View source for Clebsch–Gordan coefficients

{{Use American English|date=January 2019}}{{Short description|Coefficients in angular momentum eigenstates of quantum systems
}}
In [[physics]], the '''Clebsch–Gordan''' ('''CG''') '''coefficients''' are numbers that arise in [[angular momentum coupling]] in [[quantum mechanics]].  They appear as the expansion coefficients of [[total angular momentum]] [[eigenstate]]s in an uncoupled [[tensor product]] basis. In more mathematical terms, the CG coefficients are used in [[representation theory]], particularly of [[compact Lie group]]s, to perform the explicit [[direct sum of representations|direct sum]] decomposition of the [[tensor product of representations|tensor product]] of two [[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 [[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 (mathematics)|group]] can be defined simply in terms of integrals of products of [[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]]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 [[Paul Dirac|Dirac's]] [[bra–ket notation]] and the [[spherical harmonics#Condon–Shortley phase|Condon–Shortley phase convention]]<ref>{{harvnb|Condon|Shortley|1970}}</ref> is adopted.

== Angular momentum operators ==

Angular momentum operators are [[self-adjoint operator]]s {{math|j<sub>x</sub>}}, {{math|j<sub>y</sub>}}, and {{math|j<sub>z</sub>}} that satisfy the [[commutation relation]]s
:<math>
\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 permutation symbol|Levi-Civita<!-- never change to a dash!!! --> symbol]]. Together the three operators define a ''vector operator'', a rank one Cartesian [[tensor operator]],
:<math>
\mathbf j = (\mathrm{j_x}, \mathrm{j_y}, \mathrm{j_z}).
</math>
It 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>
\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|'''so'''(3) ≅ '''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<sub>+</sub>}}) and ''lowering'' ({{math|j<sub>−</sub>}}) operators, the so-called [[ladder operator#Angular momentum|ladder operators]],
:<math>
\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>
\begin{align}
  &[\mathbf j^2, \mathrm {j}_k] = 0 & k &\in \{\mathrm x, \mathrm y, \mathrm z\}.
\end{align}
</math>

When two [[Hermitian operator]]s 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|{{!}}''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>
\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>
  \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>
  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 [[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> 
\langle j \, m | j' \, m' \rangle = \delta_{j, j'} \delta_{m, m'}.
</math>

Here the italicized {{math|''j'' and ''m''}} denote integer or half-integer [[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.

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

== 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]] 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]] spanned by the states
:<math>
\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>
\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>
  |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>
  (\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>
  (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 [[Angular momentum operator|orbital angular momentum]] and [[spin (physics)|spin]].</ref> '''angular momentum''' operators are defined by the [[coalgebra|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} \otimes 1 + 1 \otimes \mathbf{j}   ~.
</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>
  [\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>
\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>
  |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>
  \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>
  \langle J \, M | J' \, M' \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>
{\mathbf 2}^{\otimes n} = \bigoplus_{k=0}^{\lfloor n/2 \rfloor}~
\Bigl( {n+1-2k \over n+1}  {n+1 \choose k}\Bigr)~~({\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 (2{{mvar|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 |url= |journal= Modern Physics Letters |volume=A7 |issue= 18|pages=1595–1600 |doi=10.1142/S0217732392001270|arxiv = hep-th/9203027 |bibcode = 1992MPLA....7.1595Z }}</ref>
For instance, from this formula,  addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s,  &nbsp;  <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>
  |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>
\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|⟨''j''<sub>1</sub> ''j''<sub>2</sub>; ''m''<sub>1</sub> ''m''<sub>2</sub>{{!}}''J'' ''M''⟩}}. Another common notation is
{{math|⟨''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 operator
:<math>
  \mathrm J_{\mathrm z} = \mathrm j_{\mathrm z} \otimes 1 + 1 \otimes \mathrm j_{\mathrm z}
</math>
to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when
:<math>
  M = m_1 + m_2
</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>
  \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>
\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>
\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)
  \text{.}
\end{align}
</math>
where {{math|''C''<sub>±</sub>}} was defined in {{EquationNote|1}}.  Combining these results gives recursion relations for the Clebsch–Gordan coefficients:
:<math>
  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 {{math|''M'' {{=}} ''J''}} gives initial recursion relation:
:<math>
  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 ==
{{for|an explicit expression of the Clebsch–Gordan coefficients and tables with numerical values|table of Clebsch–Gordan coefficients}}

== Orthogonality relations ==

These are most clearly written down by introducing the alternative notation
:<math>
  \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>
    \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|1 ≡ ∑<sub>''x''</sub> {{!}}''x''⟩ ⟨''x''{{!}}}}) and the second one is
:<math>
    \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>
    \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>
 \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>
    \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>
    \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>
\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>
\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>
\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 symbol]]s 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>(-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>(-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>(-1)^{a j_i + b (j_i - m_i)}</math>
where {{math|''a'' ∈ {0, 1, 2, 3<nowiki>}</nowiki>}} and {{math|''b'' ∈ {0, 1<nowiki>}</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>(-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 symbol]]s which have more convenient symmetry relations.

