In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly). The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory. From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in quantum-mechanical terms can be read directly from this approach as spherical harmonics are eigenfunctions of total angular momentum and projection thereof onto an axis, and the integrals correspond to the Hilbert space inner product. From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. There also exist complicated explicit formulas for their direct calculation. The formulas below use Dirac's bra–ket notation and the Condon–Shortley phase convention is adopted.
Review of the angular momentum operators Angular momentum operators are self-adjoint operators jx, jy, and jz 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 } , {\displaystyle {\begin{aligned}&[\mathrm {j} _{k},\mathrm {j} _{l}]\equiv \mathrm {j} _{k}\mathrm {j} _{l}-\mathrm {j} _{l}\mathrm {j} _{k}=i\hbar \varepsilon _{klm}\mathrm {j} _{m}&k,l,m&\in \{\mathrm {x,y,z} \},\end{aligned}}}
where εklm 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 ) . {\displaystyle \mathbf {j} =(\mathrm {j_{x}} ,\mathrm {j_{y}} ,\mathrm {j_{z}} ).}
It is 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 . {\displaystyle \mathbf {j} ^{2}=\mathrm {j_{x}^{2}} +\mathrm {j_{y}^{2}} +\mathrm {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 s o ( 3 , R ) ≅ s u ( 2 ) {\displaystyle {\mathfrak {so}}(3,\mathbb {R} )\cong {\mathfrak {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 . {\displaystyle \mathrm {j_{\pm }} =\mathrm {j_{x}} \pm i\mathrm {j_{y}} .}
Spherical basis for angular momentum eigenstates It can be shown from the above definitions that j2 commutes with jx, jy, and jz:
[ j 2 , j k ] = 0 k ∈ { x , y , z } . {\displaystyle {\begin{aligned}&[\mathbf {j} ^{2},\mathrm {j} _{k}]=0&k&\in \{\mathrm {x} ,\mathrm {y} ,\mathrm {z} \}.\end{aligned}}}
When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, j2 and jz are chosen. From the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted |j m⟩ where j is the angular momentum quantum number and m is the angular momentum projection onto the z-axis. They comprise the spherical basis, are complete, and 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 } . {\displaystyle {\begin{aligned}\mathbf {j} ^{2}|j\,m\rangle &=\hbar ^{2}j(j+1)|j\,m\rangle ,&j&\in \{0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots \}\\\mathrm {j_{z}} |j\,m\rangle &=\hbar m|j\,m\rangle ,&m&\in \{-j,-j+1,\ldots ,j\}.\end{aligned}}}
The raising and lowering operators can be used to alter the value of m,
j ± | j m ⟩ = ℏ C ± ( j , m ) | j ( m ± 1 ) ⟩ , {\displaystyle \mathrm {j} _{\pm }|j\,m\rangle =\hbar C_{\pm }(j,m)|j\,(m\pm 1)\rangle ,}
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 ) {\displaystyle C_{\pm }(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 ′ . {\displaystyle \langle j\,m|j'\,m'\rangle =\delta _{j,j'}\delta _{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 jx, jy, jz, j+, j−, and j2 denote operators. The δ {\displaystyle \delta } 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. Mathematically, this means that the angular momentum operators act on a space V 1 {\displaystyle V_{1}} of dimension 2 j 1 + 1 {\displaystyle 2j_{1}+1} and also on a space V 2 {\displaystyle V_{2}} of dimension 2 j 2 + 1 {\displaystyle 2j_{2}+1} . We are then going to define a family of "total angular momentum" operators acting on the tensor product space V 1 ⊗ V 2 {\displaystyle V_{1}\otimes V_{2}} , which has dimension ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) {\displaystyle (2j_{1}+1)(2j_{2}+1)} . The action of the total angular momentum operator on this space constitutes a representation of the SU(2) Lie algebra, but a reducible one. The reduction of this reducible representation into irreducible pieces is the goal of Clebsch–Gordan theory. 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 } , {\displaystyle {\begin{aligned}&|j_{1}\,m_{1}\rangle ,&m_{1}&\in \{-j_{1},-j_{1}+1,\ldots ,j_{1}\}\end{aligned}},}
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 } . {\displaystyle {\begin{aligned}&|j_{2}\,m_{2}\rangle ,&m_{2}&\in \{-j_{2},-j_{2}+1,\ldots ,j_{2}\}\end{aligned}}.}
