Álgebra de momentos angulares - unridbetan/cursonuclear... · 2020. 10. 6. · the angular...

Rodolfo M. Id Betan (Rolo) idbetan@ifir-conicet.gov.ar

Edificio Ifir, Of. 235 (Esmeralda y Ocampo) Tel. 4853200 Int. 486

Álgebra de momentos angulares

Contenido:

Definición de momento angular. Deducción de las matrices de Pauli. Acople de dos momentos angulares. Coeficientes de Clebsch-Gordan. Ejemplo de acoples. Acople de tres y cuatro momentos angulares. Cambio de acoples y símbolo de Wigner 6j y 9j. Ejemplo de acople de cuatro momentos angulares.

Introducción a la Física Nuclear

2020

Motivación

Motivación para el uso del álgebra angular

l y s desacopladosψnljm(r, s) = ϕnlj(r) χsms

Ylml(θ, ϕ)

l: momento angular orbitals: momento intrínseco de espín

Función de onda de una partícula

χsmsYlml(θ, ϕ)ϕnlj(r)

Motivación para el uso del álgebra angular

l y s acoplados

Coeficientes de Clebsch-Gordan

ψnljm(r, s) = ϕnlj(r)[χsYl(θ, ϕ)]jm

[χsYl(θ, ϕ)]jm =

sl

j

∑ms,ml

⟨smslml | jm⟩χsmsYlml

(θ, ϕ)

Función de onda de dos partículas en acople sl (ver aplicaciones al final)

〈x1x2|lalbSL, JM〉 =∑

MSML

∑

mlamlb

∑

ms1ms2

〈SMSLML|JM〉

〈lamla lbmlb |LML〉〈s1ms1s2ms2 |SMS〉φa(r1)χs1,ms1

(1)Ylamla(r1)

φb(r2)χs2,ms2(2)Ylbmlb

(r2) (3.5)

〈x1x2|lalbSL, JM〉 = φa(r1)φb(r2)∑

MSML

〈SMSLML|JM〉

[χs1(1)χs2(2)]SMS[Yla(r1)Ylb(r2)]LML

(3.6)

Finally,

〈x1x2|lalbSL, JM〉 = φa(r1)φb(r2)[

[χs1(1)χs2(2)]S [Yla(r1)Ylb (r2)]L]

JM(3.7)

with x = {r, s} and a = {na, la, ja}.

Non-antisymmetrized normalized two-body wave function with total angular momentum J injj-coupling:

〈x1x2|jajb, JM〉 =∑

ma,mb

〈jamajbmb|JM〉ψama(x1)ψbmb(x2) (3.8)

〈x1x2|jajb, JM〉 = φa(r1)φb(r2)∑

ma,mb

〈jamajbmb|JM〉

[χs1(1)Yla(r1)]jama[χs2(2)Ylb(r2)]jbmb

(3.9)

〈x1x2|jajb, JM〉 = φa(r1)φb(r2)[

[χs1(1)Yla(r1)]ja [χs2(2)Ylb(r2)]jb

]

JM(3.10)

3.2 Angular-momentum operators

The nucleons have an intrinsic spin and (in general) some orbital angular momentum. The angular momen-tum of a nucleus is built from the coupling of these angular momenta. For this reason it is important tostudy the algebra of angular momentum, which is done in this chaper. The notion of angular momentumcan be related to abstract rotations of state vectors in a abstract Hilbert space.

The angular momentum is an operator with three components which (the components) satisfy the fol-lowing commutation rules (this commutation have to be postulated in order to go from the classical conceptof angular momentum over the quantum mechanics one)

[J1, J2] = J1J2 − J2J1 = i!J3 (3.11)

[J2, J3] = i!J1 (3.12)

[J3, J1] = i!J2 (3.13)

these and the other commutation are more generally expreseed using the antisymmetric three-dimensionalLevi-Civita permutation symbol εijk

[Ji, Jj] = i!∑

k

εijkJk (3.14)

with

εijk =

0 if two of the indices are the same1 if the indices are different in cyclic permutations(c.p.):123, 231, 312−1 if the indices are different witn other differen than the above c.p.

(3.15)

We have used the numeration 1, 2, 3 to represent in more compact form the conmuation rules of the componentof the angular momentum. These indexes mean to represent the cartesian x, y, z components.

The orthonormalized eigenstates of the operator J2 are labelled by the quantum numbers (j,m)

J2|jm〉 = (J2x + J2

y + J2z )|jm〉 = j(j + 1)!2|jm〉 (3.16)

Jz |jm〉 = m!|jm〉 (3.17)

19

〈x1x2|lalbSL, JM〉 =∑

MSML

∑

mlamlb

∑

ms1ms2

〈SMSLML|JM〉

〈lamla lbmlb |LML〉〈s1ms1s2ms2 |SMS〉φa(r1)χs1,ms1

(1)Ylamla(r1)

φb(r2)χs2,ms2(2)Ylbmlb

(r2) (3.5)

〈x1x2|lalbSL, JM〉 = φa(r1)φb(r2)∑

MSML

〈SMSLML|JM〉

[χs1(1)χs2(2)]SMS[Yla(r1)Ylb(r2)]LML

(3.6)

Finally,

〈x1x2|lalbSL, JM〉 = φa(r1)φb(r2)[

[χs1(1)χs2(2)]S [Yla(r1)Ylb (r2)]L]

JM(3.7)

with x = {r, s} and a = {na, la, ja}.

Non-antisymmetrized normalized two-body wave function with total angular momentum J injj-coupling:

〈x1x2|jajb, JM〉 =∑

ma,mb

〈jamajbmb|JM〉ψama(x1)ψbmb(x2) (3.8)

〈x1x2|jajb, JM〉 = φa(r1)φb(r2)∑

ma,mb

〈jamajbmb|JM〉

[χs1(1)Yla(r1)]jama[χs2(2)Ylb(r2)]jbmb

(3.9)

〈x1x2|jajb, JM〉 = φa(r1)φb(r2)[

[χs1(1)Yla(r1)]ja [χs2(2)Ylb(r2)]jb

]

JM(3.10)

3.2 Angular-momentum operators

The nucleons have an intrinsic spin and (in general) some orbital angular momentum. The angular momen-tum of a nucleus is built from the coupling of these angular momenta. For this reason it is important tostudy the algebra of angular momentum, which is done in this chaper. The notion of angular momentumcan be related to abstract rotations of state vectors in a abstract Hilbert space.

