Álgebra de momentos angulares - unridbetan/cursonuclear... · 2020. 10. 6. · the angular...
TRANSCRIPT
Rodolfo M. Id Betan (Rolo) [email protected]
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