{{NumBlk|:|<math>\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>
\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]]s of [[spherical harmonic]]s:

:<math>
  \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 a single spherical harmonic:

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

== 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.<ref>{{harvnb|Alex|Kalus|Huckleberry|von Delft|2011}}</ref><ref>{{harvnb|Kaplan|Resnikoff|1967}}</ref> In particular, [[Clebsch-Gordan coefficient for SU(3)|SU(3) Clebsch-Gordan coefficients]] have been computed and tabulated because of their utility in characterizing hadronic decays, where a [[Flavor (particle physics)|flavor]]-SU(3) symmetry exists that relates the [[Up quark|up]], [[Down quark|down]], and [[Strange quark|strange]] quarks.<ref>{{harvnb|de Swart|1963}}</ref><ref>{{harvnb|Kaeding|1995}}</ref> A [http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/ web interface for tabulating SU(N) Clebsch–Gordan coefficients] is readily available.

==See also==
{{div col|colwidth=25em}}
* [[3-j symbol]]
* [[6-j symbol]]
* [[9-j symbol]]
* [[Racah W-coefficient]]
* [[Spherical harmonics]]
* [[Spherical basis]]
*[[Representation theory#Tensor products of representations|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}}

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

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

==References==
*{{cite journal|ref=harv|first=A.|last=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}}
*{{cite book|ref=harv|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=[[Cambridge University Press]]|location=Cambridge|isbn= 978-0-521-09209-8|chapter=Ch. 3}}
* {{cite book|ref=harv|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}}
*{{cite book|ref=harv|last1=Greiner|first1=Walter|authorlink1=Walter Greiner|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|ref=harv|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|ref=harv|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|ref=harv|last=Merzbacher|first=Eugen|authorlink=Eugen Merzbacher|title=Quantum Mechanics|edition=3rd|year=1998|publisher=John Wiley|pages=428–9|isbn= 978-0-471-88702-7}}
* [[Albert Messiah]] (1966). ''Quantum Mechanics'' (Vols. I & II), English translation from French by G. M. Temmer. North Holland,  John Wiley & Sons.
*{{cite journal|ref=harv|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=http://cds.cern.ch/record/345262|type=Submitted manuscript}}

==External links==
*{{Cite journal|last=Nakamura|first=Kenzo|last2=Hagiwara|first2=Kaoru|displayauthors=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|quote=Partial update for 2012 edition|via=}}
* [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==
* ''Quantum mechanics'', E. Zaarur, Y. Peleg, R. Pnini, Schaum's Easy Oulines Crash Course, McGraw Hill (USA), 2006, {{ISBN|978-007-145533-6}}
* ''Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles'' (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, {{ISBN|978-0-471-87373-0}}
* ''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, {{ISBN|978-0-13-146100-0}}
* ''Physics of Atoms and Molecules'', B. H. Bransden, C. J. Joachain, Longman, 1983, {{ISBN|0-582-44401-2}}
* ''The Cambridge Handbook of Physics Formulas'', G. Woan, Cambridge University Press, 2010, {{ISBN|978-0-521-57507-2}}.
* ''Encyclopaedia of Physics'' (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
* ''McGraw Hill Encyclopaedia of Physics'' (2nd Edition), C. B. Parker, 1994, {{ISBN|0-07-051400-3}}
* {{cite book |last= Biedenharn |first= L. C. |author2=Louck, J. D. |title= Angular Momentum in Quantum Physics
    |year= 1981 |publisher= Addison-Wesley |location= Reading, Massachusetts |isbn= 978-0-201-13507-7 }}
* {{cite book |last= Brink |first= D. M. |author2=Satchler, G. R.  |title= Angular Momentum
    |year= 1993 |edition= 3rd|publisher= Clarendon Press |location= Oxford |isbn= 978-0-19-851759-7 |chapter= Ch. 2 }}
* {{cite book |last= Messiah |first= Albert |title= Quantum Mechanics (Volume II) |year= 1981
    |publisher= North Holland Publishing |location= New York |isbn= 978-0-7204-0045-8
|chapter=Ch. XIII}}
* {{cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988
    |publisher= John Wiley & Sons |location= New York |isbn= 978-0-471-85892-8 |chapter= Ch. 2}}

{{Physics operator}}

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