3-j symbol

{{Short description|Coefficients coupled with angular momentum}} In quantum mechanics, the '''Wigner's 3-j symbols''', also called 3''-jm'' symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta.<ref name="Wigner1951">{{cite book |last=Wigner |first=E. P. |author-link=Eugene Wigner |title=The Collected Works of Eugene Paul Wigner |date=1993 |isbn=978-3-642-08154-5 |editor1-last=Wightman |editor1-first=Arthur S. |volume=A/1 |pages=608–654 |chapter=On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups |doi=10.1007/978-3-662-02781-3_42}}</ref> While the two approaches address exactly the same physical problem, the 3-''j'' symbols do so more symmetrically.

== Mathematical relation to Clebsch–Gordan coefficients ==

The 3-''j'' symbols are given in terms of the Clebsch–Gordan coefficients by :<math> \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} \equiv \frac{(-1)^{j_1 - j_2 - m_3}}{\sqrt{2 j_3 + 1}} \langle j_1 \, m_1 \, j_2 \, m_2 | j_3 \, (-m_3) \rangle. </math> The ''j'' and ''m'' components are angular-momentum quantum numbers, i.e., every {{math|''j''}} (and every corresponding {{math|''m''}}) is either a nonnegative integer or half-odd-integer. The exponent of the sign factor is always an integer, so it remains the same when transposed to the left, and the inverse relation follows upon making the substitution {{math|''m''3 → −''m''3}}: :<math> \langle j_1 \, m_1 \, j_2 \, m_2 | j_3 \, m_3 \rangle = (-1)^{-j_1 + j_2 - m_3} \sqrt{2 j_3 + 1} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & -m_3 \end{pmatrix}. </math>

== Explicit expression == :<math> \begin{align}\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} & \equiv \delta(m_1+m_2+m_3,0) (-1)^{j_1 - j_2 - m_3} {} \sqrt{\frac{(j_1+j_2-j_3)!(j_1-j_2+j_3)!(-j_1+j_2+j_3)!}{(j_1+j_2+j_3+1)!}}\ \times {} \\[6pt] &\times\sqrt{(j_1-m_1)!(j_1+m_1)!(j_2-m_2)!(j_2+m_2)!(j_3-m_3)!(j_3+m_3)!}\ \times {} \\[6pt] &\times\sum_{k=K}^N \frac{(-1)^k}{k!(j_1+j_2-j_3-k)!(j_1-m_1-k)!(j_2+m_2-k)!(j_3-j_2+m_1+k)!(j_3-j_1-m_2+k)!}, \end{align} </math> where <math>\delta(i,j)</math> is the Kronecker delta.

The summation is performed over those integer values {{mvar|k}} for which the argument of each factorial in the denominator is non-negative, i.e. summation limits {{mvar|K}} and {{mvar|N}} are taken equal: the lower one <math>K=\max(0, j_2-j_3-m_1, j_1-j_3+m_2),</math> the upper one <math>N=\min(j_1+j_2-j_3, j_1-m_1, j_2+m_2).</math> Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3''j'' symbol at, for example, <math>j_3>j_1+j_2</math> or <math>j_1<m_1</math> are automatically set to zero.

== Definitional relation to Clebsch–Gordan coefficients ==

The CG coefficients are defined so as to express the addition of two angular momenta in terms of a third: :<math> |j_3\, m_3\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} \langle j_1 \, m_1 \, j_2 \, m_2 | j_3 \, m_3 \rangle |j_1 \, m_1 \, j_2 \, m_2 \rangle. </math> The 3-''j'' symbols, on the other hand, are the coefficients with which three angular momenta must be added so that the resultant is zero: :<math> \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} \sum_{m_3=-j_3}^{j_3} |j_1 m_1\rangle |j_2 m_2\rangle |j_3 m_3\rangle \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = |0 \, 0\rangle. </math> Here <math>|0 \, 0\rangle</math> is the zero-angular-momentum state (<math>j = m = 0</math>). It is apparent that the 3-''j'' symbol treats all three angular momenta involved in the addition problem on an equal footing and is therefore more symmetrical than the CG coefficient.

Since the state <math>|0 \, 0\rangle</math> is unchanged by rotation, one also says that the contraction of the product of three rotational states with a 3-''j'' symbol is invariant under rotations.

