Quantum Mechanics II angular Momentum II : Addition of Angular Momentum Clebsch-Gordan Coefficients

J^→ = J^→₁ ⊗ I ⊕ I ⊗ J^→₂

abbreviated as


J^ˆ = J^ˆ₁ + J^ˆ₂

[The hat or the vector signs (which is more commonly used) indicating the operator nature of angular momentum will be frequently omitted. The commutation relations satisfied by

J₁ and J₂ are

[J_1i, J_1j] = ikϵ_ijkJ_1k

[J_2i, J_2j] = ikϵ_ijkJ_2k
[J_1i, J_2j] = 0 (1)

Since the components of J₁ commute with those of J₂, they can have common eigenstates. We can choose, J², J_1z, J² and J_2z to be a set of operators which have common eigenstates.

1 2
Let us denote the common eigenstates by | j₁, j₂; m₁, m₂⟩ ≡| j₁, m₁⟩ | j₂, m₂⟩ [ Note : This
notation is not standard, several books will denote this as | j₁, m₁, j₂, m₂⟩. We follow Sakurai here.]. We then have

1

1
J² | j₁, j₂; m₁, m₂⟩ = j₁(j₁ + 1)k² | j₁, j₂; m₁, m₂⟩ J^z | j₁, j₂; m₁, m₂⟩ = m₁k | j₁, j₂; m₁, m₂⟩

2

2
J² | j₁, j₂; m₁, m₂⟩ = j₂(j₂ + 1)k² | j₁, j₂; m₁, m₂⟩ J^z | j₁, j₂; m₁, m₂⟩ = m₂k | j₁, j₂; m₁, m₂⟩
The dimension of the space to which J₁ and J₂ belong is (2j₁ + 1)(2j₂ + 1). The set of states | j₁, j₂; m₁, m₂⟩ form a complete and orthonormal set

Σ

Σ
+j₁ +j₂
| j₁, j₂; m₁, m₂⟩⟨j₁, j₂; m₁, m₂ |= 1 (2)

m₁=−j₁ m₂=−j₂
⟨j₁, j₂; m₁, m₂ | j₁^j , j₂^j ; m^j₁, m^j₂⟩ = δ_j1,j1^′ δ_j2,j2^′ δ_m1,m^′₁ δ_m2,m2^′

(3)

k

k

2R
As mentioned for the case of the orbital and spin angular momenta, the rotation operator in this case is given by

1R
U (nˆ, θ) ⊗ U
(nˆ, θ) = exp −i^{J→1 · nˆθ} ! exp −i^{J→2 · nˆθ} !

As a consequence of (1), the components of the total angular momentum J^ˆ satisfy

[J_i, J_j] = ikϵ_ijkJ_k (4)
which shows that J^ˆ is an angular momentum operator. It is easy to check that J² com- mutes with J_z, J² and J². However, J_z does not commute with either J_1z or J_2z. As a re-
1 2

1

2
sult, it is possible to consider an alternate set of commuting operators, viz., {J²,_z , J², J²}

1

2
which would describe the same space as did our old set {J², J², J_1z, J_2z}. We denote the
simultaneous eigenstates represented by this new set by | j₁, j₂; j, m⟩. In terms of this
new set, we have


J² | j₁, j₂; j, m⟩ = j(j + 1)k² | j₁, j₂; j, m⟩ (5)
J_z | j₁, j₂; j, m⟩ = mk | j₁, j₂; j, m⟩ (6)
These kets are also eigenkets of J² and J², though not of J_1z and J_2z. They satisfy the
1 2
completeness relation
j

mΣ=−j
| j₁, j₂; j, m⟩⟨j₁, j₂; j, m |= 1 (7)

They can be taken to be orthonormalized

⟨j₁, j₂; , j, m | j₁^j , j₂^j ; j^j, m^j⟩ = δ_m,m^′ δ_j,j^′ (8)

It is tacitly assumed that j₁ and j₂ in a given problem are given and fixed . We have not yet found out what values j can take. However, the sum over j in (7) must be over all values of j consistent with given values of j₁ and j₂. Using the completeness property(2) of the old set {| j₁, j₂; m₁, m₂⟩}, we may express a member of the new set {| j₁, j₂; j, m⟩} as



Σ

Σ
j₁ j₂
| j₁, j₂; j, m⟩ = | j₁, j₂; m₁, m₂⟩⟨j₁, j₂, m₁, m₂ | j₁, j₂; j, m⟩ (9)
m₁=−j₁ m₂=−j₂
The coefficients ⟨j₁, j₂, m₁, m₂ | j₁, j₂; j, m⟩ defined in (9) are called Clebsch-Gordan (C-G)Coefficients.. The basis defined here as represented by the kets on the left hand side of (9) will be also referred to as the Coupled Representation.

