Английская Википедия:Clebsch–Gordan coefficients

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Шаблон:Short description Шаблон:Use American EnglishIn 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 Шаблон:Math, Шаблон:Math, and Шаблон:Math that satisfy the commutation relations <math display="block"> \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} </math> where Шаблон:Math is the Levi-Civita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator, <math display="block"> \mathbf j = (\mathrm{j_x}, \mathrm{j_y}, \mathrm{j_z}). </math> 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 Шаблон:Math as the inner product of Шаблон:Math with itself: <math display="block"> \mathbf j^2 = \mathrm{j_x^2} + \mathrm{j_y^2} + \mathrm{j_z^2}. </math> This is an example of a Casimir operator. It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra <math>\mathfrak{so}(3,\mathbb{R}) \cong \mathfrak{su}(2)</math>. 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 (Шаблон:Math) and lowering (Шаблон:Math) operators, the so-called ladder operators, <math display="block"> \mathrm {j_\pm} = \mathrm{j_x} \pm i \mathrm{j_y}. </math>

Spherical basis for angular momentum eigenstates

It can be shown from the above definitions that Шаблон:Math commutes with Шаблон:Math, Шаблон:Math, and Шаблон:Math: <math display="block"> \begin{align}

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

\end{align} </math>

When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, Шаблон:Math and Шаблон:Math are chosen. From the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted Шаблон:Math where Шаблон:Math is the angular momentum quantum number and Шаблон:Math is the angular momentum projection onto the z-axis.

They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations, <math display="block"> \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} </math>

The raising and lowering operators can be used to alter the value of Шаблон:Math, <math display="block">

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

</math> where the ladder coefficient is given by: Шаблон:NumBlk

In principle, one may also introduce a (possibly complex) phase factor in the definition of <math>C_\pm(j, m)</math>. 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, <math display="block"> \langle j \, m | j' \, m' \rangle = \delta_{j, j'} \delta_{m, m'}. </math>

Here the italicized Шаблон:Math and Шаблон:Math denote integer or half-integer angular momentum quantum numbers of a particle or of a system. On the other hand, the roman Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, and Шаблон:Math denote operators. The <math>\delta</math> symbols are Kronecker deltas.

Tensor product space

We now consider systems with two physically different angular momenta Шаблон:Math and Шаблон:Math. 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 <math>V_1</math> of dimension <math>2j_1+1</math> and also on a space <math>V_2</math> of dimension <math>2j_2 + 1</math>. We are then going to define a family of "total angular momentum" operators acting on the tensor product space <math>V_1 \otimes V_2</math>, which has dimension <math>(2j_1+1)(2j_2+1)</math>. 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 Шаблон:Math be the Шаблон:Math-dimensional vector space spanned by the states <math display="block"> \begin{align}

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

\end{align}, </math> and Шаблон:Math the Шаблон:Math-dimensional vector space spanned by the states <math display="block"> \begin{align}

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

\end{align}. </math>

The tensor product of these spaces, Шаблон:Math, has a Шаблон:Math-dimensional uncoupled basis <math display="block">

 |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\}.

</math> Angular momentum operators are defined to act on states in Шаблон:Math in the following manner: <math display="block">

 (\mathbf j \otimes 1) |j_1 \, m_1 \, j_2 \, m_2\rangle \equiv \mathbf j |j_1 \, m_1\rangle \otimes |j_2 \, m_2\rangle

</math> and <math display="block">

 (1 \otimes \mathrm \mathbf j) |j_1 \, m_1 \, j_2 \, m_2\rangle \equiv |j_1 \, m_1\rangle \otimes \mathbf j |j_2 \, m_2\rangle,

</math> where Шаблон:Math denotes the identity operator.

The total angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on Шаблон:Math,

Шаблон:Equation box 1 The total angular momentum operators can be shown to satisfy the very same commutation relations, <math display="block">

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

</math> where Шаблон:Math. 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, <math display="block"> \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} </math> for Шаблон:Math. Note that it is common to omit the Шаблон:Math part.

The total angular momentum quantum number Шаблон:Math must satisfy the triangular condition that <math display="block">

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

</math> 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 Шаблон:Math: <math display="block">

 \sum_{J = |j_1 - j_2|}^{j_1 + j_2} (2 J + 1) = (2 j_1 + 1) (2 j_2 + 1) ~.

</math> As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension <math>2J+1</math>, where <math>J</math> ranges from <math>|j_1 - j_2|</math> to <math>j_1 + j_2</math> in increments of 1. As an example, consider the tensor product of the three-dimensional representation corresponding to <math>j_1 = 1</math> with the two-dimensional representation with <math>j_2 = 1/2</math>. The possible values of <math>J</math> are then <math>J = 1/2</math> and <math>J = 3/2</math>. 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 Шаблон:Math: <math display="block">

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

</math>

These rules may be iterated to, e.g., combine Шаблон:Mvar doublets (Шаблон:Mvar=1/2) to obtain the Clebsch-Gordan decomposition series, (Catalan's triangle), <math display="block">

 \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})~,

</math> where <math>\lfloor n/2 \rfloor</math> is the integer floor function; and the number preceding the boldface irreducible representation dimensionality (Шаблон:Math) 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, <math>{\mathbf 2}\otimes{\mathbf 2}\otimes{\mathbf 2} = {\mathbf 4}\oplus{\mathbf 2}\oplus{\mathbf 2}</math>.

