View source for Physics:Clebsch–Gordan coefficients

{{Short description|Coefficients in angular momentum eigenstates of quantum systems
}}
 In [[Physics:Physics|physics]], the '''Clebsch–Gordan''' ('''CG''') '''coefficients''' are numbers that arise in [[Physics:Angular momentum coupling|angular momentum coupling]] in [[Physics:Quantum mechanics|quantum mechanics]]. They appear as the expansion coefficients of [[Physics:Total angular momentum|total angular momentum]] eigenstates in an uncoupled [[Tensor product|tensor product]] basis. In more mathematical terms, the CG coefficients are used in [[Representation theory|representation theory]], particularly of compact Lie groups, to perform the explicit direct sum decomposition of the [[Tensor product of representations|tensor product]] of two [[Irreducible representation|irreducible representation]]s (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 [[Biography:Alfred Clebsch|Alfred Clebsch]] and [[Biography:Paul Gordan|Paul Gordan]], who encountered an equivalent problem in [[Invariant theory|invariant theory]].

From a [[Vector calculus|vector calculus]] perspective, the CG coefficients associated with the [[SO(3)]] [[Group (mathematics)|group]] can be defined simply in terms of integrals of products of [[Physics:Spherical harmonic|spherical harmonic]]s and their complex conjugates. The addition of spins in quantum-mechanical terms can be read directly from this approach as spherical harmonics are [[Eigenfunction|eigenfunction]]s of total angular momentum and projection thereof onto an axis, and the integrals correspond to the [[Hilbert space]] inner product.<ref>{{harvnb|Greiner|Müller|1994}}</ref> 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.<ref>{{harvnb|Edmonds|1957}}</ref>

The formulas below use [[Biography:Paul Dirac|Dirac's]] [[Bra–ket notation|bra–ket notation]] and the [[Physics:Spherical harmonics#Condon–Shortley phase|Condon–Shortley phase convention]]<ref>{{harvnb|Condon|Shortley|1970}}</ref> is adopted.

== Review of the angular momentum operators ==

Angular momentum operators are [[Self-adjoint operator|self-adjoint operator]]s {{math|j<sub>x</sub>}}, {{math|j<sub>y</sub>}}, and {{math|j<sub>z</sub>}} 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 {{math|''ε''<sub>''klm''</sub>}} is the Levi-Civita<!-- never change to a dash!!! --> symbol. Together the three operators define a ''vector operator'', a rank one Cartesian [[Tensor operator|tensor operator]],
<math display="block">
\mathbf j = (\mathrm{j_x}, \mathrm{j_y}, \mathrm{j_z}).
</math>
It is also known as a [[Spherical basis|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|'''j'''<sup>2</sup>}} as the inner product of {{math|'''j'''}} 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 [[Representation theory of SU(2)|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'' ({{math|j<sub>+</sub>}}) and ''lowering'' ({{math|j<sub>−</sub>}}) operators, the so-called [[Physics:Ladder operator#Angular momentum|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 {{math|'''j'''<sup>2</sup>}} commutes with {{math|j<sub>x</sub>}}, {{math|j<sub>y</sub>}}, and {{math|j<sub>z</sub>}}:
<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, {{math|'''j'''<sup>2</sup>}} and {{math|j<sub>z</sub>}} 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 [[Wigner D-matrix#Definition of the Wigner D-matrix|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 {{math|''m''}},
<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:
{{NumBlk||<math display="block">
  C_\pm(j, m) = \sqrt{j (j + 1) - m (m \pm 1)} = \sqrt{(j \mp m)(j \pm m + 1)}.
</math>|{{EquationRef|1}}}}

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 [[Physics:Spherical harmonics#Condon–Shortley phase|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 {{math|''j''}} and {{math|''m''}} denote integer or half-integer [[Physics:Angular momentum operator|angular momentum]] quantum numbers of a particle or of a system.  On the other hand, the roman {{math|j<sub>x</sub>}}, {{math|j<sub>y</sub>}}, {{math|j<sub>z</sub>}}, {{math|j<sub>+</sub>}}, {{math|j<sub>−</sub>}}, and {{math|'''j'''<sup>2</sup>}} denote operators. The <math>\delta</math> symbols are [[Kronecker delta]]s.