== Selection rules ==

The Wigner 3-''j'' symbol is zero unless all these conditions are satisfied:

:<math>\begin{align} & m_i \in \{-j_i, -j_i + 1, -j_i + 2, \ldots, j_i\} \quad (i = 1, 2, 3), \\ & m_1 + m_2 + m_3 = 0, \\ & |j_1 - j_2| \le j_3 \le j_1 + j_2, \\ & (j_1 + j_2 + j_3) \text{ is an integer (and, moreover, an even integer if } m_1 = m_2 = m_3 = 0 \text{)}. \\ \end{align}</math>

== Symmetry properties == A 3-''j'' symbol is invariant under an even permutation of its columns: :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \begin{pmatrix} j_2 & j_3 & j_1\\ m_2 & m_3 & m_1 \end{pmatrix} = \begin{pmatrix} j_3 & j_1 & j_2\\ m_3 & m_1 & m_2 \end{pmatrix}. </math> An odd permutation of the columns gives a phase factor: :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_2 & j_1 & j_3\\ m_2 & m_1 & m_3 \end{pmatrix} </math> :<math> = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_3 & j_2\\ m_1 & m_3 & m_2 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_3 & j_2 & j_1\\ m_3 & m_2 & m_1 \end{pmatrix}. </math> Changing the sign of the <math>m</math> quantum numbers (time reversal) also gives a phase: :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}. </math> The 3-''j'' symbols also have so-called Regge symmetries, which are not due to permutations or time reversal.<ref>{{cite journal |first1=T. |last1=Regge |title=Symmetry Properties of Clebsch-Gordan Coefficients |journal=Nuovo Cimento |year=1958 |volume=10 |issue=3 |page=544 |doi=10.1007/BF02859841 |bibcode=1958NCim...10..544R|s2cid=122299161 }}</ref> These symmetries are: :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \begin{pmatrix} j_1 & \frac{j_2+j_3-m_1}{2} & \frac{j_2+j_3+m_1}{2}\\ j_3-j_2 & \frac{j_2-j_3-m_1}{2}-m_3 & \frac{j_2-j_3+m_1}{2}+m_3 \end{pmatrix}, </math> :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} \frac{j_2+j_3+m_1}{2} & \frac{j_1+j_3+m_2}{2} & \frac{j_1+j_2+m_3}{2}\\ j_1 - \frac{j_2+j_3-m_1}{2} & j_2 - \frac{j_1+j_3-m_2}{2} & j_3-\frac{j_1+j_2-m_3}{2} \end{pmatrix}. </math> With the Regge symmetries, the 3-''j'' symbol has a total of 72 symmetries. These are best displayed by the definition of a Regge symbol, which is a one-to-one correspondence between it and a 3-''j'' symbol and assumes the properties of a semi-magic square:<ref name="rasch-yu">{{Cite journal |first1=J. |last1=Rasch |first2=A. C. H. |last2=Yu |title=Efficient Storage Scheme for Pre-calculated Wigner 3''j'', 6''j'' and Gaunt Coefficients |journal=SIAM J. Sci. Comput. |volume=25 |issue=4 |year=2003 |pages=1416–1428 |doi=10.1137/s1064827503422932 }}</ref> :<math> R= \begin{array}{|ccc|} \hline -j_1+j_2+j_3 & j_1-j_2+j_3 & j_1+j_2-j_3\\ j_1-m_1 & j_2-m_2 & j_3-m_3\\ j_1+m_1 & j_2+m_2 & j_3+m_3\\ \hline \end{array}, </math> whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. These facts can be used to devise an effective storage scheme.<ref name="rasch-yu" />

== Orthogonality relations == A system of two angular momenta with magnitudes {{math|''j''1}} and {{math|''j''2}} can be described either in terms of the uncoupled basis states (labeled by the quantum numbers {{math|''m''1}} and {{math|''m''2}}), or the coupled basis states (labeled by {{math|''j''3}} and {{math|''m''3}}). The 3-''j'' symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations :<math> (2 j_3 + 1)\sum_{m_1 m_2} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j'_3\\ m_1 & m_2 & m'_3 \end{pmatrix} = \delta_{j_3, j'_3} \delta_{m_3, m'_3} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix}, </math> :<math> \sum_{j_3 m_3} (2 j_3 + 1) \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1' & m_2' & m_3 \end{pmatrix} = \delta_{m_1, m_1'} \delta_{m_2, m_2'}. </math> The ''triangular delta'' {{math|{''j''1 ''j''2 ''j''3}}} is equal to 1 when the triad (''j''1, ''j''2, ''j''3) satisfies the triangle conditions, and is zero otherwise. The triangular delta itself is sometimes confusingly called<ref name="WormerPaldus2006">{{cite journal |title=Angular Momentum Diagrams |author1=P. E. S. Wormer |author2=J. Paldus |journal=Advances in Quantum Chemistry |publisher=Elsevier |volume=51 |pages=59–124 |year=2006 |issn=0065-3276 |doi=10.1016/S0065-3276(06)51002-0 |bibcode=2006AdQC...51...59W |isbn=9780120348510}}</ref> a "3-''j'' symbol" (without the ''m'') in analogy to 6-''j'' and 9-''j'' symbols, all of which are irreducible summations of 3-''jm'' symbols where no {{mvar|m}} variables remain.

