},{1,0},{3,0},{-2,-1},{0,-1},{2,-1},{-1,-2},{1,-2},{0,-3},{1,-4}},
{Line[{{1,11},{2,5},{3,8},{4,10}}],
Text@@@{{"\!\(j\_1\)",1,{0,-1}},{"\!\(j\_2\)",2,{0,-1}},{"\!\(j\_3\)",3,{0,-1}},{"\!\(j\_4\)",4,{0,-1}},{"\!\(j\_5\)",11,{2,0}},{"\!\(j\_6\)",8,{2,0}},{"\!\(j\_7\)",10,{2,0}}}}],
GraphicsComplex[{{-3,0},{-1,0},{1,0},{3,0},{-2,-1},{0,-1},{2,-1},{-1,-2},{1,-2},{0,-3},{1,-4}},
{Line[{{1,11},{2,6},{3,8},{4,10}}],
Text@@@{{"\!\(j\_1\)",1,{0,-1}},{"\!\(j\_2\)",2,{0,-1}},{"\!\(j\_3\)",3,{0,-1}},{"\!\(j\_4\)",4,{0,-1}},{"\!\(j\_5\)",11,{2,0}},{"\!\(j\_10\)",8,{-1.5,0}},{"\!\(j\_7\)",10,{2,0}}}}]},
8CirclePoints[5]}],
Block[{arr},
arr[{p1_,p2_}]:={{GrayLevel[0.7],AbsoluteThickness[5],Arrow[{(p1+p2)/2+0.12(p1-p2),(p1+p2)/2-0.12(p1-p2)}]},Text["\!\( F\)",(p1+p2)/2,1.5{{0,1},{-1,0}}.Normalize[p2-p1]]};
Translate[{Arrowheads[0.08],arr[10CirclePoints[5][[#]]]&/@{{4,3},{3,2},{4,5},{5,1},{1,2}}},{0,-1}]]},
ImageSize->240]

(Subscript[((Subscript[F, (Subscript[j, 5])]^((Subscript[j, 6])(Subscript[j, 3])(Subscript[j, 4])))), (Subscript[j, 8])]^(Subscript[j, 7]))(Subscript[((Subscript[F, (Subscript[j, 5])]^((Subscript[j, 1])(Subscript[j, 2])(Subscript[j, 8])))), (Subscript[j, 9])]^(Subscript[j, 6]))=(∑_(Subscript[j, 10]))(Subscript[((Subscript[F, (Subscript[j, 7])]^((Subscript[j, 1])(Subscript[j, 2])(Subscript[j, 3])))), (Subscript[j, 10])]^(Subscript[j, 6]))(Subscript[((Subscript[F, (Subscript[j, 5])]^((Subscript[j, 1])(Subscript[j, 10])(Subscript[j, 4])))), (Subscript[j, 9])]^(Subscript[j, 7]))(Subscript[((Subscript[F, (Subscript[j, 9])]^((Subscript[j, 2])(Subscript[j, 3])(Subscript[j, 4])))), (Subscript[j, 8])]^(Subscript[j, 10])).

Hydrogen Atom

Classical Mechanical Background

Classical Equation of Motion

A hydrogen atom is a bound state of a single electron and a single proton.

The proton is much heavier than the electron (mass ratio (Subscript[m, p])/(Subscript[m, e])≈1836), it is essentially pinned at the center of mass.

Consider electron moving in the electric static potential created by the proton.

V(r)=-(k/r),

where k=(e^2)/(4π (Subscript[ϵ, 0])) describes the strength of the Coulomb force.

The Energy (Hamiltonian) of the electron is given by

r is the displacement from the proton and r=|r| is the electron-proton distance.

p=m (OverDot[r]) is the momentum.

m is the mass of electron (or more precisely the reduced mass of the electron in this two-body problem).

The angular momentum

The classical equation of motion:

(OverDot[r])=(p/m), (OverDot[p])=-((k r)/(r^3)).

