Clebsch–Gordan coefficients#Tensor product space

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

\begin{align}

&[\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{align}

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

\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 {{math|j²}} as the inner product of {{math|j}} with itself:

\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 $\backslash mathfrak\{so\}(3,\backslash mathbb\{R\})\; \backslash cong\; \backslash mathfrak\{su\}(2)$. 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 ({{math|j₊}}) and lowering ({{math|j₋}}) operators, the so-called ladder operators,

\mathrm {j_\pm} = \mathrm{j_x} \pm i \mathrm{j_y}.

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

\begin{align}

&[\mathbf j^2, \mathrm {j}_k] = 0 & k &\in \{\mathrm x, \mathrm y, \mathrm z\}.

\end{align}

When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, {{math|j²}} and {{math|j_z}} are chosen. From the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted {{math|{{ket|j m}}}} where {{math|j}} is the angular momentum quantum number and {{math|m}} is the angular momentum projection onto the z-axis.

They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations,

\begin{align}

\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{align}

The raising and lowering operators can be used to alter the value of {{math|m}},

\mathrm j_\pm |j \, m\rangle = \hbar C_\pm(j, m) |j \, (m \pm 1)\rangle,

where the ladder coefficient is given by:

{{NumBlk||$$$$

C_\pm(j, m) = \sqrt{j (j + 1) - m (m \pm 1)} = \sqrt{(j \mp m)(j \pm m + 1)}.

|{{EquationRef|1}}}}

In principle, one may also introduce a (possibly complex) phase factor in the definition of $C\_\backslash pm(j,\; m)$. 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,

\langle j \, m | j' \, m' \rangle = \delta_{j, j'} \delta_{m, m'}.

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

{{see also|Canonical commutation relation#Uncertainty relation for angular momentum operators}}

We now consider systems with two physically different angular momenta {{math|j₁}} and {{math|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 $V\_1$ of dimension $2j\_1+1$ and also on a space $V\_2$ of dimension $2j\_2\; +\; 1$. We are then going to define a family of "total angular momentum" operators acting on the tensor product space $V\_1\; \backslash otimes\; V\_2$, which has dimension $(2j\_1+1)(2j\_2+1)$. 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 {{math|V₁}} be the {{math|(2 j₁ + 1)}}-dimensional vector space spanned by the states

\begin{align}

&|j_1 \, m_1\rangle, & m_1 &\in \{-j_1, -j_1 + 1, \ldots, j_1\}

\end{align},

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

\begin{align}

&|j_2 \, m_2\rangle, & m_2 &\in \{-j_2, -j_2 + 1, \ldots, j_2\}

\end{align}.

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

|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 {{math|V₃}} in the following manner:

(\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 \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 {{math|1}} denotes the identity operator.

The totalThe word "total" is often overloaded to mean several different things. In this article, "total angular momentum" refers to a generic sum of two angular momentum operators {{math|j₁}} and {{math|j₂}}. It is not to be confused with the other common use of the term "total angular momentum" that refers specifically to the sum of orbital angular momentum and spin. angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on {{math|V₁⊗V₂}},

{{Equation box 1

|indent =:

|equation = $\backslash mathbf\{J\}\; \backslash equiv\; \backslash mathbf\{j\}\_1\; \backslash otimes\; 1\; +\; 1\; \backslash otimes\; \backslash mathbf\{j\}\_2~.$

|cellpadding= 6

|border

|border colour = #0070BF

|bgcolor=#FAFFFB

}}

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

[\mathrm{J}_k, \mathrm{J}_l] = i \hbar \varepsilon_{k l m} \mathrm{J}_m ~,

where {{math|k, l, m ∈ {{mset|x, y, z}}}}. Indeed, the preceding construction is the standard method{{harvnb|Hall|2015}} Section 4.3.2 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,

\begin{align}

\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{align}

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

The total angular momentum quantum number {{math|J}} must satisfy the triangular condition that

|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.{{harvnb|Merzbacher|1998}}