The angular momentum is an operator with three components which (the components) satisfy the fol-lowing commutation rules (this commutation have to be postulated in order to go from the classical conceptof angular momentum over the quantum mechanics one)

[J1, J2] = J1J2 − J2J1 = i!J3 (3.11)

[J2, J3] = i!J1 (3.12)

[J3, J1] = i!J2 (3.13)

these and the other commutation are more generally expreseed using the antisymmetric three-dimensionalLevi-Civita permutation symbol εijk

[Ji, Jj] = i!∑

k

εijkJk (3.14)

with

εijk =

0 if two of the indices are the same1 if the indices are different in cyclic permutations(c.p.):123, 231, 312−1 if the indices are different witn other differen than the above c.p.

(3.15)

We have used the numeration 1, 2, 3 to represent in more compact form the conmuation rules of the componentof the angular momentum. These indexes mean to represent the cartesian x, y, z components.

The orthonormalized eigenstates of the operator J2 are labelled by the quantum numbers (j,m)

J2|jm〉 = (J2x + J2

y + J2z )|jm〉 = j(j + 1)!2|jm〉 (3.16)

Jz |jm〉 = m!|jm〉 (3.17)

19

Función de onda de dos partículas en acople jj (ver aplicaciones al final)

Sistema Many-Body Finito

Motivación para el uso del álgebra angular

Definición de momento angular y sus autovectores

Operadores Momento Angular

J = (J1, J2, J3)

〈x1x2|lalbSL, JM〉 =∑

MSML

∑

mlamlb

∑

ms1ms2

〈SMSLML|JM〉

〈lamla lbmlb |LML〉〈s1ms1s2ms2 |SMS〉φa(r1)χs1,ms1

(1)Ylamla(r1)

φb(r2)χs2,ms2(2)Ylbmlb

(r2) (3.5)

〈x1x2|lalbSL, JM〉 = φa(r1)φb(r2)∑

MSML

〈SMSLML|JM〉

[χs1(1)χs2(2)]SMS[Yla(r1)Ylb(r2)]LML

(3.6)

Finally,

〈x1x2|lalbSL, JM〉 = φa(r1)φb(r2)[

[χs1(1)χs2(2)]S [Yla(r1)Ylb (r2)]L]

JM(3.7)

with x = {r, s} and a = {na, la, ja}.

Non-antisymmetrized normalized two-body wave function with total angular momentum J injj-coupling:

〈x1x2|jajb, JM〉 =∑

ma,mb

〈jamajbmb|JM〉ψama(x1)ψbmb(x2) (3.8)

〈x1x2|jajb, JM〉 = φa(r1)φb(r2)∑

ma,mb

〈jamajbmb|JM〉

[χs1(1)Yla(r1)]jama[χs2(2)Ylb(r2)]jbmb

(3.9)

〈x1x2|jajb, JM〉 = φa(r1)φb(r2)[

[χs1(1)Yla(r1)]ja [χs2(2)Ylb(r2)]jb

]

JM(3.10)

3.2 Angular-momentum operators

The nucleons have an intrinsic spin and (in general) some orbital angular momentum. The angular momen-tum of a nucleus is built from the coupling of these angular momenta. For this reason it is important tostudy the algebra of angular momentum, which is done in this chaper. The notion of angular momentumcan be related to abstract rotations of state vectors in a abstract Hilbert space.

The angular momentum is an operator with three components which (the components) satisfy the fol-lowing commutation rules (this commutation have to be postulated in order to go from the classical conceptof angular momentum over the quantum mechanics one)

[J1, J2] = J1J2 − J2J1 = i!J3 (3.11)

[J2, J3] = i!J1 (3.12)

[J3, J1] = i!J2 (3.13)

these and the other commutation are more generally expreseed using the antisymmetric three-dimensionalLevi-Civita permutation symbol εijk

[Ji, Jj] = i!∑

k

εijkJk (3.14)

with

εijk =

0 if two of the indices are the same1 if the indices are different in cyclic permutations(c.p.):123, 231, 312−1 if the indices are different witn other differen than the above c.p.

(3.15)

We have used the numeration 1, 2, 3 to represent in more compact form the conmuation rules of the componentof the angular momentum. These indexes mean to represent the cartesian x, y, z components.

The orthonormalized eigenstates of the operator J2 are labelled by the quantum numbers (j,m)

J2|jm〉 = (J2x + J2

y + J2z )|jm〉 = j(j + 1)!2|jm〉 (3.16)

Jz |jm〉 = m!|jm〉 (3.17)

19

〈x1x2|lalbSL, JM〉 =∑

MSML

∑

mlamlb

∑

ms1ms2

〈SMSLML|JM〉

〈lamla lbmlb |LML〉〈s1ms1s2ms2 |SMS〉φa(r1)χs1,ms1

(1)Ylamla(r1)

φb(r2)χs2,ms2(2)Ylbmlb

(r2) (3.5)

〈x1x2|lalbSL, JM〉 = φa(r1)φb(r2)∑

MSML

〈SMSLML|JM〉

[χs1(1)χs2(2)]SMS[Yla(r1)Ylb(r2)]LML

(3.6)

Finally,

〈x1x2|lalbSL, JM〉 = φa(r1)φb(r2)[

[χs1(1)χs2(2)]S [Yla(r1)Ylb (r2)]L]

JM(3.7)

with x = {r, s} and a = {na, la, ja}.

Non-antisymmetrized normalized two-body wave function with total angular momentum J injj-coupling:

〈x1x2|jajb, JM〉 =∑

ma,mb

〈jamajbmb|JM〉ψama(x1)ψbmb(x2) (3.8)

〈x1x2|jajb, JM〉 = φa(r1)φb(r2)∑

ma,mb

〈jamajbmb|JM〉

[χs1(1)Yla(r1)]jama[χs2(2)Ylb(r2)]jbmb

(3.9)

〈x1x2|jajb, JM〉 = φa(r1)φb(r2)[

[χs1(1)Yla(r1)]ja [χs2(2)Ylb(r2)]jb

]

JM(3.10)

3.2 Angular-momentum operators

The nucleons have an intrinsic spin and (in general) some orbital angular momentum. The angular momen-tum of a nucleus is built from the coupling of these angular momenta. For this reason it is important tostudy the algebra of angular momentum, which is done in this chaper. The notion of angular momentumcan be related to abstract rotations of state vectors in a abstract Hilbert space.