(OverDot[L])=(OverDot[r])×p+r×(OverDot[p])=0 from Eq. () => the angular momentum is conserved. (∵ the Hamiltonian has the SO(3) rotation symmetry)

Laplace-Runge-Lenz (LRL) Vector

The Laplace-Runge-Lenz (LRL) vector is defined to be

DynamicModule[{A=0.7,θ,r,bg},
θ=π-ArcTan[Sqrt[1-A^2]/A];
bg=Dynamic[{Cases[PolarPlot[1/(1+A Cos[α]),{α,-π,π}],_Line,∞],
If[A>0.,{Red,Arrow[{{0,0},{A,0}}]},Nothing]}];
Column@{Grid[{{Style["\!\(A \):",SingleLetterItalics->True,FontFamily->"CMU"],Slider[Dynamic[A],{0.,0.9},ImageSize->Small]},{Style["θ:",FontFamily->"CMU"],Slider[Dynamic[θ],{-π,π},ImageSize->Small]}}],
Dynamic[
r=1/(1+A Cos[θ]);Graphics[{Arrowheads[{{0.001,1,Arrowhead["Alignment"->0]}}],bg,{Arrow[{{0,0},#}],Text["\!\(r \)",#/2,Reverse[{1,-1}Normalize[#]]],Translate[{{Green,Arrow[{{0,0},Normalize[#]}],Text["\!\(m k r\&^\)",Normalize[#]/2,1.2Sign[θ]Reverse[{-1,1}Normalize[#]]]},{Blue,Arrow[{{0,0},Normalize[#]+{A,0}}],Text["\!\(p × L\)",(Normalize[#]+{A,0})/2,1.2Sign[θ]Reverse[{1,-1}Normalize[(Normalize[#]+{A,0})]]]},{Red,Arrow[{Normalize[#],Normalize[#]+{A,0}}],Text["\!\(A \)",Normalize[#]+{A,0}/2,{0,-Sign[θ]+0.1}]}},#]}&@(r{Cos[θ],Sin[θ]}),{Red,Point[{0,0}],AbsoluteThickness[1],Circle[{0,0},Offset@{6,6},{0,θ}],Text["θ",Offset[6{Cos[θ/2],Sin[θ/2]}],-0.7{Cos[θ/2],Sin[θ/2]}]}},PlotRange->{{-1/(1-A)-1,1/(1+A)+1+A},{-1/Sqrt[1-A^2]-1,1/Sqrt[1-A^2]+1}},PlotRangePadding->Scaled[0.05],ImageSize->{Automatic,180}]]}]

The LRL vector is conserved (there must be an symmetry associate to it, what is it?)

Using Eq. (),

(OverDot[A])=(OverDot[p])×L-m k ((OverDot[r])/r)+m k (r/(r^2))(OverDot[r])
=-((k )/(r^3))r×(r×p)-(k/r)p+m k (r/(r^2))(OverDot[r])
=-((k )/(r^3))(r (r·p)-(r^2)p)-(k/r)p+m k (r/(r^2))(OverDot[r])
=-m k((r )/(r^3))(r·(OverDot[r]))+m k (r/(r^2))(OverDot[r])=0

A·r=A r cos θ=r·(p×L)-m k r,

(1/r)=((m k)/(L^2))(1+(A/(m k))cos θ).

Conservation of the magnitude A=|A| => conservation of the eccentricity of the orbit (in fact e≡c/a=A/m k).

Conservation of the direction of A => conservation of the periapsis (the point of closest approach) with respect to the force center.

L and A are always perpendicular (L·A=0), they fully specifies a unique Kepler orbit (shape and orientation are both determined).

As the orbit is specified, its energy must be determined too => so the energy E is not an independent conserved quantity, in fact

Given the magnitude of L and A,

All orbits of different orientations are degenerated in energy => they are related by symmetry.