2. Properties of C-G Coefficients

   1. 
      The C-G coefficients ⟨j₁, j₂, m₁, m₂ | j₁, j₂; j, m⟩ are zero unless m = m₁ + m₂. To show this, note that J_z − J_1z − J_2z is a null operator,
      

⟨j₁, j₂, m₁, m₂ | J_z − J_1z − J_2z | j₁, j₂; j, m⟩ = 0

because J_z acts to the ket on its right giving m times the ket, while J_1z − J_2z acts to the bra on the left giving m₁ − m₂ times the bra,
(m − m₁ − m₂)⟨j₁, j₂, m₁, m₂ | j₁, j₂; j, m⟩ = 0 which shows that of m =/ m₁ + m₂, the C-G coefficient would be zero.

2. 
   By convention, the C-G coefficients are taken to be real, so that
   

⟨j₁, j₂, m₁, m₂ | j₁, j₂; j, m⟩ = ⟨j₁, j₂; jm | j₁, j₂; m₁, m₂⟩

3. 
   Orthonormality of C-G Coefficients : In the expression (8) given above, if we insert a complete set given by (2) of old states, we get
   

⟨j₁, j₂; , j, m | j₁^j , j₂^j ; j^j, m^j⟩ = δ_m,m^′ δ_j,j^′

Σ

Σ
+j₁ +j₂
⟨j₁, j₂; j, m | j₁, j₂; m₁, m₂⟩⟨j₁, j₂; m₁, m₂ | j₁^j , j₂^j ; j^j, m^j⟩ = δ_m,m^′ δ_j,j^′

m₁=−j₁ m₂=−j₂


Substituting j = j^j and m = m^j in the above, we get

(10)

(11)

In a similar way may start with the orthonormality condition on the old basis set
(3) and insert the completeness condition (7) of the new basis set to obtain
(12)

4. 
   The C-G coefficients vanish unless
   

(13)

The inequality on the right is obvious because m^max = j₁ and m^max = j₂. Hence

1 2
m^max = j₁ + j₂. Since the maximum value of m is j, the right inequality follows

and j_max = j₁ + j₂.

Σ
The inequality on the left requires a bit of working out. We know that there are (2j₁ + 1)(2j₂ + 1) number of states. For each value of j, there are 2j + 1 states. Thus we must have
^jmax
(2j + 1) = (2j₁ = 1)(2j₂ + 1)
j=j_min
Left hand side of above is an arithmetic series of finite number of terms. The sum works out as follows :

l.h.s. = ^{jmax − jmin + 1} (2j
2


max


+ 1 + 2j


min

+ 1)

= (j_max − j_min + 1)(j_max + J_min + 1)

min
= (j_max + 1)² − j²

min
= (j₁ + j₂ + 1)² − j²

min
Equating the above to (2j₁ + 1)(2j₂ + 1), we get j²
= (j₁ − j₂)², which gives

Thus the values of j satisfies
j_min =| j₁ − j₂ |

| j₁ − j₂ |≤ j ≤ j₁ + j₂ (14)

5. 
   For a given j, the possible m values are −j ≤ m ≤ +j.
   

A notation, primarily used by the nuclear physicists is known as Wigner’s 3-j symbol and is related to C-G coefficients as follows. While the C-G coefficients are for adding two angular moments, the 3-j symbols are for addition of three angular momenta such that their sum gives zero angular momentum state.

1

2

1

1

2
⟨j , j ; m , m
| j , j ; j, m⟩ = (−1)^j1−j2+m√2j + 1 ^{j1 j2 j} (15)

3.