== Tensor product space ==
We now consider systems with two physically different angular momenta {{math|''j''<sub>1</sub>}} and {{math|''j''<sub>2</sub>}}. 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|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 {{math|''V''<sub>1</sub>}} be the {{math|(2 ''j''<sub>1</sub> + 1)}}-dimensional [[Vector space|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 {{math|''V''<sub>2</sub>}} the {{math|(2 ''j''<sub>2</sub> + 1)}}-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, {{math|''V''<sub>3</sub> ≡ ''V''<sub>1</sub> ⊗ ''V''<sub>2</sub>}}, has a {{math|(2 ''j''<sub>1</sub> + 1) (2 ''j''<sub>2</sub> + 1)}}-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 {{math|''V''<sub>3</sub>}} 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 {{math|1}} denotes the identity operator.

The '''total'''<ref group=nb>The 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|'''j'''<sub>1</sub>}} and {{math|'''j'''<sub>2</sub>}}.  It is not to be confused with the other common use of the term "total angular momentum" that refers specifically to the sum of [[Physics:Angular momentum operator|orbital angular momentum]] and [[Physics:Spin|spin]].</ref> '''angular momentum''' operators are defined by the [[Tensor algebra#Coproduct|coproduct]] (or [[Representation theory#Tensor products of representations|tensor product]]) of the two representations acting on {{math|''V''<sub>1</sub>⊗''V''<sub>2</sub>}},

{{Equation box 1
|indent =:
|equation = <math>\mathbf{J} \equiv \mathbf{j}_1 \otimes 1 + 1 \otimes \mathbf{j}_2~.</math>
|cellpadding= 6
|border
|border colour = #0070BF
|bgcolor=#FAFFFB
}}
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 {{math|''k'', ''l'', ''m'' ∈ {{mset|''x'', ''y'', ''z''}}}}. Indeed, the preceding construction is the standard method<ref>{{harvnb|Hall|2015}} Section 4.3.2</ref> 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 {{math|''M'' ∈ {{mset|−''J'', −''J'' + 1, ..., ''J''}}}}.  Note that it is common to omit the {{math|[''j''<sub>1</sub> ''j''<sub>2</sub>]}} part.

The total angular momentum quantum number {{math|''J''}} 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.<ref>{{harvnb|Merzbacher|1998}}</ref>

The total number of total angular momentum eigenstates is necessarily equal to the dimension of {{math|''V''<sub>3</sub>}}:
<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.<ref>{{harvnb|Hall|2015}} Appendix C</ref> 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 {{math|''V''<sub>3</sub>}}:
<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 {{mvar|n}} doublets ({{mvar|s}}=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 ({{math|2''j''+1}}) label indicates multiplicity of that representation in the representation reduction.<ref>{{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 }}</ref> 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
{{NumBlk||<math display="block">
  |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
</math>|{{EquationRef|2}}}}
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 {{math|{{bra-ket|''j''<sub>1</sub> ''j''<sub>2</sub>; ''m''<sub>1</sub> ''m''<sub>2</sub> | ''J'' ''M''}}}}. Another common notation is
{{math|1={{bra-ket|''j''<sub>1</sub> ''m''<sub>1</sub> ''j''<sub>2</sub> ''m''<sub>2</sub> | ''J'' ''M''}} = ''C''{{su|p=''JM''|b=''j''<sub>1</sub>''m''<sub>1</sub>''j''<sub>2</sub>''m''<sub>2</sub>}}}}.

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 {{math|''C''<sub>±</sub>}} was defined in {{EquationNote|1}}:
<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><nowiki> </nowiki>

Taking the upper sign with the condition that {{math|''M'' {{=}} ''J''}} 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 {{math|⟨''j''<sub>1</sub> ''m''<sub>1</sub> ''j''<sub>2</sub> ''m''<sub>2</sub> {{!}} ''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|{{!}}[''j''<sub>1</sub> ''j''<sub>2</sub>] ''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.

== Explicit expression ==

== 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 {{math|''J'' {{=}} 0}} 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 {{math|''J'' {{=}} ''j''<sub>1</sub> + ''j''<sub>2</sub>}} and {{math|''M'' {{=}} ''J''}} 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 {{math|''j''<sub>1</sub> {{=}} ''j''<sub>2</sub> {{=}} ''J'' / 2}} and {{math|''m''<sub>1</sub> {{=}} −''m''<sub>2</sub>}} 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 {{math|''j''<sub>1</sub> {{=}} ''j''<sub>2</sub> {{=}} ''m''<sub>1</sub> {{=}} −''m''<sub>2</sub>}} 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 {{math|''j''<sub>2</sub> {{=}} 1}}, {{math|''m''<sub>2</sub> {{=}} 0}} 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 {{math|''j''<sub>2</sub> {{=}} 1/2}} 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 {{EquationNote|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 {{math|(−1)<sup>2''k''</sup>}} 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:
<math display="block">(-1)^{4 k} = 1</math>
for any angular-momentum-like quantum number {{math|''k''}}.