The total number of total angular momentum eigenstates is necessarily equal to the dimension of {{math|V₃}}:

\sum_{J = |j_1 - j_2|}^{j_1 + j_2} (2 J + 1) = (2 j_1 + 1) (2 j_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 $2J+1$, where $J$ ranges from $|j\_1\; -\; j\_2|$ to $j\_1\; +\; j\_2$ in increments of 1.{{harvnb|Hall|2015}} Appendix C As an example, consider the tensor product of the three-dimensional representation corresponding to $j\_1\; =\; 1$ with the two-dimensional representation with $j\_2\; =\; 1/2$. The possible values of $J$ are then $J\; =\; 1/2$ and $J\; =\; 3/2$. 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 {{math|V₃}}:

\left\langle J\, M | J'\, M' \right\rangle = \delta_{J, J'}\delta_{M, M'}~.

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

\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 $\backslash lfloor\; n/2\; \backslash rfloor$ is the integer floor function; and the number preceding the boldface irreducible representation dimensionality ({{math|2j+1}}) label indicates multiplicity of that representation in the representation reduction.{{cite journal |last=Zachos |first=C K|year=1992 |title=Altering the Symmetry of Wavefunctions in Quantum Algebras and Supersymmetry |journal= Modern Physics Letters A |volume=A7 |issue= 18|pages=1595–1600 |doi=10.1142/S0217732392001270|arxiv = hep-th/9203027 |bibcode = 1992MPLA....7.1595Z |s2cid=16360975 }} For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s, $\{\backslash mathbf\; 2\}\backslash otimes\{\backslash mathbf\; 2\}\backslash otimes\{\backslash mathbf\; 2\}\; =\; \{\backslash mathbf\; 4\}\backslash oplus\{\backslash mathbf\; 2\}\backslash oplus\{\backslash mathbf\; 2\}$.

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

{{NumBlk||$$$$

|J \, M\rangle =

\sum_{m_1 = -j_1}^{j_1} \sum_{m_2 = -j_2}^{j_2}

|j_1 \, m_1 \, j_2 \, m_2\rangle \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M\rangle

|{{EquationRef|2}}}}

The expansion coefficients

\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 {{math|{{bra-ket|j₁ j₂; m₁ m₂ | J M}}}}. Another common notation is

{{math|1={{bra-ket|j₁ m₁ j₂ m₂ | J M}} = C{{su|p=JM|b=j₁m₁j₂m₂}}}}.

Applying the operators

\begin{align}

\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{align}

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

\begin{align}

|j_1 - j_2| \leq J &\leq j_1 + j_2 \\

M &= m_1 + m_2.

\end{align}

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$$$$

\mathrm J_\pm = \mathrm j_\pm \otimes 1 + 1 \otimes \mathrm j_\pm

to the left hand side of the defining equation gives$$$$

\begin{align}

\mathrm J_\pm |[j_1 \, j_2] \, J \, M\rangle

&= \hbar C_\pm(J, M) |[j_1 \, j_2] \, J \, (M \pm 1)\rangle \\

&= \hbar C_\pm(J, M)

\sum_{m_1, m_2}

|j_1 \, m_1 \, j_2 \, m_2\rangle

\langle j_1 \, m_1 \, j_2 \, m_2 | J \, (M \pm 1)\rangle

\end{align}

Applying the same operators to the right hand side gives$$$$

\begin{align}

\mathrm J_\pm &\sum_{m_1, m_2}

|j_1 \, m_1 \, j_2 \, m_2\rangle

\langle j_1 \, m_1 \, j_2 \, m_2 | J \, M\rangle \\

= \hbar &\sum_{m_1, m_2} \Bigl(

C_\pm(j_1, m_1) |j_1 \, (m_1 \pm 1) \, j_2 \, m_2\rangle

+ C_\pm(j_2, m_2) |j_1 \, m_1 \, j_2 \, (m_2 \pm 1)\rangle

\Bigr) \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M\rangle \\

= \hbar &\sum_{m_1, m_2} |j_1 \, m_1 \, j_2 \, m_2\rangle \Bigl(

C_\pm(j_1, m_1 \mp 1) \langle j_1 \, (m_1 \mp 1) \, j_2 \, m_2 | J \, M\rangle

+ C_\pm(j_2, m_2 \mp 1) \langle j_1 \, m_1 \, j_2 \, (m_2 \mp 1) | J \, M\rangle

\Bigr) .