Each orbit orientation ⇔ a pair of perpendicular unit vectors (L/L,A/A) ⇔ a particular rotation in 3D space, i.e., a group element of SO(3) ⇔ a point on (half of) a 4D sphere! (more precisely a point on (RP^3))

An explicit construction: every 4-dim unit vector ((Subscript[n, 1]),(Subscript[n, 2])(Subscript[n, 3]),(Subscript[n, 4])) ⇔ two orthogonal 3-dim unit vectors

L/L=((Subscript[n, 1]^2)+(Subscript[n, 2]^2)-(Subscript[n, 3]^2)-(Subscript[n, 4]^2),2((Subscript[n, 2])(Subscript[n, 3])-(Subscript[n, 1])(Subscript[n, 4])),2((Subscript[n, 1])(Subscript[n, 3])+(Subscript[n, 2])(Subscript[n, 4]))),
A/A=(2((Subscript[n, 2])(Subscript[n, 3])+(Subscript[n, 1])(Subscript[n, 4])),(Subscript[n, 1]^2)-(Subscript[n, 2]^2)+(Subscript[n, 3]^2)-(Subscript[n, 4]^2),2((Subscript[n, 3])(Subscript[n, 4])-(Subscript[n, 1])(Subscript[n, 2]))).

The symmetry that keeps the 4D sphere invariant is SO(4) (more precisely, to keep (RP^3) invariant, the symmetry is actually SO(4)/(Subscript[ℤ, 2])).

SO(4) group has six generators, all of them must be conserved => corresponding to the conservation of both L and A.

Quantum Mechanical Treatment

Operator Algebra

Coordinate r and momentum p operators, satisfying [(Subscript[r, a]),(Subscript[p, b])]=i (Subscript[δ, a b]). See Eq. ().

Angular momentum operator L=r×p. See Eq. ().

Quantum LRL operator

A=(1/2)(p×L-L×p)-m k (r/r).

Compared with the classical case Eq. (), the additional “anti-symmetrization” is to ensure A to be Hermitian.

Hamiltonian operator

H=((p^2)/(2m))-(k/r).

[H,(Subscript[L, a])]=0,
[H,(Subscript[A, a])]=0,
[(Subscript[L, a]),(Subscript[L, b])]=i (Subscript[ϵ, a b c])(Subscript[L, c]),
[(Subscript[L, a]),(Subscript[A, b])]=i (Subscript[ϵ, a b c])(Subscript[A, c]),
[(Subscript[A, a]),(Subscript[A, b])]=-i 2m (Subscript[ϵ, a b c])(Subscript[L, c])H,

Block[{ϵ,r,p,L,A,mH},
ϵ=LeviCivitaTensor[3];
r={(Subscript[𝓇, 1]),(Subscript[𝓇, 2]),(Subscript[𝓇, 3])};
p={(Subscript[𝓅, 1]),(Subscript[𝓅, 2]),(Subscript[𝓅, 3])};
L=r⋀p;
A=(p⋀L-L⋀p)/2-m k (1/𝓇)**r;
mH=Tr[p**p]/2-m k(1/𝓇);
Timing[MatrixForm@OpReduce[𝓇,𝓅][#]]&/@{
Com[mH,L],
Com[mH,A],
Com[L,L]-I ϵ.L,
Com[L,A]-I ϵ.A,
Com[A,A]+I 2ϵ.(L**mH)}]

Verification.

A·L=L·A=0,
(A^2)=2m H((L^2)+1)+(m^2) (k^2).