The angular momentum is an operator with three components which (the components) satisfy the fol-lowing commutation rules (this commutation have to be postulated in order to go from the classical conceptof angular momentum over the quantum mechanics one)

[J1, J2] = J1J2 − J2J1 = i!J3 (3.11)

[J2, J3] = i!J1 (3.12)

[J3, J1] = i!J2 (3.13)

these and the other commutation are more generally expreseed using the antisymmetric three-dimensionalLevi-Civita permutation symbol εijk

[Ji, Jj] = i!∑

k

εijkJk (3.14)

with

εijk =

0 if two of the indices are the same1 if the indices are different in cyclic permutations(c.p.):123, 231, 312−1 if the indices are different witn other differen than the above c.p.

(3.15)

We have used the numeration 1, 2, 3 to represent in more compact form the conmuation rules of the componentof the angular momentum. These indexes mean to represent the cartesian x, y, z components.

The orthonormalized eigenstates of the operator J2 are labelled by the quantum numbers (j,m)

J2|jm〉 = (J2x + J2

y + J2z )|jm〉 = j(j + 1)!2|jm〉 (3.16)

Jz |jm〉 = m!|jm〉 (3.17)

19

Levi-Civita

ϵ121 = ϵ112 = ϵ212 = … = 0

Relaciones de conmutación

ϵ123 = ϵ231 = ϵ312 = 1ϵ132 = ϵ321 = ϵ213 = − 1

Ejemplo

[J1, J2] = iℏ3

∑k=1

ϵ12kJk

[J1, J2] = iℏJ3

Autovalores y autovectores

j: entero o semi-entero

J = (Jx, Jy, Jz)

J2 | jm⟩ = ℏ2j( j + 1) | jm⟩Jz | jm⟩ = ℏm | jm⟩

m=-j, -j+1,…, j-1, j

Autovalores de J2

Autovectores

Autovalores de Jz

| jm⟩

Operadores

J2 = J2x + J2

y + J2z

Autovalores y autovectores

J = (Jx, Jy, Jz) Autovectores

| jm⟩

Operadores

J2 = J2x + J2

y + J2z

I = ∑jm

| jm⟩⟨jm |

⟨jm | j′ m′ ⟩ = δjj′ δmm′

Ortonormalidad y completitud| f ⟩ ⟶ | f ⟩ = I | f ⟩

| f ⟩ = ∑lml

| lml⟩⟨lml | f ⟩

Base

| f ⟩ = ∑lml

flm | lml⟩

Ejemplo: momento angular orbital

l = (lx, ly, lz) → l2 = l2x + l2

y + l2z

l2 | lml⟩ = ℏ2l(l + 1) | lml⟩

lz | lml⟩ = ℏml | lml⟩ ml = − l, − l + 1, ⋯, 0,⋯, l

l = 0, 1, 2, ⋯

Representación abstracta

Autovectores | lml⟩

Ejemplo: momento angular orbital

∫ dΩ⟨lml |θϕ⟩⟨θϕ | l′ m′ l⟩

⟨θϕ | lml⟩ = Ylml(θ, ϕ)

Representación coordenadas

⟨lm | l′ m′ ⟩ = δll′ δmm′ →

| lml⟩ ⟶

Ortogonalidad

= ∫ dΩY*lml(θϕ)Yl′ m′ l

(θϕ) = δll′ δmlm′ l

Armónicos esféricos

Ejemplo: momento angular orbital

I = ∑lml

| lml⟩⟨lml | ⟶

⟨θϕ | lml⟩ = Ylml(θ, ϕ)

⟨θϕ | I |θ′ ϕ′ ⟩ = ∑lml

⟨θϕ | lml⟩⟨lml |θ′ ϕ′ ⟩

⟨θϕ |θ′ ϕ′ ⟩ = δ(θ − θ′ )δ(ϕ − ϕ′ )

Representación coordenadas

| lml⟩ ⟶

Completitud

= ∑lml

Ylml(θϕ)Y*lml

(θ′ ϕ′ )

Originalidad en el espacio de coordenada

Ejemplo: momento angular orbital

I = ∑lml

| lml⟩⟨lml | →

⟨θϕ | lml⟩ = Ylml(θ, ϕ)

Representación coordenadas

| lml⟩ ⟶

Base

| f ⟩ = ∑lml

| lml⟩⟨lml | f ⟩ = ∑lml

flm | lml⟩

→ ⟨θϕ | f ⟩ = ∑lml

flm⟨θϕ | lml⟩ = ∑lml

flmYlml(θ, ϕ)

→ f(θ, ϕ) = ∑lml

flmYlml(θ, ϕ) Comparar con la versión abstracta

| f ⟩ = ∑lml

flm | lml⟩

Operadores de crecimiento

Operador de crecimientoRaising and lowering operators

[J+, J−] = 2ℏJ3

J+ = J1 + i J2

J− = J1 − i J2

Relación de conmutación

J = (J1, J2, J3)

( | jm⟩)

Veamos… [J+, J−] = [J1 + i J2, J1 − i J2] =

= [J1, J1] − i[J1, J2] + i[J2, J1] − i2[J2, J2] =

= 0 − i(iℏJ3) + i(−iℏJ3) − 0 == − 2i2ℏJ3 = 2ℏJ3

Operador de crecimientoRaising and lowering operators

[J+, J−] = 2ℏJ3

J+ = J1 + i J2

J− = J1 − i J2

[J+, J3] = − ℏJ+

[J−, J3] = ℏJ−

[J+, J2] = 0

Relación de conmutación

( | jm⟩)

[J−, J2] = 0

J = (J1, J2, J3)

Uso de los operadores de crecimiento

J2(J+ | jm⟩) = ℏ2j( j + 1)(J+ | jm⟩)