==Relation to spherical harmonics; Gaunt coefficients== The 3-''jm'' symbols give the integral of the products of three spherical harmonics<ref>{{Cite journal |last=Cruzan |first=Orval R. |date=1962 |title=Translational addition theorems for spherical vector wave functions |url=https://www.ams.org/qam/1962-20-01/S0033-569X-1962-0132851-2/ |journal=Quarterly of Applied Mathematics |language=en |volume=20 |issue=1 |pages=33–40 |doi=10.1090/qam/132851 |issn=0033-569X|doi-access=free }}</ref> :<math> \begin{align} & \int Y_{l_1 m_1}(\theta, \varphi) Y_{l_2 m_2}(\theta, \varphi) Y_{l_3 m_3}(\theta, \varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi \\ &\quad = \sqrt{\frac{(2l_1 + 1)(2l_2 + 1)(2l_3 + 1)}{4\pi}} \begin{pmatrix} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \end{align} </math> with <math>l_1</math>, <math>l_2</math> and <math>l_3</math> integers. These integrals are called Gaunt coefficients.

=== Relation to integrals of spin-weighted spherical harmonics ===

Similar relations exist for the spin-weighted spherical harmonics if <math>s_1 + s_2 + s_3 = 0</math>: :<math> \begin{align} & \int d\mathbf{\hat n} \,_{s_1}\!Y_{j_1 m_1}(\mathbf{\hat n}) \,_{s_2}\!Y_{j_2 m_2}(\mathbf{\hat n}) \,_{s_3}\!Y_{j_3 m_3}(\mathbf{\hat n}) \\ &\quad = \sqrt{\frac{(2j_1 + 1)(2j_2 + 1)(2j_3 + 1)}{4\pi}} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3\\ -s_1 & -s_2 & -s_3 \end{pmatrix}. \end{align} </math>

== Recursion relations == :<math> \begin{align} & {-}\sqrt{(l_3 \mp s_3)(l_3 \pm s_3 + 1)} \begin{pmatrix} l_1 & l_2 & l_3 \\ s_1 & s_2 & s_3 \pm 1 \end{pmatrix}= \\ &\quad = \sqrt{(l_1 \mp s_1)(l_1 \pm s_1 + 1)} \begin{pmatrix} l_1 & l_2 & l_3 \\ s_1 \pm 1 & s_2 & s_3 \end{pmatrix} + \sqrt{(l_2 \mp s_2)(l_2 \pm s_2 + 1)} \begin{pmatrix} l_1 & l_2 & l_3 \\ s_1 & s_2 \pm 1 & s_3 \end{pmatrix}. \end{align} </math>

== Asymptotic expressions == For <math>l_1 \ll l_2, l_3</math> a non-zero 3-''j'' symbol is :<math> \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3+m_3} \frac{d^{l_1}_{m_1, l_3 - l_2}(\theta)}{\sqrt{2l_3 + 1}}, </math> where <math>\cos(\theta) = -2m_3 / (2l_3 + 1)</math>, and <math>d^l_{mn}</math> is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by :<math> \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{l_2+l_3+1}}, </math> where <math>\cos(\theta) = (m_2 - m_3)/(l_2 + l_3 + 1)</math>.

== Metric tensor ==

The following quantity acts as a metric tensor in angular-momentum theory and is also known as a ''Wigner 1-jm symbol'':<ref name="Wigner1951"/> :<math>\begin{pmatrix} j \\ m \quad m' \end{pmatrix} := \sqrt{2 j + 1} \begin{pmatrix} j & 0 & j \\ m & 0 & m' \end{pmatrix} = (-1)^{j - m'} \delta_{m, -m'}. </math> It can be used to perform time reversal on angular momenta.

== Special cases and other properties == :<math>\sum_m (-1)^{j - m} \begin{pmatrix} j & j & J \\ m & -m & 0 \end{pmatrix} = \sqrt{2 j + 1} \, \delta_{J, 0}. </math>

From equation (3.7.9) in <ref>{{cite book |last1=Edmonds |first1=Alan |title=Angular Momentum in Quantum Mechanics |date=1957 |publisher=Princeton University Press}}</ref> :<math> \begin{pmatrix} j & j & 0 \\ m & -m & 0 \end{pmatrix} = \frac{1}{\sqrt{2 j + 1}} (-1)^{j - m}. </math>

:<math> \frac{1}{2} \int_{-1}^1 P_{l_1}(x) P_{l_2}(x) P_{l}(x) \, dx = \begin{pmatrix} l & l_1 & l_2 \\ 0 & 0 & 0 \end{pmatrix}^2, </math> where ''P'' are Legendre polynomials.

