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In physics, the Clebsch–Gordan coefficients are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics.

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 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 (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem in invariant theory.

In terms of classical mathematics, the CG coefficients, or at least those associated to the group SO(3), may be defined much more directly, by means of formulae for the multiplication of spherical harmonics. The addition of spins in quantum-mechanical terms can be read directly from this approach. The formulas below use Dirac's bra-ket notation.

Clebsch–Gordan coefficients

Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis.

Below, this definition is made precise by defining angular momentum operators, angular momentum eigenstates, and tensor products of these states.

From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. To find numerical values for the coefficients a phase convention must be adopted. Below the Condon–Shortley phase convention is chosen.
Angular momentum operators

Angular momentum operators are self-adjoint operators \textrm{j}_x, \textrm{j}_y, and \textrm{j}_z that satisfy the commutation relations

$$[\textrm{j}_k,\textrm{j}_l] = \textrm{j}_k \textrm{j}_l - \textrm{j}_l \textrm{j}_k = i\hbar \sum_m \varepsilon_{kl m}\textrm{j}_m, \quad\mathrm{where}\quad k,l,m \in (x,y,z)$$

where $$\varepsilon_{klm} is the Levi-Civita symbol. Together the three operators define a "vector operator": \( \mathbf{j} = [\textrm{j}_x,\textrm{j}_y,\textrm{j}_z]$$

By developing this concept further, one can define an operator as an "inner product" of $${\mathbf j} with itself: \( \mathbf{j}^2 = \textrm{j}_x^2+\textrm{j}_y^2+\textrm{j}_z^2. \,$$

It is an example of a Casimir operator.

We also define raising $$(\textrm{j}_{+}) and lowering \( (\textrm{j}_{-})$$ operators:

$$\textrm{j}_\pm = \textrm{j}_x \pm i \textrm{j}_y. \,$$

Angular momentum states

It can be shown from the above definitions that $$\mathbf{j}^2$$ commutes with $$\textrm{j}_x, \textrm{j}_y and \textrm{j}_z$$

$$[\mathbf{j}^2, \textrm{j}_k] = 0\ \mathrm{for}\ k = x,y,z$$

When two Hermitian operators commute a common set of eigenfunctions exists. Conventionally $$\mathbf{j}^2$$ and $$\textrm{j}_z$$ are chosen. From the commutation relations the possible eigenvalues can be found. The result is

\begin{alignat}{2} \mathbf{j}^2 |j\,m\rangle = \hbar^2 j(j+1) |j\,m\rangle & \;\;\; j=0,\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\\ \textrm{j}_z|j\,m\rangle = \hbar m |j\,m\rangle & \;\;\; m = -j, -j+1, \ldots , j. \end{alignat}

The raising and lowering operators change the value of m

$$\textrm{j}_\pm |j\,m\rangle = C_\pm(j,m) |j\,m\pm 1\rangle$$

with

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

A (complex) phase factor could be included in the definition of $$C_\pm(j,m)$$ The choice made here is in agreement with the Condon and Shortley phase conventions. The angular momentum states must be orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and they are assumed to be normalized

$$\langle j_1\,m_1 | j_2\,m_2 \rangle = \delta_{j_1,j_2}\delta_{m_1,m_2}.$$

Tensor product space

Let $$V_1$$ be the $$2j_1+1$$ dimensional vector space spanned by the states

$$|j_1 m_1\rangle,\quad m_1=-j_1,-j_1+1,\ldots j_1$$

and $$V_2 the \( 2j_2+1 dimensional vector space spanned by \( |j_2 m_2\rangle,\quad m_2=-j_2,-j_2+1,\ldots j_2.$$

The tensor product of these spaces, $$V_{12}\equiv V_1\otimes V_2$$, has a $$(2j_1+1)(2j_2+1)$$ dimensional uncoupled basis

$$|j_1 m_1\rangle|j_2 m_2\rangle \equiv |j_1 m_1\rangle \otimes |j_2 m_2\rangle, \quad m_1=-j_1,\ldots j_1, \quad m_2=-j_2,\ldots j_2.$$