[J+, Jz] = − ℏJ+

J+ = Jx + i Jy J− = Jx − i Jy

J = (Jx, Jy, Jz) J2 | jm⟩ = ℏ2j( j + 1) | jm⟩

[J±, J2] = 0

Generación del autovector (j,m+1):

Jz(J+ | jm⟩) = ℏ(m + 1)(J+ | jm⟩)

Veamos…

Veamos… consideremos el vector y veamos que es autovector de y

J+ | jm⟩J2 Jz

Uso de los operadores de crecimiento

J+ = Jx + i Jy J− = Jx − i Jy

J = (Jx, Jy, Jz) J2 | jm⟩ = ℏ2j( j + 1) | jm⟩

Generación del autovector (j,m+1):…luego…

J+ | jm⟩ = ℏ j( j + 1) − m(m + 1) | jm + 1⟩

J− | jm⟩ = ℏ j( j + 1) − m(m − 1) | jm − 1⟩

Condon-Shortley phase convention assumed (ver más adelante)

Deducción de las matrices de Pauli

… ver notas de clase…

Deducción de las matrices de Pauli

σz = (1 00 −1)

σx = (0 11 0)

σy = (0 −ii 0 )

S2 |sms⟩ = ℏ2s(s + 1) |sms⟩

Sz |sms⟩ = ℏms |sms⟩

S = (Sx, Sy, Sz)Espín

Autovectores

s =12

ms = −12

,12

σ = (σx, σy, σz)

S =ℏ2

σ

Matriz S_z

Sz |sms⟩ = ℏms |sms⟩

S = (Sx, Sy, Sz)S =ℏ2

σ σ = (σx, σy, σz)

Sz =⟨ 1

2 , 12 |Sz | 1

2 , 12 ⟩ ⟨ 1

2 , 12 |Sz | 1

2 , − 12 ⟩

⟨ 12 , − 1

2 |Sz | 12 , 1

2 ⟩ ⟨ 12 , − 1

2 |Sz | 12 , − 1

2 ⟩

Sz =ℏ2 0

0 − ℏ2

=ℏ2

σz σz = (1 00 −1)

⟨sms |s′ m′ s⟩ = δss′ δmsm′ S

Matriz S_x

S = (Sx, Sy, Sz)S =ℏ2

σ σ = (σx, σy, σz)

Sx =⟨ 1

2 , 12 |Sx | 1

2 , 12 ⟩ ⟨ 1

2 , 12 |Sx | 1

2 , − 12 ⟩

⟨ 12 , − 1

2 |Sx | 12 , 1

2 ⟩ ⟨ 12 , − 1

2 |Sx | 12 , − 1

2 ⟩

Sx =0 ℏ

2ℏ2 0

=ℏ2

σx σx = (0 11 0)

⟨sms |s′ m′ s⟩ = δss′ δmsm′ S

S+ = Sx + i SyS− = Sx − i Sy

Sx =S+ + S−

2S± |s, ms⟩ = ℏ s(s + 1) − ms(ms ± 1) |s, ms ± 1⟩

Matriz S_y

S = (Sx, Sy, Sz)S =ℏ2

σ σ = (σx, σy, σz)

Sy =⟨ 1

2 , 12 |Sy | 1

2 , 12 ⟩ ⟨ 1

2 , 12 |Sy | 1

2 , − 12 ⟩

⟨ 12 , − 1

2 |Sy | 12 , 1

2 ⟩ ⟨ 12 , − 1

2 |Sy | 12 , − 1

2 ⟩

Sy = − i0 ℏ

2

− ℏ2 0

=ℏ2

σy σy = (0 −ii 0 )

S+ = Sx + i SyS− = Sx − i Sy

Sy =S+ − S−

2iS± |s, ms⟩ = ℏ s(s + 1) − ms(ms ± 1) |s, ms ± 1⟩

Suma de momentos angulares

Suma de dos momentos angulares

j21 | j1m1⟩ = j1( j1 + 1)ℏ2 | j1m1⟩

[j1, j2] = 0

j1,z | j1m1⟩ = m1ℏ | j1m1⟩m1 = − j1, ⋯, j1

Momentos angulares j1, j2

j22 | j2m2⟩ = j2( j2 + 1)ℏ2 | j2m2⟩

j2,z | j2m2⟩ = m2ℏ | j2m2⟩m2 = − j2, ⋯, j2

{ | j1m1⟩, | j2m2⟩}

Suma de dos momentos angulares

Composición (suma)

J = (Jx, Jy, Jz)

j21 | j1m1⟩ = j1( j1 + 1)ℏ2 | j1m1⟩

[j1, j2] = 0j1,z | j1m1⟩ = m1ℏ | j1m1⟩

J = j1 + j2

Momentos angulares j1, j2

Jx = j1,x + j2,x

Jy = j1,y + j2,y

Jz = j1,z + j2,z

j22 | j2m2⟩ = j2( j2 + 1)ℏ2 | j2m2⟩

j2,z | j2m2⟩ = m2ℏ | j2m2⟩{ | j1m1⟩, | j2m2⟩}

Suma de dos momentos angulares

Momento angular total

J = (Jx, Jy, Jz)

[j1, j2] = 0

J = j1 + j2

Momentos angulares j1, j2

Jx = j1,x + j2,x

Jy = j1,y + j2,y

Jz = j1,z + j2,z

{ | j1m1⟩, | j2m2⟩}

j2k | jkmk⟩ = jk( jk + 1)ℏ2 | jkmk⟩

jk,z | jkmk⟩ = mkℏ | jkmk⟩k = 1,2

[Jα, Jβ] = i ℏ∑γ

ϵαβγJγ

α, β, γ = (x, y, z)