Block[{ϵ,r,p,L,A,mH},
ϵ=LeviCivitaTensor[3];
r={(Subscript[𝓇, 1]),(Subscript[𝓇, 2]),(Subscript[𝓇, 3])};
p={(Subscript[𝓅, 1]),(Subscript[𝓅, 2]),(Subscript[𝓅, 3])};
L=r⋀p;
A=(p⋀L-L⋀p)/2-m k (1/𝓇)**r;
mH=Tr[p**p]/2-m k(1/𝓇);
OpReduce[𝓇,𝓅]@{
Tr[A**L],
Tr[L**A],
Tr[A**A]-(2 mH**(Tr[L**L]+ℐ)+(m k)^2ℐ)}]

Verification.

SO(4) Generators

(OverTilde[A])=(1/(Sqrt[-2m E]))A,

[H,(Subscript[L, a])]=0,
[H,(Subscript[A, a])]=0,
[(Subscript[L, a]),(Subscript[L, b])]=i (Subscript[ϵ, a b c])(Subscript[L, c]),
[(Subscript[L, a]),(Subscript[(OverTilde[A]), b])]=i (Subscript[ϵ, a b c])(Subscript[(OverTilde[A]), c]),
[(Subscript[(OverTilde[A]), a]),(Subscript[(OverTilde[A]), b])]=i  (Subscript[ϵ, a b c])(Subscript[L, c]).

T=(1/2)(L+(OverTilde[A])), S=(1/2)(L-(OverTilde[A])).

How do we know they form SO(4) generators? We check their commutation relations

[H,(Subscript[T, a])]=0,
[H,(Subscript[S, a])]=0,
[(Subscript[T, a]),(Subscript[T, b])]=i (Subscript[ϵ, a b c])(Subscript[T, c]),
[(Subscript[S, a]),(Subscript[S, b])]=i  (Subscript[ϵ, a b c])(Subscript[S, c]),
[(Subscript[T, a]),(Subscript[S, b])]=0.

A·L=L·A=0 further implies (T^2)=(S^2).

T and S are two sets of independent “angular-momentum-like” operators.

Common eigen states of T and S are labeled by three quantum numbers t, (Subscript[m, t]), (Subscript[m, s]):

t=0,(1/2),1,(3/2),2,…. For each fixed t => (((2t+1))^2) degeneracy.

(Subscript[m, t]),(Subscript[m, s])=-t,-t+1,…,t-1,t.

Energy Levels

4(T^2)=(L^2)+((OverTilde[A])^2)
=(L^2)+(1/(-2m E))(A^2)
=(L^2)-(1/(2m E))(2m E((L^2)+1)+(m^2) (k^2))
=-1-((m (k^2))/(2E)).

Acting on the state (Ket[t,(Subscript[m, t]),(Subscript[m, s])]), (T^2)(Ket[t,(Subscript[m, t]),(Subscript[m, s])])=t(t+1)(Ket[t,(Subscript[m, t]),(Subscript[m, s])]),

E=-((m (k^2))/(2(4(T^2)+1)))=-((m (k^2))/(2(4t(t+1)+1)))=-((m (k^2))/(2(((2t+1))^2))).

Given t=0,(1/2),1,(3/2),2,…, define the principal quantum number

n=2t+1=1,2,3,4,5,…

The energy levels are given by n∈(SubPlus[ℤ])

The corresponding eigenstates (Ket[t,(Subscript[m, t]),(Subscript[m, s])]) (with t=(n-1)/2) are labeled by (Subscript[m, t]),(Subscript[m, s])=-t,-t+1,…,t-1,t => level degeneracy: (((2t+1))^2)=(n^2).

Ground State

The energy (Subscript[E, n])=-m (k^2)/(2(n^2)) is minimized when n=1 (or t=0).

When t=0, (Subscript[m, t]) and (Subscript[m, s]) has only one choice, i.e. (Subscript[m, t])=(Subscript[m, s])=0.

So the ground state corresponds to the (Ket[0,0,0]) state.

H (Subscript[ψ, 1s])=(Subscript[E, 1])(Subscript[ψ, 1s]),

(((p^2)/(2m))-(k/r))(Subscript[ψ, 1s])=-((m (k^2))/2)(Subscript[ψ, 1s]).