== Relation to Racah {{math|''V''}}-coefficients ==

Wigner 3-''j'' symbols are related to Racah {{mvar|V}}-coefficients<ref>{{Cite journal |first=G. |last=Racah |title=Theory of Complex Spectra II |journal=Physical Review |volume=62 |issue=9–10 |pages=438–462 |year=1942 |doi=10.1103/PhysRev.62.438 |bibcode = 1942PhRv...62..438R }}</ref> by a simple phase:

:<math> V(j_1 \, j_2 \, j_3; m_1 \, m_2 \, m_3) = (-1)^{j_1 - j_2 - j_3} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix}. </math>

== Relation to group theory == This section essentially recasts the definitional relation in the language of group theory.

A group representation of a group is a homomorphism of the group into a group of linear transformations over some vector space. The linear transformations can be given by a group of matrices with respect to some basis of the vector space.

The group of transformations leaving angular momenta invariant is the three dimensional rotation group SO(3). When "spin" angular momenta are included, the group is its double covering group, SU(2).

A reducible representation is one where a change of basis can be applied to bring all the matrices into block diagonal form. A representation is irreducible (irrep) if no such transformation exists.

For each value of ''j'', the 2''j''+1 kets form a basis for an irreducible representation (irrep) of SO(3)/SU(2) over the complex numbers. Given two irreps, the tensor direct product can be reduced to a sum of irreps, giving rise to the Clebsch-Gordon coefficients, or by reduction of the triple product of three irreps to the trivial irrep '''1''' giving rise to the 3j symbols.

== 3j symbols for other groups == The <math>3j</math> symbol has been most intensely studied in the context of the coupling of angular momentum. For this, it is strongly related to the group representation theory of the groups SU(2) and SO(3) as discussed above. However, many other groups are of importance in physics and chemistry, and there has been much work on the <math>3j</math> symbol for these other groups. In this section, some of that work is considered.

=== Simply reducible groups === The original paper by Wigner<ref name=Wigner1951 /> was not restricted to SO(3)/SU(2) but instead focussed on simply reducible (SR) groups. These are groups in which * all classes are ambivalent i.e. if <math>X</math> is a member of a class then so is <math>X^{-1}</math> * the Kronecker product of two irreps is multiplicity free i.e. does not contain any irrep more than once. For SR groups, every irrep is equivalent to its complex conjugate, and under permutations of the columns the absolute value of the symbol is invariant and the phase of each can be chosen so that they at most change sign under odd permutations and remain unchanged under even permutations.

=== General compact groups === Compact groups form a wide class of groups with topological structure. They include the finite groups with added discrete topology and many of the Lie groups.

General compact groups will neither be ambivalent nor multiplicity free. Derome and Sharp<ref> {{Cite journal |first1=J-R |last1=Derome |first2=W. T. |last2=Sharp |title=Racah Algebra for an Arbitrary Group |journal=J. Math. Phys. |volume=6 |issue=10 |year=1965 |pages=1584–1590 |doi=10.1063/1.1704698 |bibcode=1965JMP.....6.1584D }} </ref> and Derome<ref> {{Cite journal |first=J-R |last=Derome |title=Symmetry Properties of the 3j Symbols for an Arbitrary Group |journal=J. Math. Phys. |volume=7 |issue=4 |year=1966 |pages=612–615 |doi=10.1063/1.1704973 |bibcode=1966JMP.....7..612D }} </ref> examined the <math>3j</math> symbol for the general case using the relation to the Clebsch-Gordon coefficients of :<math> \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} \equiv \frac{1}{[j_3]} \langle j_1 \, m_1 \, j_2 \, m_2 | j_3^* \, m_3 \rangle. </math> where <math>[j]</math> is the dimension of the representation space of <math>j</math> and <math>j_3^*</math> is the complex conjugate representation to <math>j_3</math>.