\end{align}

Combining these results gives recursion relations for the Clebsch–Gordan coefficients, where {{math|C_±}} was defined in {{EquationNote|1}}:

C_\pm(J, M) \langle j_1 \, m_1 \, j_2 \, m_2 | J \, (M \pm 1)\rangle

= C_\pm(j_1, m_1 \mp 1) \langle j_1 \, (m_1 \mp 1) \, j_2 \, m_2 | J \, M\rangle

+ C_\pm(j_2, m_2 \mp 1) \langle j_1 \, m_1 \, j_2 \, (m_2 \mp 1) | J \, M\rangle.

Taking the upper sign with the condition that {{math|M {{=}} J}} gives initial recursion relation:$$$$

0 = C_+(j_1, m_1 - 1) \langle j_1 \, (m_1 - 1) \, j_2 \, m_2 | J \, J\rangle

+ C_+(j_2, m_2 - 1) \langle j_1 \, m_1 \, j_2 \, (m_2 - 1) | J \, J\rangle.

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

:$\backslash langle\; j\_1\; \backslash ,\; j\_1\; \backslash ,\; j\_2\; \backslash ,\; (J\; -\; j\_1)\; |\; J\; \backslash ,\; J\backslash rangle\; >\; 0$

(and is therefore also real).

The Clebsch–Gordan coefficients {{math|⟨j₁ m₁ j₂ m₂ {{!}} J M⟩}} 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 {{math|{{!}}[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 {{math|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.

{{for|an explicit expression of the Clebsch–Gordan coefficients and tables with numerical values|table of Clebsch–Gordan coefficients}}

These are most clearly written down by introducing the alternative notation

\langle J \, M | j_1 \, m_1 \, j_2 \, m_2 \rangle \equiv \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle

The first orthogonality relation is

\sum_{J = |j_1 - j_2|}^{j_1 + j_2} \sum_{M = -J}^J

\langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle

\langle J \, M | j_1 \, m_1' \, j_2 \, m_2' \rangle

= \langle j_1 \, m_1 \, j_2 \, m_2 | j_1 \, m_1' \, j_2 \, m_2' \rangle

= \delta_{m_1, m_1'} \delta_{m_2, m_2'}

(derived from the fact that $\backslash mathbf\; 1\; =\; \backslash sum\_x\; |x\backslash rangle\; \backslash langle\; x|$) and the second one is

\sum_{m_1, m_2}

\langle J \, M | j_1 \, m_1 \, j_2 \, m_2 \rangle

\langle j_1 \, m_1 \, j_2 \, m_2 | J' \, M' \rangle

= \langle J \, M | J' \, M' \rangle

= \delta_{J, J'} \delta_{M, M'}.

For {{math|J {{=}} 0}} the Clebsch–Gordan coefficients are given by

\langle j_1 \, m_1 \, j_2 \, m_2 | 0 \, 0 \rangle

= \delta_{j_1, j_2} \delta_{m_1, -m_2} \frac{(-1)^{j_1 - m_1}}{\sqrt{2 j_1 + 1}}.