ϵ123 = ϵ231 = ϵ312 = 1ϵ132 = ϵ321 = ϵ213 = − 1

Recordemos Levi-Civita

ϵαβγ = 0 resto

Autovectores acoplados y

desacoplados

Base desacoplada

Base

j2k | jkmk⟩ = jk( jk + 1)ℏ2 | jkmk⟩

[j1, j2] = 0jk,z | jkmk⟩ = mkℏ | jkmk⟩J = j1 + j2

= δj1 j′ 1δm1m′ 1

δj2 j′ 2δm2m′ 2

k = 1,2

j1, j2

Momentos angulares

| j1m1 j2m2⟩ = | j1m1⟩ | j2m2⟩

mk = − jk, ⋯, jk

I = ∑m1m2

| j1m1 j2m2⟩⟨ j1m1 j2m2 |

{ | j1m1 j2m2⟩}

Ortogonalidad

Completitud

⟨j1m1 j2m2 | j′ 1m′ 1 j′ 2m′ 2⟩ =

Base desacoplada

Base

j2k | jkmk⟩ = jk( jk + 1)ℏ2 | jkmk⟩

[j1, j2] = 0jk,z | jkmk⟩ = mkℏ | jkmk⟩J = j1 + j2

k = 1,2j1, j2

Momentos angulares

Sistema completo de operadores que conmutan:

{j21, j2

2, j1,z, j2,z}

{ | j1m1 j2m2⟩ = | j1m1⟩ | j2m2⟩}

Base acoplada

[j1, j2] = 0

J = j1 + j2

j1, j2

Momentos angulares

Base { | j1j2 jm⟩}

| j1 − j2 | ≤ j ≤ j1 + j2

m = − j, − j + 1, ⋯, j − 1, j

⟨ j1 j2 jm | j1 j2 j′ m′ ⟩ = δjj′ δmm′

I = ∑jm

| j1 j2 jm⟩⟨ j1 j2 jm |

Ortogonalidad y completitud

J

j2j1

Base acoplada

[j1, j2] = 0

J = j1 + j2

j1, j2

Momentos angulares

j2k | j1 j2 jm⟩ = jk( jk + 1)ℏ2 | j1 j2 jm⟩

Jz | j1 j2 jm⟩ = mℏ | j1 j2 jm⟩

J2 | j1 j2 jm⟩ = j( j + 1)ℏ2 | j1 j2 jm⟩Notar que hay cuatro ecuaciones de autovectores como antes

j2k | jkmk⟩ = jk( jk + 1)ℏ2 | jkmk⟩

jk,z | jkmk⟩ = mkℏ | jkmk⟩ k = 1,2

k = 1,2

Base { | j1j2 jm⟩}

| j1 − j2 | ≤ j ≤ j1 + j2m = − j, − j + 1, ⋯, j − 1, j

Autovalores y autovectores

J

j2j1

Base acoplada

[j1, j2] = 0

J = j1 + j2

j1, j2

Momentos angulares

j2k | j1 j2 jm⟩ = jk( jk + 1)ℏ2 | j1 j2 jm⟩

Jz | j1 j2 jm⟩ = mℏ | j1 j2 jm⟩

J2 | j1 j2 jm⟩ = j( j + 1)ℏ2 | j1 j2 jm⟩

k = 1,2

Base { | j1j2 jm⟩}

Sistema completo de operadores que conmutan:

{j21, j2

2, J2, Jz}

J

j2j1

Construcción de vectores acoplados

[j1, j2] = 0 J = j1 + j2j1, j2

Momentos angulares

J

j2j1

Base desacoplada

{ | j1m1 j2m2⟩}Base acoplada

{ | j1 j2 jm⟩}

I = ∑m1m2

| j1m1 j2m2⟩⟨ j1m1 j2m2 | I = ∑jm

| j1 j2 jm⟩⟨ j1 j2 jm |

CompletitudCompletitud

Veamos…

Construcción de vectores acoplados

[j1, j2] = 0 J = j1 + j2j1, j2

Momentos angulares

J

j2j1Base desacoplada

{ | j1m1 j2m2⟩}

Base acoplada

| j1 j2 jm⟩ = I | j1 j2 jm⟩

= ∑m1m2

| j1m1 j2m2⟩⟨ j1m1 j2m2 | j1 j2 jm⟩

{ | j1 j2 jm⟩}

← I = ∑m1m2

| j1m1 j2m2⟩⟨ j1m1 j2m2 |

⟨ j1m1 j2m2 | jm⟩ ≡ ⟨ j1m1 j2m2 | j1 j2 jm⟩Coeficientes de Clebsch-Gordan= ∑

m1m2

⟨ j1m1 j2m2 | jm⟩ | j1m1 j2m2⟩

Propiedades de los Clebsch-Gordan

J

j2 j1

⟨ j1m1 j2m2 | jm⟩ = 0 m1 + m2 ≠ m

| j1 − j2 | ≤ j ≤ j1 + j2

m = − j, − j + 1, ⋯, j − 1, j

⟨ j1m1 j2m2 | jm⟩ = ( − ) j1+j2−j⟨ j2m2 j1m1 | jm⟩

J

j2j1

J

j2j1= ( − ) j1+j2−j

Propiedades de los Clebsch-Gordan

J

j2j1

i) The projection quantum numbers have to fulfill the addition law 〈j1m1j2m2|jm〉 = 0 unless m1+m2 = m.

ii) The coupled angular momenta have to fulfill the triangular condition |j1 − j2| ≤ j ≤ j1 + j2, denoted as∆(j1j2j).

iii) The allowed values of the total angular momentum comes from the relation j1 + j2 + j =integer, i.e.j = |j1 − j2|, |j1 − j2|+ 1, · · · , j1 + j2 − 1, j1 + j2, with j1 and j2 integer or half-integer.

iv) The Clebsch-Gordan coefficients are chosen to be real, and so that 〈j1j1j2j2|j1 + j2j1 + j2〉 = +1,〈j1m1j2 − j2|jm〉 ≥ 0.

These conditions fix the phases of all Clebsch-Gordan coefficients.Clebsh-Gordan coefficients are orthogonal (demonstrate it)

∑

m1m2

〈j1m1j2m2|jm〉〈j1m1j2m2|j′m′〉 = δjj′δmm′ (2.76)

and complete∑

jm

〈j1m1j2m2|jm〉〈j1m′1j2m

′2|jm〉 = δm1m′

1δm2m′

2(2.77)

The uncouple states |j1m1j2m2〉 in term of the coupled basis are give by (demostrar: multiplicar por∑

jm〈j1m′1j2m

′2|jm〉 y usar 2.77)

|j1m1j2m2〉 =∑

jm

〈j1m1j2m2|jm〉|j1j2jm〉 (2.78)

Racah [?] showed that the Clebsch-Gordan coefficients can be written in the form of a finite series,

〈j1m1j2m2|j3m3〉 = j3

√

(j1 + j2 − j3)!(j1 − j2 + j3)!(−j1 + j2 + j3)!