Angular momentum operators acting on $$V_{12}$$ can be defined by

$$(\textrm{j}_i \otimes 1)|j_1 m_1\rangle|j_2 m_2\rangle \equiv (j_i|j_1m_1\rangle) \otimes |j_2m_2\rangle$$

and

$$(1 \otimes \textrm{j}_i) |j_1 m_1\rangle|j_2 m_2\rangle \equiv |j_1m_1\rangle \otimes j_i|j_2m_2\rangle \quad\mathrm{for}\quad i = x,y,z.$$

Total angular momentum operators are defined by

$$\textrm{J}_i = \textrm{j}_i \otimes 1 + 1 \otimes \textrm{j}_i\quad\mathrm{for}\quad i = x,y,z.$$

The total angular momentum operators satisfy the required commutation relations

$$[\textrm{J}_k,\textrm{J}_l] = i\hbar\epsilon_{klm}\textrm{J}_m, \quad \mathrm{where}\quad k,l,m \in (x,y,z)$$

and hence total angular momentum eigenstates exist

\begin{align} \mathbf{J}^2 |(j_1j_2)JM\rangle &= \hbar^2 J(J+1) |(j_1j_2)JM\rangle \\ \textrm{J}_z |(j_1j_2)JM\rangle &= \hbar M |(j_1j_2)JM\rangle,\quad \mathrm{for}\quad M=-J,\ldots,J. \end{align}

It can be derived that J must satisfy the triangular condition

$$|j_1-j_2| \leq J \leq j_1+j_2.$$

The total number of total angular momentum eigenstates is equal to the dimension of $$V_{12}$$

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

The total angular momentum states form an orthonormal basis of $$V_{12}$$

$$\langle J_1 M_1 | J_2 M_2 \rangle = \delta_{J_1J_2}\delta_{M_1M_2}.$$

Formal definition of Clebsch–Gordan coefficients

The total angular momentum states can be expanded with the use of the completeness relation in the uncoupled basis

$$|(j_1j_2)JM\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} |j_1m_1j_2m_2\rangle \langle j_1m_1j_2m_2|JM\rangle$$

The expansion coefficients $$\langle j_1m_1j_2m_2|JM\rangle$$ are called Clebsch–Gordan coefficients.

Applying the operator

$$\textrm{J}_z = \textrm{j}_z \otimes 1 + 1 \otimes \textrm{j}_z$$

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 the physicist Giulio Racah. Applying the total angular momentum raising and lowering operators

$$\textrm{J}_\pm = \textrm{j}_\pm \otimes 1 + 1 \otimes \textrm{j}_\pm$$

to the left hand side of the defining equation gives

$$\textrm{J}_\pm|(j_1j_2)JM\rangle = C_\pm(J,M) |(j_1j_2)JM\pm 1\rangle = C_\pm(J,M)\sum_{m_1m_2}|j_1m_1\rangle|j_2m_2\rangle \langle j_1 m_1 j_2 m_2|J M\pm 1\rangle.$$

Applying the same operators to the right hand side gives

\begin{align} \textrm{J}_\pm & \sum_{m_1m_2} |j_1m_1\rangle|j_2m_2\rangle \langle j_1m_1j_2m_2|JM\rangle\\ & =\sum_{m_1m_2}\left[ C_\pm(j_1,m_1)|j_1 m_1\pm 1\rangle |j_2m_2\rangle +C_\pm(j_2,m_2)|j_1 m_1\rangle |j_2 m_2\pm 1\rangle \right] \langle j_1 m_1 j_2 m_2|J M\rangle \\ &= \sum_{m_1m_2} |j_1m_1\rangle|j_2m_2\rangle \left[ 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 \right]. \end{align}

where

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

Combining these results gives recursion relations for the Clebsch–Gordan coefficients

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

Taking the upper sign with M=J gives

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

In the Condon and Shortley phase convention the coefficient $$\langle j_1 j_1 j_2 J-j_1|J J\rangle$$ is taken real and positive. With the last equation all other Clebsch–Gordan coefficients $$\langle j_1 m_1 j_2 m_2|J J\rangle$$ can be found. The normalization is fixed by the requirement that the sum of the squares, which corresponds to the norm of the state $$|(j_1j_2)JJ\rangle$$ 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 and Shortley phase convention).
Explicit expression

For an explicit expression of the Clebsch–Gordan coefficients and tables with numerical values, see table of Clebsch–Gordan coefficients.
Orthogonality relations