For {{math|J {{=}} j₁ + j₂}} and {{math|M {{=}} J}} we have

$$\backslash langle\; j\_1\; \backslash ,\; j\_1\; \backslash ,\; j\_2\; \backslash ,\; j\_2\; |\; (j\_1\; +\; j\_2)\; \backslash ,\; (j\_1\; +\; j\_2)\; \backslash rangle\; =\; 1.$$

For {{math|j₁ {{=}} j₂ {{=}} J / 2}} and {{math|m₁ {{=}} −m₂}} we have

\langle j_1 \, m_1 \, j_1 \, (-m_1) | (2 j_1) \, 0 \rangle

= \frac{(2 j_1)!^2}{(j_1 - m_1)! (j_1 + m_1)! \sqrt{(4 j_1)!}}.

For {{math|j₁ {{=}} j₂ {{=}} m₁ {{=}} −m₂}} we have

\langle j_1 \, j_1 \, j_1 \, (-j_1) | J \, 0 \rangle

= (2 j_1)! \sqrt{\frac{2 J + 1}{(J + 2 j_1 + 1)! (2 j_1 - J)!}}.

For {{math|j₂ {{=}} 1}}, {{math|m₂ {{=}} 0}} we have

\begin{align}

\langle j_1 \, m \, 1 \, 0 | (j_1 + 1) \, m \rangle &= \sqrt{\frac{(j_1 - m + 1) (j_1 + m + 1)}{(2 j_1 + 1) (j_1 + 1)}} \\

\langle j_1 \, m \, 1 \, 0 | j_1 \, m \rangle &= \frac{m}{\sqrt{j_1 (j_1 + 1)}} \\

\langle j_1 \, m \, 1 \, 0 | (j_1 - 1) \, m \rangle &= -\sqrt{\frac{(j_1 - m) (j_1 + m)}{j_1 (2 j_1 + 1)}}

\end{align}

For {{math|j₂ {{=}} 1/2}} we have

\begin{align}

\left\langle j_1 \, \left( M - \frac{1}{2} \right) \, \frac{1}{2} \, \frac{1}{2} \Bigg| \left( j_1 \pm \frac{1}{2} \right) \, M \right\rangle &= \pm \sqrt{\frac{1}{2} \left( 1 \pm \frac{M}{j_1 + \frac{1}{2}} \right)} \\

\left\langle j_1 \, \left( M + \frac{1}{2} \right) \, \frac{1}{2} \, \left( -\frac{1}{2} \right) \Bigg| \left( j_1 \pm \frac{1}{2} \right) \, M \right\rangle &= \sqrt{\frac{1}{2} \left( 1 \mp \frac{M}{j_1 + \frac{1}{2}} \right)}

\end{align}

\begin{align}

\langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle

&= (-1)^{j_1 + j_2 - J} \langle j_1 \, (-m_1) \, j_2 \, (-m_2) | J \, (-M)\rangle \\

&= (-1)^{j_1 + j_2 - J} \langle j_2 \, m_2 \, j_1 \, m_1 | J \, M \rangle \\

&= (-1)^{j_1 - m_1} \sqrt{\frac{2 J + 1}{2 j_2 + 1}} \langle j_1 \, m_1 \, J \, (-M)| j_2 \, (-m_2) \rangle \\

&= (-1)^{j_2 + m_2} \sqrt{\frac{2 J + 1}{2 j_1 + 1}} \langle J \, (-M) \, j_2 \, m_2| j_1 \, (-m_1) \rangle \\

&= (-1)^{j_1 - m_1} \sqrt{\frac{2 J + 1}{2 j_2 + 1}} \langle J \, M \, j_1 \, (-m_1) | j_2 \, m_2 \rangle \\

&= (-1)^{j_2 + m_2} \sqrt{\frac{2 J + 1}{2 j_1 + 1}} \langle j_2 \, (-m_2) \, J \, M | j_1 \, m_1 \rangle

\end{align}

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

Care is needed when simplifying phase factors: a quantum number may be a half-integer rather than an integer, therefore {{math|(−1)^2k}} is not necessarily {{math|1}} for a given quantum number {{math|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 {{math|k}}.

Nonetheless, a combination of {{math|j_i}} and {{math|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 {{math|j_i}} or {{math|m_i}} or both is reversed.