By examining permutations of columns of the <math>3j</math> symbol, they showed three cases: * if all of <math>j_1, j_2, j_3</math> are inequivalent then the <math>3j</math> symbol may be chosen to be invariant under any permutation of its columns * if exactly two are equivalent, then transpositions of its columns may be chosen so that some symbols will be invariant while others will change sign. An approach using a wreath product of the group with <math>S_3</math><ref> {{Cite journal |first=J. D. |last=Newmarch |title=On the 3j symmetries |journal=J. Math. Phys. |volume=24 |issue=4 |year=1983 |pages=757–764 |doi=10.1063/1.525771 |bibcode=1983JMP....24..757N }} </ref> showed that these correspond to the representations <math>[2]</math> or <math>[1^2]</math> of the symmetric group <math>S_2</math>. Cyclic permutations leave the <math>3j</math> symbol invariant. * if all three are equivalent, the behaviour is dependent on the representations of the symmetric group<math>S_3</math>. Wreath group representations corresponding to <math>[3]</math> are invariant under transpositions of the columns, corresponding to <math>[1^3]</math> change sign under transpositions, while a pair corresponding to the two dimensional representation <math>[21]</math> transform according to that.

Further research into <math>3j</math> symbols for compact groups has been performed based on these principles. <ref> {{Cite journal |first1=P. H. |last1=Butler |first2=B. G. |last2=Wybourne |title=Calculation of ''j'' and ''jm'' Symbols forArbitrary Compact Groups. I. Methodology |journal=Int. J. Quantum Chem. |volume=X |year=1976 |issue=4 |pages=581–598 |doi=10.1002/qua.560100404 |bibcode= }} </ref>

=== SU(n) === The Special unitary group SU(n) is the Lie group of n × n unitary matrices with determinant 1.

The group SU(3) is important in particle theory. There are many papers dealing with the <math>3j</math> or equivalent symbol <ref> {{Cite journal |first1=Marcos |last1=Moshinsky |title=Wigner coefficients for the SU3 group and some applications |journal=Rev. Mod. Phys. |volume=34 |year=1962 |page=813 |doi=10.1103/RevModPhys.34.813 |issue=4 |bibcode = 1962RvMP...34..813M }} </ref> <ref> {{Cite journal |first1=S. J. |last1=P. McNamee |first2=Frank |last2=Chilton |title=Tables of Clebsch-Gordan coefficients of SU3 |journal=Rev. Mod. Phys. |volume=36 |year=1964 |page=1005 |doi=10.1103/RevModPhys.36.1005 |issue=4 |bibcode=1964RvMP...36.1005M }} </ref> <ref> {{Cite journal |first1=J. P. |last1=Draayer |first2=Yoshimi |last2=Akiyama |title=Wigner and Racah coefficients for SU3 |journal=J. Math. Phys. |volume=14 |year=1973 |page=1904 |doi=10.1063/1.1666267 |issue=12 |bibcode = 1973JMP....14.1904D |url=https://deepblue.lib.umich.edu/bitstream/2027.42/70151/2/JMAPAQ-14-12-1904-1.pdf |hdl=2027.42/70151 |hdl-access=free }} </ref> <ref> {{Cite journal |first1=Yoshimi |last1=Akiyama |first2=J. P. |last2=Draayer |title=A users' guide to fortran programs for Wigner and Racah coefficients of SU3 |journal=Comput. Phys. Commun. |volume=5 |issue=6 |year=1973 |page=405 |doi=10.1016/0010-4655(73)90077-5 |bibcode = 1973CoPhC...5..405A |hdl=2027.42/24983 |hdl-access=free }} </ref> <ref> {{Cite journal |first1=R. P. |last1=Bickerstaff |first2=P. H. |last2=Butler |first3=M. B. |last3=Butts |first4=R. w. |last4=Haase |first5=M. F. |last5=Reid |title=3jm and 6j tables for some bases of SU6 and SU3 |journal=J. Phys. A |volume=15 |issue=4 |year=1982 |page=1087 |doi=10.1088/0305-4470/15/4/014 |bibcode=1982JPhA...15.1087B }} </ref> <ref> {{Cite journal |first1=J. J. |last1=Swart de |title=The octet model and its Glebsch-Gordan coefficients |journal=Rev. Mod. Phys. |volume=35 |year=1963 |page=916 |doi=10.1103/RevModPhys.35.916 |issue=4 |bibcode = 1963RvMP...35..916D |url=https://cds.cern.ch/record/345262 }} </ref> <ref> {{Cite journal |first=J-R |last=Derome |title=Symmetry Properties of the 3j Symbols for SU(3) |journal=J. Math. Phys. |volume=8 |issue=4 |year=1967 |pages=714–716 |doi=10.1063/1.1705269 |bibcode=1967JMP.....8..714D }} </ref> <ref> {{Cite journal |first1=K. T. |last1=Hecht |title=SU3 recoupling and fractional parentage in the 2s-1d shell |journal=Nucl. Phys. |volume=62 |year=1965 |page=1 |doi=10.1016/0029-5582(65)90068-4 |issue=1 |bibcode=1965NucPh..62....1H |hdl=2027.42/32049 |hdl-access=free }} </ref>