(j1 + j2 + j3 + 1)!√

√

√

√

3∏

i=1

(ji +mi)!(ji −mi)!

∑

k

(−)k

k!(j1 + j2 − j3 − k)!(j1 −m1 − k)!(j2 +m2 − k)!

1

(j3 − j2 +m1 + k)!(j3 − j1 −m2 + k)!(2.79)

Some symmetry properties of the Clebsch-Gordan coefficients (from Ref. [?, 11]):

1. 〈j1m1j2m2|jm〉 = (−)j1+j2−j〈j2m2j1m1|jm〉

2. 〈j1m1j2m2|jm〉 = (−)j1+j2−j〈j1 −m1j2 −m2|j −m〉

3. 〈j10j20|j0〉 = 0 unless j1 + j2 + j =even.

4. 〈j1m1j2m2|jm〉 = (−)j1−m1 jj2〈j1m1j −m|j2 −m2〉

5. 〈jmj −m|00〉 = (−)j−m

j

6. 〈jm00|jm〉 = 1

where j =√2j + 1

2.3.2 Spin-Spin coupling

• Desarrollar en la clase.

• Dar el desarrollo formal del singlete y el triplete

• Dar la forma exlicita del singlete para mostrar que la combinacion antisimetrica de los espines.

χSMS (σ1σ2) =[

χ1/2(σ1)χ1/2(σ2)]

SMS(2.80)

21

j = 2j + 1

| j1 − j2 | ≤ j ≤ j1 + j2

m = − j, − j + 1, ⋯, j − 1, j

Aplicación: partícula con spin

Uso del álgebra angular Función de onda desacoplada

⟨ r | lml⟩ = Ylml( r) ⟶ | lml⟩

⟨σ |sms⟩ = χsms(σ) ⟶ |sms⟩

Kets

ϕnlmlms(r, s) = Rnl(r) Ylml

( r) χsms

⟨r |nl⟩ = Rnl(r) ⟶ |nl⟩

Base desacoplada

{ |nlmlsms⟩ = |nl⟩ | lml⟩ |sms⟩}

Momentos angulares orbital: l

⟨ r | lml⟩ = Ylml( r) ⟶ | lml⟩

Kets

ϕnlmlms(r, s) = Rnl(r) Ylml

( r) χsms

l 2 | lml⟩ = l(l + 1) | lml⟩

lz | lml⟩ = ml | lml⟩

(ℏ = 1)

Uso del álgebra angular Función de onda desacoplada

ml = − l, ⋯, l

ϕnlmlms(r, s) = ⟨r |nlmlsms⟩

Momentos angulares orbital: l

⟨ r | lml⟩ = Ylml( r) ⟶ | lml⟩

⟨σ |sms⟩ = χsms(σ) ⟶ |sms⟩

Kets

ϕnlmlms(r, s) = Rnl(r) Ylml

( r) χsms

l2 | lml⟩ = l(l + 1) | lml⟩lz | lml⟩ = ml | lml⟩

s2 |sms⟩ = x(s + 1) |sms⟩ =34

|sms⟩

sz |sms⟩ = ms |sms⟩

(ℏ = 1)

Spin: s

ms = −12

,12

(ℏ = 1)

Uso del álgebra angular Función de onda desacoplada

| l − s | ≤ j ≤ l + s ⟶

Acople s-l

s l

j

ψnljm(r, s) = Rnlj(r)[χsYl(θ, ϕ)]jm

[χsYl(θ, ϕ)]jm = ∑ms,ml

⟨smslml | jm⟩χsmsYlml

(θ, ϕ)

m = − j, − j + 1,⋯, j − 1,j

[χsYl(θ, ϕ)]jm = 𝒴ljm( r)

j = l ± 12

Uso del álgebra angular Función de onda acoplada

Kets

⟨ r |sl, jm⟩ = 𝒴ljm( r) ⟶ |sl, jm⟩

Base acoplada

⟨r |nlj⟩ = Rnl(r) → |nlj⟩

ψnljm(r) = ⟨r |nljm⟩ ⟶ |nljm⟩

{ |nljm⟩ = |nlj⟩ | ljm⟩}

s l

j

ψnljm(r, s) = Rnlj(r)[χsYl(θ, ϕ)]jm

Uso del álgebra angular Función de onda acoplada

| ljm⟩ ≡ |sl, jm⟩

l2 |nljm⟩ = l(l + 1) |nljm⟩

s2 |nljm⟩ = s(s + 1) |nljm⟩ =34

|nljm⟩

j2 |nljm⟩ = j( j + 1) |nljm⟩

jz |nljm⟩ = m |nljm⟩

ψnljm(r) = ⟨r |nljm⟩

Autovectores

s l

j

ψnljm(r, s) = Rnlj(r)[χsYl(θ, ϕ)]jm

Uso del álgebra angular Función de onda acoplada

{l2, s2, j2, jz}

ψnljm(r) = ⟨r |nljm⟩

Sistema completo de observables que conmutan

ψnljm(r, s) = Rnlj(r)[χsYl(θ, ϕ)]jm

Uso del álgebra angular Funciones de onda desacoplada y acoplada

ϕnlmlms(r, s) = Rnl(r) Ylml

( r) χsms

ϕnlmlms(r, s) = ⟨r |nlmlsms⟩

Función de onda desacoplada Función de onda acoplada

Base desacoplada Base acoplada

{l2, s2, lz, mz}

l2 |nljm⟩ = l(l + 1) |nljm⟩s2 |nljm⟩ = s(s + 1) |nljm⟩

j2 |nljm⟩ = j( j + 1) |nljm⟩

jz |nljm⟩ = m |nljm⟩

Base acoplada

l

Acople s-l

s

j

j2 = s2 + l2 + 2l ⋅ s

l ⋅ s =j2 − s2 − l2

2

[l, s] = 0 l ⋅ s |nljm⟩ =j( j + 1) − l(l + 1) − 3/4

2|nljm⟩

Uso del álgebra angular Autovalores del producto escalar l ⋅ s

s2k |skmsk

⟩ =34

|skmsk⟩s1 s2

Ssk,z |skmsk

⟩ = mk |skmsk⟩

k = 1,2

0 ≤ S ≤ 1 MS = − S,0,S

⟨σ |sms⟩ = χsms(σ)

|SMS⟩ = ∑ms1

,ms2

⟨s1ms1s2ms2

|SMS⟩ |s1ms1⟩ |s2ms2

⟩

⟨σ1σ2 |SMS⟩ = ∑ms1

,ms2

⟨s1ms1s2ms2

|SMS⟩⟨σ1 |s1ms1⟩⟨σ2 |s2ms2

⟩

χSMS(σ1, σ2) = ∑

ms1,ms2

⟨s1ms1s2ms2

|SMS⟩χs1ms1(σ1)χs2ms2

(σ2)

