Clebsch–Gordan coefficients

Angular momentum operators

Angular momentum operators are self-adjoint operators j_x, j_y, and j_z that satisfy the commutation relations

[ j k , j l ] ≡ j k j l − j l j k = i ℏ ε k l m j m k , l , m ∈ { x , y , z } ,

where ε_klm is the Levi-Civita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator,

j = ( j x , j y , j z ) .

It also known as a spherical vector, since it is also a spherical tensor operator. It is only for rank one that spherical tensor operators coincide with the Cartesian tensor operators.

By developing this concept further, one can define another operator j² as the inner product of j with itself:

j 2 = j x 2 + j y 2 + j z 2 .

This is an example of a Casimir operator. It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra so(3) ≅ su(2). This is physically interpreted as the square of the total angular momentum of the states on which the representation acts.

One can also define raising (j₊) and lowering (j₋) operators, the so-called ladder operators,

j ± = j x ± i j y .

Angular momentum states

It can be shown from the above definitions that j² commutes with j_x, j_y, and j_z:

[ j 2 , j k ] = 0 k ∈ { x , y , z } .

When two Hermitian operators commute, a common set of eigenfunctions exists. Conventionally j² and j_z are chosen. From the commutation relations the possible eigenvalues can be found. These states are denoted |j m⟩ where j is the angular momentum quantum number and m is the angular momentum projection onto the z-axis. They satisfy the following eigenvalue equations:

j 2 | j m ⟩ = ℏ 2 j ( j + 1 ) | j m ⟩ , j ∈ { 0 , 1 2 , 1 , 3 2 , … } j z | j m ⟩ = ℏ m | j m ⟩ , m ∈ { − j , − j + 1 , … , j } .

The raising and lowering operators can be used to alter the value of m:

j ± | j m ⟩ = ℏ C ± ( j , m ) | j ( m ± 1 ) ⟩ ,

where the ladder coefficient is given by:

In principle, one may also introduce a (possibly complex) phase factor in the definition of C ± ( j , m ) . The choice made in this article is in agreement with the Condon–Shortley phase convention. The angular momentum states are orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized:

⟨ j m | j ′ m ′ ⟩ = δ j , j ′ δ m , m ′ .

Here the italicized j and m denote integer or half-integer angular momentum quantum numbers of a particle or of a system. On the other hand, the roman j_x, j_y, j_z, j₊, j₋, and j² denote operators. The δ symbols are Kronecker deltas.

Tensor product space

We now consider systems with two physically different angular momenta j₁ and j₂. Examples include the spin and the orbital angular momentum of a single electron, or the spins of two electrons, or the orbital angular momenta of two electrons. These comprise two different irreducible representations of the same group, SO(3) (SU(2)), which combine together to another representation of the same group, but a reducible one. The reduction of this combined representation into a direct sum of irreducible ones comprises the Clebsch–Gordan series.

Let V₁ be the (2 j₁ + 1)-dimensional vector space spanned by the states

| j 1 m 1 ⟩ , m 1 ∈ { − j 1 , − j 1 + 1 , … , j 1 } ,

and V₂ the (2 j₂ + 1)-dimensional vector space spanned by the states

| j 2 m 2 ⟩ , m 2 ∈ { − j 2 , − j 2 + 1 , … , j 2 } .

The tensor product of these spaces, V₃ ≡ V₁ ⊗ V₂, has a (2 j₁ + 1) (2 j₂ + 1)-dimensional uncoupled basis

| j 1 m 1 j 2 m 2 ⟩ ≡ | j 1 m 1 ⟩ ⊗ | j 2 m 2 ⟩ , m 1 ∈ { − j 1 , − j 1 + 1 , … , j 1 } , m 2 ∈ { − j 2 , − j 2 + 1 , … , j 2 } .

Angular momentum operators are defined to act on states in V₃ in the following manner:

( j ⊗ 1 ) | j 1 m 1 j 2 m 2 ⟩ ≡ j | j 1 m 1 ⟩ ⊗ | j 2 m 2 ⟩

and

( 1 ⊗ j ) | j 1 m 1 j 2 m 2 ⟩ ≡ | j 1 m 1 ⟩ ⊗ j | j 2 m 2 ⟩ ,

where 1 denotes the identity operator.