The <math>3j</math> symbol for the group SU(4) has been studied <ref> {{Cite journal |first1=K. T. |last1=Hecht |first2=Sing Ching |last2=Pang |title=On the Wigner Supermultiplet Scheme |journal=J. Math. Phys. |volume=10 |year=1969 |page=1571 |doi=10.1063/1.1665007 |issue=9 |bibcode = 1969JMP....10.1571H |hdl=2027.42/70485 |url=https://deepblue.lib.umich.edu/bitstream/2027.42/70485/2/JMAPAQ-10-9-1571-1.pdf |hdl-access=free }} </ref> <ref> {{Cite journal |first1=E. M. |last1=Haacke |first2=J. W. |last2=Moffat |first3=P. |last3=Savaria |title=A calculation of SU(4) Glebsch-Gordan coefficients |journal=J. Math. Phys. |volume=17 |year=1976 |page=2041 |doi=10.1063/1.522843 |issue=11 |bibcode = 1976JMP....17.2041H }} </ref> while there is also work on the general SU(n) groups <ref> {{Cite journal |first1=G. E. |last1=Baird |first2=L. C. |last2=Biedenharn |title=On the representation of the semisimple Lie Groups. II |journal=J. Math. Phys. |volume=4 |issue=12 |year=1963 |page=1449 |doi=10.1063/1.1703926 |bibcode=1963JMP.....4.1449B }} </ref> <ref> {{Cite journal |first1=G. E. |last1=Baird |first2=L. C. |last2=Biedenharn |title=On the representations of the semisimple Lie Groups. III. The explicit conjugation Operation for SUn |journal=J. Math. Phys. |volume=5 |issue=12 |year=1964 |page=1723 |doi=10.1063/1.1704095 |bibcode = 1964JMP.....5.1723B }} </ref>

=== Crystallographic point groups === There are many papers dealing with the <math>3j</math> symbols or Clebsch-Gordon coefficients for the finite crystallographic point groups and the double point groups The book by Butler <ref> {{Cite book |first=P. H. |last=Butler |title=Point Group Symmetry Applications: methods and tables |publisher=Plenum Press, New York |year=1981 }} </ref> references these and details the theory along with tables.

=== Magnetic groups === Magnetic groups include antilinear operators as well as linear operators. They need to be dealt with using Wigner's theory of corepresentations of unitary and antiunitary groups. A significant departure from standard representation theory is that the multiplicity of the irreducible corepresentation <math>j_3^*</math> in the direct product of the irreducible corepresentations <math>j_1 \otimes j_2</math> is generally smaller than the multiplicity of the trivial corepresentation in the triple product <math>j_1 \otimes j_2 \otimes j_3</math>, leading to significant differences between the Clebsch-Gordon coefficients and the <math>3j</math> symbol.

The <math>3j</math> symbols have been examined for the grey groups <ref> {{Cite thesis |last=Newmarch |first=J. D. |date=1981 |title=The Racah Algebra for Groups with Time Reversal Symmetry |url=http://unsworks.unsw.edu.au/fapi/datastream/unsworks:71130/SOURCE01?view=true |publisher=University of New South Wales }} </ref> <ref> {{Cite journal |first1=J. D. |last1=Newmarch |first2=R. M. |last2=Golding |title=The Racah Algebra for Groups with Time Reversal Symmetry |journal=J. Math. Phys. |volume=22 |issue=2 |year=1981 |pages=233–244 |doi=10.1063/1.524894 |bibcode=1981JMP....22..233N |hdl=1959.4/69692 |url=https://unsworks.unsw.edu.au/bitstreams/eec97fbd-242e-49ed-9f1b-63ef0782be6b/download |hdl-access=free }} </ref> and for the magnetic point groups <ref> {{Cite journal |first1=J. N. |last1=Kotsev |first2=M. I. |last2=Aroyo |first3=M. N. |last3=Angelova |title=Tables of Spectroscopic Coefficients for Magnetic Point Group Symmetry |journal=J. Mol. Structure |volume=115 |year=1984 |pages=123–128 |doi=10.1016/0022-2860(84)80030-7 }} </ref>