(Subscript[ψ, 1s])(r)∝(e^(-r/a)),

Block[{r,ψ},
r=Sqrt[x^2+y^2+z^2];
ψ=Exp[-r/a];
Simplify[-Laplacian[ψ,{x,y,z}]/(2m)-(k/r)ψ+(m k^2/2) ψ]]

Verification.

a=1/(k m) is called the Bohr radius.

The ground state energy is given by

as if the electron in orbiting the proton in a circular orbital of radius a (the classical picture).

The electron is observed randomly in a probability cloud, called the electron cloud.

p(r)=((|ψ(r)|)^2)∝(e^(-2r/a)).

Excited States

States with the same principal quantum numbers are degenerated in energy.

Within each degenerated sectors, the states (Ket[t,(Subscript[m, t]),(Subscript[m, s])]) form a complete set of basis (with (Subscript[m, t]),(Subscript[m, s])=-t,…,t), which can be viewed as a combined system of two “spins” of the same angular momentum t.

(Ket[n,l,m])

n - principal quantum number (n=1,2,…),

l - angular quantum number (l=0,1,…,n-1)

m - magnetic quantum number (m=-l,-l+1,…,l-1,l)

Code

Fusion Category of SU(2) Representations

Off[ClebschGordan::phy];

SetAttributes[Hom,HoldAll];
Hom[HoldPattern[j1_->j2_]]:=
If[j1==j2,{SparseArray[{i_,i_}->1,{2j1+1,2j1+1}]},{}];
Hom[HoldPattern[j1_⊗j2_->j_]]:=
If[Abs[j1-j2]≤j≤j1+j2,
{SparseArray[
Flatten@Table[{j+m+1,(j1+m1)(2j2+1)+j2+m2+1}->
ClebschGordan[{j1,m1},{j2,m2},{j,m}],
{m,-j,j},{m1,-j1,j1},{m2,-j2,j2}],
{(2j+1),(2j1+1)(2j2+1)}]},
{}];

Prod[h1_List,h2_List]:=Flatten[Outer[ArrayFlatten[#1⊗#2]&,h1,h2,1],1];

Comp[h1_List,h2_List]:=Flatten[Outer[#1.#2&,h1,h2,1],1];

Hom[HoldPattern[(j1_⊗j2_)⊗j3_->j_]]:=
Flatten[Table[Comp[Hom[j4⊗j3->j],Prod[Hom[j1⊗j2->j4],Hom[j3->j3]]],
{j4,Max[Abs[j1-j2],Abs[j3-j]],Min[j1+j2,j3+j]}],1];
Hom[HoldPattern[j1_⊗(j2_⊗j3_)->j_]]:=
Flatten[Table[Comp[Hom[j1⊗j4->j],Prod[Hom[j1->j1],Hom[j2⊗j3->j4]]],
{j4,Max[Abs[j2-j3],Abs[j1-j]],Min[j2+j3,j1+j]}],1];

Ass[h1_List,h2_List]:=
Outer[With[{p=#1 #2},Total[p,Infinity]/Length[p]]&,h1,h2,1];

Operator Algebra

BeginPackage["OpAlg`"];
Com::usage="Com[A, B] represents the commutator A**B - B**A.";
OpReduce::usage="ComReduce[{𝓇,𝓅}][expr] reduces the operator expression expr, given the symbols 𝓇 for coordinate and 𝓅 for momentum.";
ℐ::usage ="ℐ represents the identity operator.";
Begin["`Private`"];