The total angular momentum operators are defined by the coproduct of the two representations acting on V₁⊗V₂,

The total angular momentum operators can be shown to satisfy the very same commutation relations,

[ J k , J l ] = i ℏ ε k l m J m   ,

where k, l, m ∈ {x, y, z}.

Hence, a set of coupled eigenstates exist for the total angular momentum operator as well,

J 2 | [ j 1 j 2 ] J M ⟩ = ℏ 2 J ( J + 1 ) | [ j 1 j 2 ] J M ⟩ J z | [ j 1 j 2 ] J M ⟩ = ℏ M | [ j 1 j 2 ] J M ⟩

for M ∈ {−J, −J + 1, …, J}. Note that it is common to omit the [j₁ j₂] part.

The total angular momentum quantum number J must satisfy the triangular condition that

| j 1 − j 2 | ≤ J ≤ j 1 + j 2 ,

such that the three nonnegative integer or half-integer values could correspond to the three sides of a triangle.

The total number of total angular momentum eigenstates is necessarily equal to the dimension of V₃:

∑ J = | j 1 − j 2 | j 1 + j 2 ( 2 J + 1 ) = ( 2 j 1 + 1 ) ( 2 j 2 + 1 )   .

The total angular momentum states form an orthonormal basis of V₃:

⟨ J M | J ′ M ′ ⟩ = δ J , J ′ δ M , M ′   .

These rules may be iterated to, e.g., combine n doublets (s=1/2) to obtain the Clebsch-Gordan decomposition series, (Catalan's triangle),

2 ⊗ n = ⨁ k = 0 ⌊ n / 2 ⌋   ( n + 1 − 2 k n + 1 ( n + 1 k ) )     ( n + 1 − 2 k )   ,

where ⌊ n / 2 ⌋ is the integer floor function; and the number preceding the boldface irreducible representation dimensionality (2j+1) label indicates multiplicity of that representation in the representation reduction. For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s,   2 ⊗ 2 ⊗ 2 = 4 ⊕ 2 ⊕ 2 .

Formal definition of Clebsch–Gordan coefficients

The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis

The expansion coefficients

⟨ j 1 m 1 j 2 m 2 | J M ⟩

are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as ⟨j₁ j₂; m₁ m₂|J M⟩.

Applying the operator

J z = j z ⊗ 1 + 1 ⊗ j z

to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when

M = m 1 + m 2 .

Recursion relations

The recursion relations were discovered by physicist Giulio Racah from the Hebrew University of Jerusalem in 1941.

Applying the total angular momentum raising and lowering operators

J ± = j ± ⊗ 1 + 1 ⊗ j ±

to the left hand side of the defining equation gives

J ± | [ j 1 j 2 ] J M ⟩ = ℏ C ± ( J , M ) | [ j 1 j 2 ] J ( M ± 1 ) ⟩ = ℏ C ± ( J , M ) ∑ m 1 , m 2 | j 1 m 1 j 2 m 2 ⟩ ⟨ j 1 m 1 j 2 m 2 | J ( M ± 1 ) ⟩

Applying the same operators to the right hand side gives

J ± ∑ m 1 , m 2 | j 1 m 1 j 2 m 2 ⟩ ⟨ j 1 m 1 j 2 m 2 | J M ⟩ = ℏ ∑ m 1 , m 2 ( C ± ( j 1 , m 1 ) | j 1 ( m 1 ± 1 ) j 2 m 2 ⟩ + C ± ( j 2 , m 2 ) | j 1 m 1 j 2 ( m 2 ± 1 ) ⟩ ) ⟨ j 1 m 1 j 2 m 2 | J M ⟩ = ℏ ∑ m 1 , m 2 | j 1 m 1 j 2 m 2 ⟩ ( C ± ( j 1 , m 1 ∓ 1 ) ⟨ j 1 ( m 1 ∓ 1 ) j 2 m 2 | J M ⟩ + C ± ( j 2 , m 2 ∓ 1 ) ⟨ j 1 m 1 j 2 ( m 2 ∓ 1 ) | J M ⟩ ) .

where C_± was defined in 1. Combining these results gives recursion relations for the Clebsch–Gordan coefficients:

C ± ( J , M ) ⟨ j 1 m 1 j 2 m 2 | J ( M ± 1 ) ⟩ = C ± ( j 1 , m 1 ∓ 1 ) ⟨ j 1 ( m 1 ∓ 1 ) j 2 m 2 | J M ⟩ + C ± ( j 2 , m 2 ∓ 1 ) ⟨ j 1 m 1 j 2 ( m 2 ∓ 1 ) | J M ⟩ .