Nonetheless, a combination of {{math|''j<sub>i</sub>''}} and {{math|''m<sub>i</sub>''}} 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 {{math|''j<sub>i</sub>''}} or {{math|''m<sub>i</sub>''}} or both is reversed.

It is useful to observe that any phase factor for a given {{math|(''j<sub>i</sub>'', ''m<sub>i</sub>'')}} pair can be reduced to the canonical form:
<math display="block">(-1)^{a j_i + b (j_i - m_i)}</math>
where {{math|''a'' ∈ {0, 1, 2, 3}<nowiki/>}} and {{math|''b'' ∈ {0, 1}<nowiki/>}} (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<sub>i</sub>'', ''m<sub>i</sub>'')}} pairs such as the one described in the next paragraph.)

An additional rule holds for combinations of {{math|''j''<sub>1</sub>}}, {{math|''j''<sub>2</sub>}}, and {{math|''j''<sub>3</sub>}} 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 {{math|''j<sub>i</sub>''}} is reversed, or if any of them are substituted with an {{math|''m<sub>i</sub>''}} instead.

== {{anchor|Relation_to_3-jm_symbols}} Relation to Wigner 3-j symbols ==

Clebsch–Gordan coefficients are related to Wigner 3-j symbols which have more convenient symmetry relations.
{{NumBlk||<math display="block">\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}</math>|{{EquationRef|3}}}}

The factor {{math|(−1)<sup>2 ''j''<sub>2</sub></sup>}} is due to the Condon–Shortley constraint that {{math|⟨''j''<sub>1</sub> ''j''<sub>1</sub> ''j''<sub>2</sub> (''J'' − ''j''<sub>1</sub>){{!}}''J J''⟩ > 0}}, while {{math|(–1)<sup>''J'' − ''M''</sup>}} is due to the time-reversed nature of {{math|{{!}}''J M''⟩}}.

== Relation to Wigner D-matrices ==
{{main|Wigner D-matrix}}
<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 [[Integral|integral]]s of [[Physics:Spherical harmonic|spherical harmonic]]s:
<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|special unitary group]] SU(''n'') are known.<ref>{{harvnb|Alex|Kalus|Huckleberry|von Delft|2011}}</ref><ref>{{harvnb|Kaplan|Resnikoff|1967}}</ref> 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 [[Physics:Up quark|up]], [[Physics:Down quark|down]], and [[Physics:Strange quark|strange]] quarks.<ref>{{harvnb|de Swart|1963}}</ref><ref>{{harvnb|Kaeding|1995}}</ref><ref>{{cite web |last1=Coleman |first1=Sidney |title=Fun with SU(3) |url=https://inspirehep.net/literature/1720680 |website=INSPIREHep}}</ref> A [http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/ web interface for tabulating SU(N) Clebsch–Gordan coefficients] is readily available.

Clebsch–Gordan coefficients for symmetric group are also known as [[Kronecker coefficient]]s.

==See also==
{{div col|colwidth=25em}}
* [[Physics:3-j symbol|3-j symbol]]
* [[Physics:6-j symbol|6-j symbol]]
* [[Physics:9-j symbol|9-j symbol]]
* [[Racah W-coefficient]]
* [[Physics:Spherical harmonics|Spherical harmonics]]
* [[Spherical basis]]
* [[Representation theory#Tensor products of representations|Tensor products of representations]]
* [[Physics:Associated Legendre polynomials|Associated Legendre polynomials]]
* [[Physics:Angular momentum|Angular momentum]]
* [[Physics:Angular momentum coupling|Angular momentum coupling]]
* [[Physics:Total angular momentum quantum number|Total angular momentum quantum number]]
* [[Physics:Azimuthal quantum number|Azimuthal quantum number]]
* [[Table of Clebsch–Gordan coefficients]]
* [[Wigner D-matrix]]
* [[Wigner–Eckart theorem]]
* [[Physics:Angular momentum diagrams (quantum mechanics)|Angular momentum diagrams (quantum mechanics)]]
* Clebsch–Gordan coefficient for SU(3)
* Littlewood–Richardson coefficient
{{div col end}}