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 } . {\displaystyle |j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle \equiv |j_{1}\,m_{1}\rangle \otimes |j_{2}\,m_{2}\rangle ,\quad m_{1}\in \{-j_{1},-j_{1}+1,\ldots ,j_{1}\},\quad m_{2}\in \{-j_{2},-j_{2}+1,\ldots ,j_{2}\}.}
Angular momentum operators are defined to act on states in V3 in the following manner:
( j 1 ⊗ 1 ) | j 1 m 1 j 2 m 2 ⟩ ≡ j 1 | j 1 m 1 ⟩ ⊗ | j 2 m 2 ⟩ {\displaystyle (\mathbf {j} _{1}\otimes 1)|j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle \equiv \mathbf {j} _{1}|j_{1}\,m_{1}\rangle \otimes |j_{2}\,m_{2}\rangle }
and
( 1 ⊗ j 2 ) | j 1 m 1 j 2 m 2 ⟩ ≡ | j 1 m 1 ⟩ ⊗ j 2 | j 2 m 2 ⟩ , {\displaystyle (1\otimes \mathrm {\mathbf {j} } _{2})|j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle \equiv |j_{1}\,m_{1}\rangle \otimes \mathbf {j} _{2}|j_{2}\,m_{2}\rangle ,}
where 1 denotes the identity operator. The total angular momentum operators are defined by the coproduct (or tensor product) 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 , {\displaystyle [\mathrm {J} _{k},\mathrm {J} _{l}]=i\hbar \varepsilon _{klm}\mathrm {J} _{m}~,}
where k, l, m ∈ {x, y, z}. Indeed, the preceding construction is the standard method for constructing an action of a Lie algebra on a tensor product representation. 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 ⟩ {\displaystyle {\begin{aligned}\mathbf {J} ^{2}|[j_{1}\,j_{2}]\,J\,M\rangle &=\hbar ^{2}J(J+1)|[j_{1}\,j_{2}]\,J\,M\rangle \\\mathrm {J_{z}} |[j_{1}\,j_{2}]\,J\,M\rangle &=\hbar M|[j_{1}\,j_{2}]\,J\,M\rangle \end{aligned}}}
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 , {\displaystyle |j_{1}-j_{2}|\leq J\leq 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 ) . {\displaystyle \sum _{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}(2J+1)=(2j_{1}+1)(2j_{2}+1)~.}
As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension 2 J + 1 {\displaystyle 2J+1} , where J {\displaystyle J} ranges from | j 1 − j 2 | {\displaystyle |j_{1}-j_{2}|} to j 1 + j 2 {\displaystyle j_{1}+j_{2}} in increments of 1. As an example, consider the tensor product of the three-dimensional representation corresponding to j 1 = 1 {\displaystyle j_{1}=1} with the two-dimensional representation with j 2 = 1 / 2 {\displaystyle j_{2}=1/2} . The possible values of J {\displaystyle J} are then J = 1 / 2 {\displaystyle J=1/2} and J = 3 / 2 {\displaystyle J=3/2} . Thus, the six-dimensional tensor product representation decomposes as the direct sum of a two-dimensional representation and a four-dimensional representation. The goal is now to describe the preceding decomposition explicitly, that is, to explicitly describe basis elements in the tensor product space for each of the component representations that arise. The total angular momentum states form an orthonormal basis of V3:
⟨ J M | J ′ M ′ ⟩ = δ J , J ′ δ M , M ′ . {\displaystyle \left\langle J\,M|J'\,M'\right\rangle =\delta _{J,J'}\delta _{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 ) , {\displaystyle \mathbf {2} ^{\otimes n}=\bigoplus _{k=0}^{\lfloor n/2\rfloor }~\left({\frac {n+1-2k}{n+1}}{n+1 \choose k}\right)~(\mathbf {n} +\mathbf {1} -\mathbf {2} \mathbf {k} )~,}
where ⌊ n / 2 ⌋ {\displaystyle \lfloor n/2\rfloor } 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 {\displaystyle {\mathbf {2} }\otimes {\mathbf {2} }\otimes {\mathbf {2} }={\mathbf {4} }\oplus {\mathbf {2} }\oplus {\mathbf {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 ⟩ {\displaystyle \langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle }
are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as ⟨j1 j2; m1 m2 | J M⟩. Another common notation is ⟨j1 m1 j2 m2 | J M⟩ = CJMj1m1j2m2. Applying the operators
J = j ⊗ 1 + 1 ⊗ j J z = j z ⊗ 1 + 1 ⊗ j z {\displaystyle {\begin{aligned}\mathrm {J} &=\mathrm {j} \otimes 1+1\otimes \mathrm {j} \\\mathrm {J} _{\mathrm {z} }&=\mathrm {j} _{\mathrm {z} }\otimes 1+1\otimes \mathrm {j} _{\mathrm {z} }\end{aligned}}}
to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when
| j 1 − j 2 | ≤ J ≤ j 1 + j 2 M = m 1 + m 2 . {\displaystyle {\begin{aligned}|j_{1}-j_{2}|\leq J&\leq j_{1}+j_{2}\\M&=m_{1}+m_{2}.\end{aligned}}}
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 ± {\displaystyle \mathrm {J} _{\pm }=\mathrm {j} _{\pm }\otimes 1+1\otimes \mathrm {j} _{\pm }}
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