Taking the upper sign with the condition that M = J gives initial recursion relation:

0 = C + ( j 1 , m 1 − 1 ) ⟨ j 1 ( m 1 − 1 ) j 2 m 2 | J J ⟩ + C + ( j 2 , m 2 − 1 ) ⟨ j 1 m 1 j 2 ( m 2 − 1 ) | J J ⟩ .

In the Condon–Shortley phase convention, one adds the constraint that

⟨ j 1 j 1 j 2 ( J − j 1 ) | J J ⟩ > 0

(and is therefore also real).

The Clebsch–Gordan coefficients ⟨j₁ m₁ j₂ m₂ | J J⟩ can then be found from these recursion relations. The normalization is fixed by the requirement that the sum of the squares, which equivalent to the requirement that the norm of the state |[j₁ j₂] J J⟩ must be one.

The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with M = J − 1. Repeated use of that equation gives all coefficients.

This procedure to find the Clebsch–Gordan coefficients shows that they are all real in the Condon–Shortley phase convention.

Orthogonality relations

These are most clearly written down by introducing the alternative notation

⟨ J M | j 1 m 1 j 2 m 2 ⟩ ≡ ⟨ j 1 m 1 j 2 m 2 | J M ⟩

The first orthogonality relation is

∑ J = | j 1 − j 2 | j 1 + j 2 ∑ M = − J J ⟨ j 1 m 1 j 2 m 2 | J M ⟩ ⟨ J M | j 1 m 1 ′ j 2 m 2 ′ ⟩ = ⟨ j 1 m 1 j 2 m 2 | j 1 m 1 ′ j 2 m 2 ′ ⟩ = δ m 1 , m 1 ′ δ m 2 , m 2 ′

(derived from the fact that 1 ≡ ∑_x |x⟩ ⟨x|) and the second one is

∑ m 1 , m 2 ⟨ J M | j 1 m 1 j 2 m 2 ⟩ ⟨ j 1 m 1 j 2 m 2 | J ′ M ′ ⟩ = ⟨ J M | J ′ M ′ ⟩ = δ J , J ′ δ M , M ′ .

Special cases

For J = 0 the Clebsch–Gordan coefficients are given by

⟨ j 1 m 1 j 2 m 2 | 0 0 ⟩ = δ j 1 , j 2 δ m 1 , − m 2 ( − 1 ) j 1 − m 1 2 j 2 + 1 .

For J = j₁ + j₂ and M = J we have

⟨ j 1 j 1 j 2 j 2 | ( j 1 + j 2 ) ( j 1 + j 2 ) ⟩ = 1 .

For j₁ = j₂ = J / 2 and m₁ = −m₂ we have

⟨ j 1 m 1 j 1 ( − m 1 ) | ( 2 j 1 ) 0 ⟩ = ( 2 j 1 ) ! 2 ( j 1 − m 1 ) ! ( j 1 + m 1 ) ! ( 4 j 1 ) ! .

For j₁ = j₂ = m₁ = −m₂ we have

⟨ j 1 j 1 j 1 ( − j 1 ) | J 0 ⟩ = ( 2 j 1 ) ! 2 J + 1 ( J + 2 j 1 + 1 ) ! ( 2 j 1 − J ) ! .

For j₂ = 1, m₂ = 0 we have

⟨ j 1 m 1 0 | ( j 1 + 1 ) m ⟩ = ( j 1 − m + 1 ) ( j 1 + m + 1 ) ( 2 j 1 + 1 ) ( j 1 + 1 ) ⟨ j 1 m 1 0 | j 1 m ⟩ = m j 1 ( j 1 + 1 ) ⟨ j 1 m 1 0 | ( j 1 − 1 ) m ⟩ = − ( j 1 − m ) ( j 1 + m ) j 1 ( 2 j 1 + 1 )