Formal definition of Clebsch–Gordan coefficients

The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis Шаблон:NumBlk The expansion coefficients <math display="block"> \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle </math> are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as Шаблон:Math. Another common notation is Шаблон:Math.

Applying the operators <math display="block"> \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} </math> to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when <math display="block"> \begin{align}

 |j_1 - j_2| \leq J &\leq j_1 + j_2   \\
                  M &= m_1 + m_2.

\end{align} </math>

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<math display="block">

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

</math> to the left hand side of the defining equation gives<math display="block"> \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} </math> Applying the same operators to the right hand side gives<math display="block"> \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} </math>

Combining these results gives recursion relations for the Clebsch–Gordan coefficients, where Шаблон:Math was defined in Шаблон:EquationNote: <math display="block">

   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.

</math>

Taking the upper sign with the condition that Шаблон:Math gives initial recursion relation:<math display="block">

 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.

</math>In the Condon–Shortley phase convention, one adds the constraint that

<math>\langle j_1 \, j_1 \, j_2 \, (J - j_1) | J \, J\rangle > 0</math>

(and is therefore also real). The Clebsch–Gordan coefficients Шаблон:Math 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 Шаблон:Math must be one.

The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with Шаблон:Math. 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 <math display="block">

 \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

</math>

The first orthogonality relation is <math display="block">

   \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'}

</math> (derived from the fact that <math display="inline">\mathbf 1 = \sum_x |x\rangle \langle x|</math>) and the second one is <math display="block">

   \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'}.

</math>

Special cases

For Шаблон:Math the Clebsch–Gordan coefficients are given by <math display="block">

   \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}}.

</math>

For Шаблон:Math and Шаблон:Math we have <math display="block"> \langle j_1 \, j_1 \, j_2 \, j_2 | (j_1 + j_2) \, (j_1 + j_2) \rangle = 1.</math>

For Шаблон:Math and Шаблон:Math we have <math display="block">

   \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)!}}.

</math>

For Шаблон:Math we have <math display="block">

   \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)!}}.

</math>

For Шаблон:Math, Шаблон:Math we have <math display="block"> \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} </math>

For Шаблон:Math we have <math display="block"> \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} </math>

Symmetry properties

<math display="block"> \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} </math>

A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using Шаблон:EquationNote. 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 Шаблон:Math is not necessarily Шаблон:Math for a given quantum number Шаблон:Math unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule: <math display="block">(-1)^{4 k} = 1</math> for any angular-momentum-like quantum number Шаблон:Math.

Nonetheless, a combination of Шаблон:Math and Шаблон:Math is always an integer, so the stronger rule applies for these combinations: <math display="block">(-1)^{2 (j_i - m_i)} = 1</math> This identity also holds if the sign of either Шаблон:Math or Шаблон:Math or both is reversed.

It is useful to observe that any phase factor for a given Шаблон:Math pair can be reduced to the canonical form: <math display="block">(-1)^{a j_i + b (j_i - m_i)}</math> where Шаблон:Math and Шаблон:Math (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 Шаблон:Math pairs such as the one described in the next paragraph.)

An additional rule holds for combinations of Шаблон:Math, Шаблон:Math, and Шаблон:Math that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol: <math display="block">(-1)^{2 (j_1 + j_2 + j_3)} = 1</math> This identity also holds if the sign of any Шаблон:Math is reversed, or if any of them are substituted with an Шаблон:Math instead.

Шаблон:Anchor Relation to Wigner 3-j symbols

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

The factor Шаблон:Math is due to the Condon–Shortley constraint that Шаблон:Math, while Шаблон:Math is due to the time-reversed nature of Шаблон:Math.

Relation to Wigner D-matrices

Шаблон:Main <math display="block"> \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} </math>

Relation to spherical harmonics

In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics: <math display="block">

 \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

</math>

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: <math display="block">

 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)

</math>

Other properties

<math display="block">\sum_m (-1)^{j - m} \langle j \, m \, j \, (-m) | J \, 0 \rangle = \delta_{J, 0} \sqrt{2 j + 1}</math>

Clebsch–Gordan coefficients for specific groups

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. 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.

Clebsch–Gordan coefficients for symmetric group are also known as Kronecker coefficients.

See also

• 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

References

• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Citation
• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Cite book
• Albert Messiah (1966). Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
• Шаблон:Cite journal

External links

• Шаблон:Cite journal
• Clebsch–Gordan, 3-j and 6-j Coefficient Web Calculator
• Downloadable Clebsch–Gordan Coefficient Calculator for Mac and Windows
• Web interface for tabulating SU(N) Clebsch–Gordan coefficients

Further reading

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book