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

$$(-1)^\{a\; j\_i\; +\; b\; (j\_i\; -\; m\_i)\}$$

where {{math|a ∈ {0, 1, 2, 3}}} and {{math|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 {{math|(j_i, m_i)}} pairs such as the one described in the next paragraph.)

An additional rule holds for combinations of {{math|j₁}}, {{math|j₂}}, and {{math|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 {{math|j_i}} is reversed, or if any of them are substituted with an {{math|m_i}} instead.

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

{{NumBlk||$$\backslash begin\{align\}$$

\langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle

&= (-1)^{-j_1 + j_2 - M} \sqrt{2 J + 1}

\begin{pmatrix}

j_1 & j_2 & J \\

m_1 & m_2 & -M

\end{pmatrix} \\

&= (-1)^{2 j_2} (-1)^{J - M} \sqrt{2 J + 1}

\begin{pmatrix}

j_1 & J & j_2 \\

m_1 & -M & m_2

\end{pmatrix}

\end{align}|{{EquationRef|3}}}}

The factor {{math|(−1)^{2 j₂}}} is due to the Condon–Shortley constraint that {{math|⟨j₁ j₁ j₂ (J − j₁){{!}}J J⟩ > 0}}, while {{math|(−1)^{J − M}}} is due to the time-reversed nature of {{math|{{!}}J M⟩}}.

This allows to reach the general expression:

:$$

\begin{align}

\langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle

& \equiv \delta(m_1+m_2,M) \sqrt{\frac{(2J+1)(j_1+j_2-J)!(j_1-j_2+J)!(-j_1+j_2+J)!}{(j_1+j_2+J+1)!}}\ \times {} \\[6pt]

&\times\sqrt{(j_1-m_1)!(j_1+m_1)!(j_2-m_2)!(j_2+m_2)!(J-M)!(J+M)!}\ \times {} \\[6pt]

&\times\sum_{k=K}^N \frac{(-1)^k}{k!(j_1+j_2-J-k)!(j_1-m_1-k)!(j_2+m_2-k)!(J-j_2+m_1+k)!(J-j_1-m_2+k)!},

\end{align}.

The summation is performed over those integer values {{mvar|k}} for which the argument of each factorial in the denominator is non-negative, i.e. summation limits {{mvar|K}} and {{mvar|N}} are taken equal: the lower one $K=\backslash max(0,\; j\_2-J-m\_1,\; j\_1-J+m\_2),$ the upper one $N=\backslash min(j\_1+j\_2-J,\; j\_1-m\_1,\; j\_2+m\_2).$ Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example, $j\_3>j\_1+j\_2$ or

{{main|Wigner D-matrix}}

\begin{align}

&\int_0^{2 \pi} d \alpha \int_0^\pi \sin \beta \, d\beta \int_0^{2 \pi} d \gamma \,

D^J_{M, K}(\alpha, \beta, \gamma)^*

D^{j_1}_{m_1, k_1}(\alpha, \beta, \gamma)

D^{j_2}_{m_2, k_2}(\alpha, \beta, \gamma) \\

{}={} &\frac{8 \pi^2}{2 J + 1}

\langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle

\langle j_1 \, k_1 \, j_2 \, k_2 | J \, K \rangle

\end{align}

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

\int_{4 \pi} Y_{\ell_1}^{m_1}{}^*(\Omega) Y_{\ell_2}^{m_2}{}^*(\Omega) Y_L^M (\Omega) \, d \Omega

= \sqrt{\frac{(2 \ell_1 + 1) (2 \ell_2 + 1)}{4 \pi (2 L + 1)}}

\langle \ell_1 \, 0 \, \ell_2 \, 0 | L \, 0 \rangle

\langle \ell_1 \, m_1 \, \ell_2 \, m_2 | L \, M \rangle

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 of a single spherical harmonic:

Y_{\ell_1}^{m_1}(\Omega) Y_{\ell_2}^{m_2}(\Omega)