==References== <references />  <div class="references"> *L. C. Biedenharn and J. D. Louck, ''Angular Momentum in Quantum Physics'', volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading, 1981. * D. M. Brink and G. R. Satchler, ''Angular Momentum'', 3rd edition, Clarendon, Oxford, 1993. * A. R. Edmonds, ''Angular Momentum in Quantum Mechanics'', 2nd edition, Princeton University Press, Princeton, 1960. *{{dlmf|id=34 |title=3j,6j,9j Symbols|first=Leonard C.|last= Maximon}} * {{cite book |first1=D. A. |last1=Varshalovich |first2=A. N. |last2=Moskalev |first3=V. K. |last3=Khersonskii |title=Quantum Theory of Angular Momentum |publisher=World Scientific Publishing Co. |year=1988 }} *{{ cite journal |first1=T. |last1=Regge |title=Symmetry Properties of Clebsch-Gordon's Coefficients |journal=Nuovo Cimento |year=1958 |volume=10 |issue=3 |doi=10.1007/BF02859841 |pages=544–545 |bibcode=1958NCim...10..544R |s2cid=122299161 }}

*{{Cite journal |first1=Hisashi |last1=Horie |title=Representations of the symmetric group and the fractional parentage coefficients |journal=J. Phys. Soc. Jpn. |volume=19 |issue=10 |year=1964 |page=1783 |doi=10.1143/JPSJ.19.1783 |bibcode = 1964JPSJ...19.1783H }} *{{Cite journal |first1=C. |last1=Itzykson |first2=M. |last2=Nauenberg |title=Unitary groups: representations and decompositions |journal=Rev. Mod. Phys. |volume=38 |year=1966 |page=95 |doi=10.1103/RevModPhys.38.95 |issue=1 |bibcode=1966RvMP...38...95I |osti=1444219 }} *{{Cite journal |first1=P. |last1=Kramer |title=Orbital fractional parentage coefficients for the harmonic oscillator shell model |journal=Z. Phys. |volume=205 |year=1967 |page=181 |doi=10.1007/BF01333370 |issue=2 |bibcode = 1967ZPhy..205..181K |s2cid=122879812 }} *{{Cite journal |first1=P. |last1=Kramer |title=Recoupling coefficients of the symmetric group for shell and cluster model configurations |journal=Z. Phys. |volume=216 |year=1968 |page=68 |doi=10.1007/BF01380094 |issue=1 |bibcode = 1968ZPhy..216...68K |s2cid=121508850 }} *{{Cite journal |first1=K. J. |last1=Lezuo |title=The symmetric group and the Gel'fand basis of U(3). Generalizations of the Dirac identity |journal=J. Math. Phys. |volume=13 |year=1972 |page=1389 |doi=10.1063/1.1666151 |issue=9 |bibcode = 1972JMP....13.1389L }} *{{Cite journal |first1=Josef |last1=Paldus |title=Group theoretical approach to the configuration interaction and perturbation theory calculations for atomic and molecular systems |journal=J. Chem. Phys. |volume=61 |year=1974 |page=5321 |doi=10.1063/1.1681883 |issue=12 |bibcode = 1974JChPh..61.5321P }} *{{cite journal |first1=Klaus |last1=Schulten |first2=Roy G. |last2=Gordon |title=Exact recursive evaluation of 3j and 6j-coefficients for quantum mechanical coupling of angular momenta |journal=J. Math. Phys. |volume=16 |issue=10 |pages=1961–1970 |year=1975 |doi=10.1063/1.522426 |bibcode = 1975JMP....16.1961S }} *{{Cite journal |first1=Josef |last1=Paldus |title=Unitary-group approach to the many-electron correlation problem: Relation of Gelfand and Weyl tableau formulations |journal=Phys. Rev. A |volume=14 |year=1976 |page=1620 |doi=10.1103/PhysRevA.14.1620 |issue=5 |bibcode = 1976PhRvA..14.1620P }} *{{Cite journal |first1=Jacques |last1=Raynal |title=On the definition and properties of generalized 3-j symbols |journal=J. Math. Phys. |volume=19 |year=1978 |page=467 |doi=10.1063/1.523668 |issue=2 }} *{{Cite journal |first1=C. R. |last1=Sarma |first2=G. G. |last2=Sahasrabudhe |title=Permutational symmetry of many particle states |journal=J. Math. Phys. |volume=21 |year=1980 |page=638 |doi=10.1063/1.524509 |issue=4 |bibcode=1980JMP....21..638S }} *{{Cite journal |first1=Jin-Quan |last1=Chen |first2=Mei-Juan |last2=Gao |title=A new approach to permutation group representation |journal=J. Math. Phys. |volume=23 |issue=6 |year=1982 |page=928 |doi=10.1063/1.525460 |bibcode=1982JMP....23..928C }} *{{Cite journal |first1=C. R. |last1=Sarma |title=Determination of basis for the irreducible representations of the unitary group for U(p+q)↓U(p)×U(q) |journal=J. Math. Phys. |volume=23 |year=1982 |page=1235 |doi=10.1063/1.525507 |issue=7 |bibcode=1982JMP....23.1235S }} *{{Cite journal |first1=J.-Q. |last1=Chen |first2=X.-G. |last2=Chen |title=The Gel'fand basis and matrix elements of the graded unitary group U(m/n) |journal=J. Phys. A |volume=16 |year=1983 |page=3435 |doi=10.1088/0305-4470/16/15/010 |issue=15 |bibcode = 1983JPhA...16.3435C }} *{{Cite journal |first1=R. S. |last1=Nikam |first2=K. V. |last2=Dinesha |first3=C. R. |last3=Sarma |title=Reduction of inner-product representations of unitary groups |journal=J. Math. Phys. |volume=24 |year=1983 |page=233 |doi=10.1063/1.525698 |issue=2 |bibcode = 1983JMP....24..233N }} *{{Cite journal |first1=Jin-Quan |last1=Chen |first2=David F. |last2=Collinson |first3=Mei-Juan |last3=Gao |title=Transformation coefficients of permutation groups |journal=J. Math. Phys. |volume=24 |issue=12 |year=1983 |page=2695 |doi=10.1063/1.525668 |bibcode=1983JMP....24.2695C }} *{{Cite journal |first1=Jin-Quan |last1=Chen |first2=Mei-Juan |last2=Gao |first3=Xuan-Gen |last3=Chen |title=The Clebsch-Gordan coefficient for SU(m/n) Gel'fand basis |journal=J. Phys. A |volume=17 |year=1984 |page=481 |doi=10.1088/0305-4470/17/3/011 |issue=3 |bibcode=1984JPhA...17..727K }} *{{ cite journal |first1=K. |last1=Srinivasa Rao |title=Special topics in the quantum theory of angular momentum |year=1985 |volume=24 | number=1 | pages=15–26 |journal=Pramana |doi=10.1007/BF02894812 |bibcode = 1985Prama..24...15R |s2cid=120663002 }} *{{ cite journal |first1=Liqiang |last1=Wei |title=Unified approach for exact calculation of angular momentum coupling and recoupling coefficients |journal=Comput. Phys. Commun. |year=1999 |volume=120 |issue=2–3 |pages=222–230 |doi=10.1016/S0010-4655(99)00232-5 |bibcode=1999CoPhC.120..222W }} *{{Cite journal |first1=J. |last1=Rasch |first2=A. C. H. |last2=Yu |title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients |journal=SIAM J. Sci. Comput. |volume=25 |issue=4 |year=2003 |pages=1416–1428 |doi=10.1137/s1064827503422932 }} </div>