OpReduce[𝓇_,𝓅_][expr_]:=With[{FactorQ=FreeQ[#,𝓇|𝓅|ℐ]&,d=3},
FixedPoint[Expand[#/.{
Com[A_+B_,C_]:>Com[A,C]+Com[B,C],
Com[A_,B_+C_]:>Com[A,B]+Com[A,C],
Com[x_?FactorQ A_,B_]:> x Com[A,B],
Com[A_,x_?FactorQ B_]:>x Com[A,B],
Com[A_**B_,C_]:>A**Com[B,C]+Com[A,C]**B,
Com[A_,B_**C_]:>B**Com[A,C]+Com[A,B]**C,
(A_+B_)**C_:>A**C+B**C,
A_**(B_+C_):>A**B+A**C,
(x_?FactorQ A_)**B_:>x A**B,
A_**(x_?FactorQ B_):>x A**B,
_**0:>0,
0**_:>0,
A_**ℐ:>A,
ℐ**A_:>A,
Com[(Subscript[𝓇, a_]),(Subscript[𝓅, b_])]:>I KroneckerDelta[a,b] ℐ,
Com[(Subscript[𝓅, a_]),(Subscript[𝓇, b_])]:>-I  KroneckerDelta[a,b] ℐ,
Com[𝓇^(n_:1),(Subscript[𝓅, a_])]:>I n If[n≠2,𝓇^(n-2),ℐ]**(Subscript[𝓇, a]),
Com[(Subscript[𝓅, a_]),𝓇^(n_:1)]:>-I n If[n≠2,𝓇^(n-2),ℐ]**(Subscript[𝓇, a]),
Com[(Subscript[𝓇, a_]),𝓅^(n_:1)]:>I n If[n≠2,𝓅^(n-2),ℐ]**(Subscript[𝓅, a]),
Com[𝓅^(n_:1),(Subscript[𝓇, a_])]:>-I n If[n≠2,𝓅^(n-2),ℐ]**(Subscript[𝓅, a]),
Com[_,_]:>0,
(Subscript[𝓇, a_])**(Subscript[𝓇, b_])/;a>b:>(Subscript[𝓇, b])**(Subscript[𝓇, a]),
(Subscript[𝓅, a_])**(Subscript[𝓅, b_])/;a>b:>(Subscript[𝓅, b])**(Subscript[𝓅, a]),
(Subscript[𝓅, a_])**(Subscript[𝓇, b_]):>(Subscript[𝓇, b])**(Subscript[𝓅, a])+Com[(Subscript[𝓅, a]),(Subscript[𝓇, b])],
(Subscript[𝓅, a_])**𝓇^(n_:1):>𝓇^n**(Subscript[𝓅, a])+Com[(Subscript[𝓅, a]),𝓇^n],
𝓅^(n_:1)**(Subscript[𝓇, a_]):>(Subscript[𝓇, a])**𝓅^n+Com[𝓅^n,(Subscript[𝓇, a])],
𝓅^(n_:1)**𝓇^(m_:1):>If[n≠2,𝓅^(n-2),ℐ]**𝓇^m**𝓅^2-m(m-2+d)If[m≠2,𝓇^(m-2),ℐ],
(Subscript[o_, a_])**o_^(n_:1):>o^n**(Subscript[o, a]),
(Subscript[o_, d])**(Subscript[o_, d]):>o^2-Sum[(Subscript[o, i])**(Subscript[o, i]),{i,1,d-1}],
𝓇^(n_:1)**𝓇^(m_:1):>If[n+m≠0,𝓇^(n+m),ℐ],
𝓅^(n_:1)**𝓅^(m_:1):>If[n+m≠0,𝓅^(n+m),ℐ]}]&,
expr,256]];

Unprotect[NonCommutativeMultiply];
As_List**B_:=(#**B&/@As);
A_**Bs_List:=(A**#&/@Bs);
Protect[NonCommutativeMultiply];
Com[As_List,B_]:=(Com[#,B]&/@As);
Com[A_,Bs_List]:=(Com[A,#]&/@Bs);

Wedge[a_,b_]:=Normal@TensorContract[
LeviCivitaTensor[3]⊗(a**b),
{{2,4},{3,5}}];

End[];
EndPackage[];