S = 0,1

Acople de dos fermiones

s1 s2

SMS = 0

χ0,0(σ1, σ2) = ∑ms1

,ms2

⟨s1ms1s2ms2

|0,0⟩χs1ms1(σ1)χs2ms2

(σ2)

SingleteS = 0 ⟨s1ms1

s2ms2|0,0⟩ = δs1,s2

δm1,−m2

( − )s1−m1

s1

s = 2s + 1 = 2

χ0,0(σ1, σ2) =1

2(χs,1/2(σ1)χs,−1/2(σ2) − χs,1/2(σ2)χs,−1/2(σ1)) Impar

σ1 ⇆ σ2

TripletePar

σ1 ⇆ σ2MS = 0S = 1

Acople de tres y cuatros momentos

angulares

J1

J2

J

Acople de tres momentos angulares

J12J3

(jm)

j_1

j_2j_3

j_12

Figure 2.2: Coupling of three angular momenta.

3. The number of linearly independent states in all three coupling is the same and each of the set form acomplete set

I =∑

J12

|j1j2(j12)j3; jm〉〈j1j2(j12)j3; jm| (2.86)

I =∑

J23

|j1j2j3(j23); jm〉〈j1j2j3(j23); jm| (2.87)

I =∑

J13

|j1j3(j13)j2; jm〉〈j1j3(j13)j2; jm| (2.88)

As for the case of two angular momenta, one can change from one basis to the other, for example thechange for the basis |j1j2(j12)j3; jm〉 to the basis |j1j2j3(j23); jm〉 is given by

|j1j2j3(j23); jm〉 =∑

J12

|j1j2(j12)j3; jm〉〈j1j2(j12)j3; jm|j1j2j3(j23); jm〉

=∑

J12

(−)j1+j2+j3+j j12j23

{

j1 j2 j12j3 j j23

}

|j1j2(j12)j3; jm〉

where the six j enclosed in the braces {· · · } is called the 6j symbol. It is proportional to the overlap of twostate vectors related to two different coupling schemes of three angular momenta. They are members of theunitary transformation since the norm of the states are preserved. 6j symbols are related to a transformationbetween two basis sets where all the states have a good total angular momentum.

The following explicit expression of the 6j symbol can be obtained from is definition in the basis trans-formation

{

j1 j2 j12j3 j j23

}

=∑

m1m2m3

∑

m12m23m

(−)j3+j+j23−m3−m−m23

(

j1 j2 j12m1 m2 m12

)(

j1 j j23m1 −m m23

)

(

j3 j2 j23m3 m2 −m23

)(

j3 j j13−m3 m m12

)

(2.89)

Symmetry:

1. The angular momenta involved in a 6j symbol have to satisfy certain triangular conditions given by(analysis theses conditions with the above expression)

{

j1 j2 j3l1 l2 l3

}

= 0 unless

∆(j1j2j3)∆(l1l2j3)∆(l1j2l3)∆(j1l2l3)

(2.90)

2. Exchange of any two columns leave the value of the symbol unchanged:

{

j1 j2 j3l1 l2 l3

}

=

{

j2 j1 j3l2 l1 l3

}

=

{

j3 j1 j2l3 l1 l2

}

= · · · (2.91)

23

2.3.3 Singlet particle wave function

The single particle wave function is written as

φslmlms(rσ) = R(r) Ylml(r) χsms(σ) (2.81)

The couple single particle wave function reads

ψsljmj (rσ) = R(r) Yljmj (rσ) (2.82)

with (dar los detalles en la clase)Yljmj (rσ) = [Yl(r)χs(σ)]jm (2.83)

where [. . . ]jm means couple to jm. Definir el rango de j

Utilidad autovalor |jm〉: Calcular la accion de l.s sobre Yljmj (rσ).

2.3.4 The Wigner 3j symbol

The phase factor which appears in the firs two previous relations can be done more symmetric by definingthe so-called 3j symbols,

(

j1 j2 j3m1 m2 m3

)

=(−)j1−j2−m3

j3〈j1m1j2m2|j3 −m3〉 (2.84)

The Clebsch-Gordan in terms of the 3j reads

〈j1m1j2m2|j3m3〉 = (−)j2−j1−m3 j3

(

j1 j2 j3m1 m2 −m3

)

(2.85)

The following are its basic symmetric properties (from Ref. [3, 6])

•(

j1 j2 j3m1 m2 m3

)

=

(

j2 j3 j1m2 m3 m1

)

=

(

j3 j1 j2m3 m1 m2

)

= (−)j1+j2+j3

(

j1 j3 j2m1 m3 m2

)

•(

j1 j2 j3−m1 −m2 −m3

)

= (−)j1+j2+j3

(

j1 j2 j3m1 m2 m3

)

•(

j1 j2 j3m1 m2 m3

)

= 0 unless

{

∆(j1j2j3)m1 +m2 +m3 = 0

•(

j1 j2 0m1 m2 0

)

= (−)j1−m1

j1δj1j2δm1−m2

•(

j1 j2 j30 0 0

)

= 0 unless j1 + j2 + j3 =even.

where ∆(j1j2j3) denotes the triangular condition |j1 − j2| ≤ j3 ≤ j1 + j2

2.4 Coupling of three angular momenta. The Wigner 6j symbol

The coupling of three commuting angular momentum vectors J1, J2 and J3 so that J = J1 + J2 + J3 isthe total angular momentum can be done in the following three different ways

• J12 = J1 + J2, J = J12 + J3

• J23 = J2 + J3, J = J23 + J1

• J13 = J1 + J3, J = J13 + J2

with the following properties

1. The values of the quantum number j (corresponding to J) do not depend on the coupling order

2. The states corresponding to different coupling schemes are no the same

22

Opciones de acoples

Bases

Cambio de base

J1

J2

J

J12

J3

(jm)

j_1

j_2j_3

j_12

Figure 2.2: Coupling of three angular momenta.