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

$$\sum_{J=|j_1-j_2|}^{j_1+j_2} \sum_{M=-J}^{J} \langle j_1 m_1 j_2 m_2|J M\rangle\langle J M|j_1 m_1' j_2 m_2'\rangle= \langle j_1 m_1 j_2 m_2 | j_1 m_1' j_2 m_2'\rangle = \delta_{m_1,m_1'}\delta_{m_2,m_2'}$$

(using the completeness relation that $$1\equiv \sum_x | x \rangle\langle x|$$ )and the second

$$\sum_{m_1m_2} \langle J M|j_1 m_1 j_2 m_2\rangle \langle j_1 m_1 j_2 m_2|J' M'\rangle = \langle J M | J' M'\rangle = \delta_{J,J'}\delta_{M,M'}.$$

Special cases

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

$$\langle j_1 m_1 j_2 m_2 | 0 0\rangle = \delta_{j_1,j_2}\delta_{m_1,-m_2} \frac{(-1)^{j_1-m_1}}{\sqrt{2j_2+1}}.$$

For $$J=j_1+j_2$$ and M=J we have

$$\langle j_1 j_1 j_2 j_2 | (j_1+j_2) (j_1+j_2)\rangle = 1.$$

For $$j_1 = j_2 = J/2 and \( m_2 = -m_1$$ we have

$$\langle j_1 m_1 j_1 -m_1 | 2j_1 0\rangle = \frac{(2j_1)!^2}{(j_1 - m_1)! (j_1 + m_1)! \sqrt{(4 j_1)!}}.$$

For $$j_1 = j_2 = m_1 = -m_2$$ we have

$$\langle j_1 j_1 j_1 -j_1 | J 0\rangle = (2j_1)! \sqrt{\frac{2J+1}{(J+2j_1+1)!(2j_1 - J)!}}.$$

Symmetry properties

\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 3-jm symbols using the equation given below. The symmetry properties of 3-jm symbols are much simpler. Care is needed when simplifying phase factors, because the quantum numbers can be integer or half integer, e.g., $$(-1)^{2j}$$ is equal to 1 for integer j and equal to −1 for half-integer j. The following relations, however, are valid in either case:

$$(-1)^{4j} = (-1)^{2(j-m)} = 1$$

and for $$j_1, j_2,$$ and J appearing in the same Clebsch–Gordan coefficient:

$$(-1)^{2(j_1+j_2+J)} = (-1)^{2(m_1+m_2+M)} = 1.$$

Relation to 3-jm symbols

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

$$\langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle = (-1)^{j_1-j_2+m_3}\sqrt{2j_3+1} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & -m_3 \end{pmatrix}.$$

Relation to Wigner D-matrices

$$\int_0^{2\pi} d\alpha \int_0^\pi \sin\beta d\beta \int_0^{2\pi} d\gamma D^J_{MK}(\alpha,\beta,\gamma)^\ast D^{j_1}_{m_1k_1}(\alpha,\beta,\gamma) D^{j_2}_{m_2k_2}(\alpha,\beta,\gamma) = \frac{8\pi^2}{2J+1} \langle j_1 m_1 j_2 m_2 | J M \rangle \langle j_1 k_1 j_2 k_2 | J K \rangle.$$

Other Properties

$$\sum_m (-1)^{j-m} \langle j m j {-m} | J 0 \rangle = \sqrt{2j+1} ~ \delta_{J0}$$

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.[1] A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.

3-jm symbol
Racah W-coefficient
6-j symbol
9-j symbol
Spherical harmonics
Associated Legendre polynomials
Angular momentum
Angular momentum coupling
Total angular momentum quantum number
Azimuthal quantum number
Table of Clebsch–Gordan coefficients
Wigner D-matrix

References

^ Alex, A.; M. Kalus, A. Huckleberry, and J. von Delft (February 2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients". J. Math. Phys. 82: 023507. Bibcode 2011JMP....52b3507A. doi:10.1063/1.3521562. Retrieved 2011-04-13.

PDF Table of Clebsch–Gordan Coefficients, Spherical Harmonics, and d-Functions
Clebsch–Gordan, 3-j and 6-j Coefficient Web Calculator
Web interface for tabulating SU(N) Clebsch–Gordan coefficients

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