= \sum_{L, M}

\sqrt{\frac{(2 \ell_1 + 1) (2 \ell_2 + 1)}{4 \pi (2 L + 1)}}

\langle \ell_1 \, 0 \, \ell_2 \, 0 | L \, 0 \rangle

\langle \ell_1 \, m_1 \, \ell_2 \, m_2 | L \, M \rangle

Y_L^M (\Omega)

$$\backslash sum\_m\; (-1)^\{j\; -\; m\}\; \backslash langle\; j\; \backslash ,\; m\; \backslash ,\; j\; \backslash ,\; (-m)\; |\; J\; \backslash ,\; 0\; \backslash rangle\; =\; \backslash delta\_\{J,\; 0\}\; \backslash sqrt\{2\; j\; +\; 1\}$$

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 SU(n) are known.{{harvnb|Alex|Kalus|Huckleberry|von Delft|2011}}{{harvnb|Kaplan|Resnikoff|1967}} 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.{{harvnb|de Swart|1963}}{{harvnb|Kaeding|1995}}{{cite web |last1=Coleman |first1=Sidney |title=Fun with SU(3) |url=https://inspirehep.net/literature/1720680 |website=INSPIREHep}} A [http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/ web interface for tabulating SU(N) Clebsch–Gordan coefficients] is readily available.

{{div col|colwidth=25em}}

• 3-j symbol
• 6-j symbol
• 9-j symbol
• Racah W-coefficient
• Spherical harmonics
• Spherical basis
• Tensor products of representations
• Associated Legendre polynomials
• Angular momentum
• Angular momentum coupling
• Total angular momentum quantum number
• Azimuthal quantum number
• Table of Clebsch–Gordan coefficients
• Wigner D-matrix
• Wigner–Eckart theorem
• Angular momentum diagrams (quantum mechanics)
• Clebsch–Gordan coefficient for SU(3)
• Littlewood–Richardson coefficient

{{div col end}}

{{reflist|group=nb}}

{{reflist|30em}}

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• {{cite journal|last=Kaeding|first=Thomas|arxiv=nucl-th/9502037|title=Tables of SU(3) isoscalar factors|year=1995 |doi=10.1006/adnd.1995.1011 |volume=61|issue=2|journal=Atomic Data and Nuclear Data Tables|pages=233–288 |bibcode=1995ADNDT..61..233K}}
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• Albert Messiah (1966). Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
• {{cite journal|first1=J. J.|last1=de Swart|title=The Octet model and its Clebsch-Gordan coefficients |year=1963 |doi=10.1103/RevModPhys.35.916|journal=Rev. Mod. Phys.|volume=35|issue=4|page=916 |bibcode=1963RvMP...35..916D |url=https://cds.cern.ch/record/345262|type=Submitted manuscript}}

• {{Cite journal|last1=Nakamura|first1=Kenzo|last2=Hagiwara|first2=Kaoru|display-authors=1|date=2010|title=Review of Particle Physics: Clebsch-Gordan coefficients, spherical harmonics, and d functions|url=http://pdg.lbl.gov/2011/reviews/rpp2011-rev-clebsch-gordan-coefs.pdf|journal=Journal of Physics G: Nuclear and Particle Physics|volume=37|issue=75021|page=368|doi=10.1088/0954-3899/37/7A/075021 |bibcode=2010JPhG...37g5021N |quote=Partial update for 2012 edition}}
• [http://www.volya.net/index.php?id=vc Clebsch–Gordan, 3-j and 6-j Coefficient Web Calculator]
• [http://phys.csuchico.edu/C-G/ Downloadable Clebsch–Gordan Coefficient Calculator for Mac and Windows]
• [http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/ Web interface for tabulating SU(N) Clebsch–Gordan coefficients]

{{refbegin}}

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{{refend}}

{{Physics operator}}

{{DEFAULTSORT:Clebsch-Gordan coefficients}}

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