==External links== * {{cite web |first1=Anthony |last1=Stone |url=http://www-stone.ch.cam.ac.uk/wigner.shtml |title=Wigner coefficient calculator }} * {{cite web |first1 = A. |last1 = Volya |url = http://www.volya.net/vc/vc.php |archive-url = https://web.archive.org/web/20070929011249/http://www.volya.net/vc/vc.php |url-status = dead |archive-date = 2007-09-29 |title = Clebsch-Gordan, 3-''j'' and 6-''j'' Coefficient Web Calculator }} (Numerical) * {{cite journal |first1=Paul |last1=Stevenson |url=http://personal.ph.surrey.ac.uk/~phs3ps/cleb.html |title=Clebsch-O-Matic |journal=Computer Physics Communications |volume=147 |issue=3 |pages=853–858 |doi=10.1016/S0010-4655(02)00462-9 |bibcode = 2002CoPhC.147..853S |year=2002 |url-access=subscription }} * [http://plasma-gate.weizmann.ac.il/369j.html 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science] (Numerical) * [http://geoweb.princeton.edu/people/simons/software.html Frederik J Simons: Matlab software archive, the code THREEJ.M] * [http://www.sagemath.org/ Sage (mathematics software)] Gives exact answer for any value of j, m * {{cite web |first1=H. T. |last1=Johansson |first2=C. |last2=Forssén |title=(WIGXJPF) |url=http://fy.chalmers.se/subatom/wigxjpf/ }} (accurate; C, fortran, python) * {{cite web |first1=H. T. |last1=Johansson |title=(FASTWIGXJ) |url=http://fy.chalmers.se/subatom/fastwigxj/ }} (fast lookup, accurate; C, fortran)

Category:Rotational symmetry Category:Representation theory of Lie groups Category:Quantum mechanics