Symmetry properties

⟨ j 1 m 1 j 2 m 2 | J M ⟩ = ( − 1 ) j 1 + j 2 − J ⟨ j 1 ( − m 1 ) j 2 ( − m 2 ) | J ( − M ) ⟩ = ( − 1 ) j 1 + j 2 − J ⟨ j 2 m 2 j 1 m 1 | J M ⟩ = ( − 1 ) j 1 − m 1 2 J + 1 2 j 2 + 1 ⟨ j 1 m 1 J ( − M ) | j 2 ( − m 2 ) ⟩ = ( − 1 ) j 2 + m 2 2 J + 1 2 j 1 + 1 ⟨ J ( − M ) j 2 m 2 | j 1 ( − m 1 ) ⟩ = ( − 1 ) j 1 − m 1 2 J + 1 2 j 2 + 1 ⟨ J M j 1 ( − m 1 ) | j 2 m 2 ⟩ = ( − 1 ) j 2 + m 2 2 J + 1 2 j 1 + 1 ⟨ j 2 ( − m 2 ) J M | j 1 m 1 ⟩

A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using 3. The symmetry properties of Wigner 3-j symbols are much simpler.

Rules for phase factors

Care is needed when simplifying phase factors: a quantum number may be a half-integer rather than an integer, therefore (−1)^2k is not necessarily 1 for a given quantum number k unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule:

( − 1 ) 4 k = 1

for any angular-momentum-like quantum number k.

Nonetheless, a combination of j_i and m_i is always an integer, so the stronger rule applies for these combinations:

( − 1 ) 2 ( j i − m i ) = 1

This identity also holds if the sign of either j_i or m_i or both is reversed.

It is useful to observe that any phase factor for a given (j_i, m_i) pair can be reduced to the canonical form:

( − 1 ) a j i + b ( j i − m i )

where a ∈ {0, 1, 2, 3} and b ∈ {0, 1} (other conventions are possible too). Converting phase factors into this form makes it easy to tell whether two phase factors are equivalent. (Note that this form is only locally canonical: it fails to take into account the rules that govern combinations of (j_i, m_i) pairs such as the one described in the next paragraph.)

An additional rule holds for combinations of j₁, j₂, and j₃ that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol:

( − 1 ) 2 ( j 1 + j 2 + j 3 ) = 1

This identity also holds if the sign of any j_i is reversed, or if any of them are substituted with an m_i instead.

Relation to Wigner 3-j symbols

Clebsch–Gordan coefficients are related to Wigner 3-j symbols which have more convenient symmetry relations.

Relation to Wigner D-matrices

∫ 0 2 π d α ∫ 0 π sin ⁡ β d β ∫ 0 2 π d γ D M , K J ( α , β , γ ) ∗ D m 1 , k 1 j 1 ( α , β , γ ) D m 2 , k 2 j 2 ( α , β , γ ) = 8 π 2 2 J + 1 ⟨ j 1 m 1 j 2 m 2 | J M ⟩ ⟨ j 1 k 1 j 2 k 2 | J K ⟩

Relation to spherical harmonics

In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics:

∫ 4 π Y ℓ 1 m 1 ∗ ( Ω ) Y ℓ 2 m 2 ∗ ( Ω ) Y L M ( Ω ) d Ω = ( 2 ℓ 1 + 1 ) ( 2 ℓ 2 + 1 ) 4 π ( 2 L + 1 ) ⟨ ℓ 1 0 ℓ 2 0 | L 0 ⟩ ⟨ ℓ 1 m 1 ℓ 2 m 2 | L M ⟩

It follows from this and orthonormality of the spherical harmonics that CG coefficients are in fact the expansion coefficients of a product of two spherical harmonics in terms a single spherical harmonic:

Y ℓ 1 m 1 ( Ω ) Y ℓ 2 m 2 ( Ω ) = ∑ L , M ( 2 ℓ 1 + 1 ) ( 2 ℓ 2 + 1 ) 4 π ( 2 L + 1 ) ⟨ ℓ 1 0 ℓ 2 0 | L 0 ⟩ ⟨ ℓ 1 m 1 ℓ 2 m 2 | L M ⟩ Y L M ( Ω )

Other Properties

∑ m ( − 1 ) j − m ⟨ j m j ( − m ) | J 0 ⟩ = δ J , 0 2 j + 1

SU(N) Clebsch–Gordan coefficients

For arbitrary groups and their representations, Clebsch–Gordan coefficients are not known in general. However, algorithms to produce Clebsch–Gordan coefficients for the special unitary group are known. In particular, SU(3) Clebsch-Gordan coefficients have been computed and tabulated because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists that relates the up, down, and strange quarks. A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.