3. The number of linearly independent states in all three coupling is the same and each of the set form acomplete set

I =∑

J12

|j1j2(j12)j3; jm〉〈j1j2(j12)j3; jm| (2.86)

I =∑

J23

|j1j2j3(j23); jm〉〈j1j2j3(j23); jm| (2.87)

I =∑

J13

|j1j3(j13)j2; jm〉〈j1j3(j13)j2; jm| (2.88)

As for the case of two angular momenta, one can change from one basis to the other, for example thechange for the basis |j1j2(j12)j3; jm〉 to the basis |j1j2j3(j23); jm〉 is given by

|j1j2j3(j23); jm〉 =∑

J12

|j1j2(j12)j3; jm〉〈j1j2(j12)j3; jm|j1j2j3(j23); jm〉

=∑

J12

(−)j1+j2+j3+j j12j23

{

j1 j2 j12j3 j j23

}

|j1j2(j12)j3; jm〉

where the six j enclosed in the braces {· · · } is called the 6j symbol. It is proportional to the overlap of twostate vectors related to two different coupling schemes of three angular momenta. They are members of theunitary transformation since the norm of the states are preserved. 6j symbols are related to a transformationbetween two basis sets where all the states have a good total angular momentum.

The following explicit expression of the 6j symbol can be obtained from is definition in the basis trans-formation

{

j1 j2 j12j3 j j23

}

=∑

m1m2m3

∑

m12m23m

(−)j3+j+j23−m3−m−m23

(

j1 j2 j12m1 m2 m12

)(

j1 j j23m1 −m m23

)

(

j3 j2 j23m3 m2 −m23

)(

j3 j j13−m3 m m12

)

(2.89)

Symmetry:

1. The angular momenta involved in a 6j symbol have to satisfy certain triangular conditions given by(analysis theses conditions with the above expression)

{

j1 j2 j3l1 l2 l3

}

= 0 unless

∆(j1j2j3)∆(l1l2j3)∆(l1j2l3)∆(j1l2l3)

(2.90)

2. Exchange of any two columns leave the value of the symbol unchanged:

{

j1 j2 j3l1 l2 l3

}

=

{

j2 j1 j3l2 l1 l3

}

=

{

j3 j1 j2l3 l1 l2

}

= · · · (2.91)

23

Coeficiente de Wigner 6j

= ∑J12

⟨j1 j2( j12)j3; jm | j1 j2 j3( j23); jm⟩J1

J2

J

J23 J3

Acople de cuatro momentos angulares

Acople jj

s1 l1 l2

j2

[s1l1]j1 [s2l2]j2

[ j1 j2]J

s2

j1

s2

s1

l1

l2j2

J

j1

Partícula 2: (s2, l2)(s1, l1)Partícula 1:

Acople de cuatro momentos angulares

s1 l1 l2

[SL]J

s2

S

Acople SL

L

J

s1

s2

S

l1

l2L

Partícula 2: (s2, l2)(s1, l1)Partícula 1:

Acople de cuatro momentos angulares

s1 l1 l2

[SL]JM

s2

S

Acople SL

L

J

s1

s2

S

l1

l2L

Acople jj

l1

j2

[s1l1]j1 [s2l2]j2

s2

j1

s2

s1

l1

l2j2

J

j1

[ j1 j2]JM

Partícula 2: (s2, l2)(s1, l1)Partícula 1:

Acople jj Acople SL

Acople jj con funciones de onda

Acople jj

s1 l1 l2

j2

[s1l1]j1 [s2l2]j2

[ j1 j2]J

s2

j1

s2

s1

l1

l2j2

J

j1

(s1, l1)Partícula 1: (s2, l2)Partícula 2:

[ χs1Yl1(θ1, ϕ1)]j1m1

= ∑ms1

,ml1

⟨s1ms1l1ml1 | j1m1⟩χs1ms1

Yl1ml1(θ1, ϕ1)

[ χs2Yl2

(θ2, ϕ2)]j2m2= ∑

ms2,ml2

⟨s2ms2l2ml2

| j2m2⟩χs2ms2Yl2ml2

(θ2, ϕ2)

⟨ r1 r2 | j1 j2, jm⟩ = ∑m1,m2

⟨ j1m1 j2m2 | jm⟩[ χs1Yl1]j1m1

[ χs2Yl2

]j2m2

s1 l1 l2

[SL]J

s2

S

(s1, l1)Partícula 1: (s2, l2)Partícula 2:

Acople SL

L

J

s1

s2

S

l1

l2L

Acople SL con funciones de onda

χSMS(σ1, σ2) = ∑

ms1,ms2

⟨s1ms1s2ms2

|SMS⟩χs1ms1(σ1)χs2ms2

(σ2)

YLML( r1, r2) = ∑

ml1,ml2

⟨l1ml1l2ml2|LML⟩Yl1ml1

( r1)Yl2ml2( r2)

⟨ r1 r2 |SL, jm⟩ = ∑MS,ML

⟨SMSLML | jm⟩χSMSYLML

Cambio de base SL —> jj

Coeficiente de Wigner 9j

= ∑S,L

⟨s1l1( j1), s2l2( j2); jm |s1s2(S)l1l2(L); jm⟩

J

s1

s2

S

l1

l2Lj1

s2

s1

l1

l2j2

J

|s1l1( j1), s2l2( j2); jm⟩|s1s2(S)l1l2(L); jm⟩

|s1l1( j1), s2l2( j2); jm⟩ = ∑S,L

j1j2

SLs1 s2 Sl1 l2 Lj1 j2 j

|s1s2(S)l1l2(L); jm⟩

Preguntas + Fin