==Remarks==
{{reflist|group=nb}}

==Notes==
{{reflist|30em}}

==References==
*{{cite journal|first1=A.|last1=Alex|first2=M.|last2=Kalus|first3=A.|last3=Huckleberry|first4=J.|last4=von Delft |title=A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients|journal=J. Math. Phys.|volume=82|issue=2 |year=2011|pages=023507|doi=10.1063/1.3521562|bibcode=2011JMP....52b3507A|arxiv=1009.0437|s2cid=55572438 }}
*{{cite book|last1=Condon|first1=Edward U.|last2=Shortley|first2=G. H.|title=The Theory of Atomic Spectra|url=https://archive.org/details/in.ernet.dli.2015.212979|year=1970|publisher={{wipe|Cambridge University Press}}|location=Cambridge|isbn= 978-0-521-09209-8| chapter=Ch. 3}}
* {{cite book|last=Edmonds|first=A. R.|title=Angular Momentum in Quantum Mechanics|year=1957|publisher=Princeton University Press|location=Princeton, New Jersey|isbn=978-0-691-07912-7|url-access=registration|url=https://archive.org/details/angularmomentumi0000edmo}}
*{{cite book|last1=Greiner|first1=Walter|last2=Müller|first2=Berndt|title=Quantum Mechanics: Symmetries|url=https://archive.org/details/quantummechanics0001grei|url-access=registration|edition=2nd|year=1994|publisher=Springer Verlag|isbn=978-3540580805}}
* {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition= 2nd|series=Graduate Texts in Mathematics|volume=222 |publisher=Springer|year=2015|isbn=978-3319134666}}
*{{cite journal|first1=L. M.|last1=Kaplan|first2=M.|last2=Resnikoff|title=Matrix products and explicit 3, 6, 9, and 12j coefficients of the regular representation of SU(n)|year=1967|doi=10.1063/1.1705141|journal=J. Math. Phys.|volume=8|issue=11 |page=2194|bibcode = 1967JMP.....8.2194K }}
*{{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}}
*{{cite book|last=Merzbacher|first=Eugen|title=Quantum Mechanics|url=https://archive.org/details/quantummechanics00merz_136|url-access=limited|edition=3rd|year=1998|publisher=John Wiley|pages=[https://archive.org/details/quantummechanics00merz_136/page/n441 428]–9|isbn= 978-0-471-88702-7}}
* [[Biography:Albert Messiah|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}}

==External links==
*{{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]

==Further reading==
{{refbegin}}
*{{cite book |first1=E. |last1=Zaarur |first2=Y. |last2=Peleg |first3=R. |last3=Pnini |title=Quantum mechanics |publisher=McGraw Hill |series=Schaum's Easy Oulines Crash Course |date=2006 |isbn=978-007-145533-6}}
*{{cite book |edition=2nd |first1=R. |last1=Eisberg |first2=R. |last2=Resnick |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles |publisher=Wiley |date=1985 |isbn=978-0-471-87373-0}}
*{{cite book |first=E. |last=Abers |title=Quantum Mechanics |publisher=Prentice Hall |date=2004 |isbn=978-0-13-146100-0}}
*{{cite book |first1=B.H. |last1=Bransden |first2=C.J. |last2=Joachain |title=Physics of Atoms and Molecules |publisher=Longman |date=1983 |isbn=0-582-44401-2}}
*{{cite book |first=G. |last=Woan |title=The Cambridge Handbook of Physics Formulas |publisher=Cambridge University Press |date=2010 |isbn=978-0-521-57507-2}}
*{{cite book |edition=2nd |first1=R.G. |last1=Lerner |first2=G.L. |last2=Trigg |title=Encyclopaedia of Physics |publisher=VHC publishers |date=1991 |isbn=0-89573-752-3}}
*{{cite book |first=C.B. |last=Parker |title=McGraw Hill Encyclopaedia of Physics |edition=2nd |date=1994 |isbn=0-07-051400-3}}
*{{cite book |last1= Biedenharn |first1=L.C. |last2=Louck |first2=J.D. |title= Angular Momentum in Quantum Physics |year= 1981 |publisher= Addison-Wesley |location= Reading, Massachusetts |isbn= 978-0-201-13507-7 }}
*{{cite book |last1= Brink |first1= D.M. |last2= Satchler |first2=G.R. |title= Angular Momentum |year= 1993 |edition= 3rd |publisher= Clarendon Press |isbn= 978-0-19-851759-7 |chapter=2. Representations of the Rotation Group |url-access= registration |url= https://archive.org/details/angularmomentum0000brin }}
*{{cite book |last= Messiah |first= Albert |title= Quantum Mechanics |volume=II |year= 1981 |publisher= North Holland |isbn= 978-0-7204-0045-8 |chapter=XIII. Angular Momentum in Quantum Mechanics |pages=507– |chapter-url={{GBurl|VR93vUk8d_8C|p=507}}}}
*{{cite book |last= Zare |first= Richard N. |title= Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics |year=1988 | publisher=Wiley |isbn=978-0-471-85892-8 |chapter=2. Coupling of Two Angular Momentum Vectors |pages=43–}}
{{refend}}

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