LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle \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}. }

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{2}&j_{3}&j_{1}\\m_{2}&m_{3}&m_{1}\end{pmatrix}}={\begin{pmatrix}j_{3}&j_{1}&j_{2}\\m_{3}&m_{1}&m_{2}\end{pmatrix}}.}$

... is translated to the CAS output ...

Semantic latex: \Wignerthreejsym{j_1}{j_2}{j_3}{m_1}{m_2}{m_3} = \Wignerthreejsym{j_2}{j_3}{j_1}{m_2}{m_3}{m_1} = \Wignerthreejsym{j_3}{j_1}{j_2}{m_3}{m_1}{m_2}

Confidence: 0.66574040393935

Mathematica

Translation: ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}, {Subscript[m, 2], Subscript[m, 1]}] == ThreeJSymbol[{Subscript[j, 3], Subscript[m, 3]}, {Subscript[j, 1], Subscript[m, 1]}, {Subscript[m, 3], Subscript[m, 2]}]

Information

Sub Equations

ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] = ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}, {Subscript[m, 2], Subscript[m, 1]}]

ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}, {Subscript[m, 2], Subscript[m, 1]}] = ThreeJSymbol[{Subscript[j, 3], Subscript[m, 3]}, {Subscript[j, 1], Subscript[m, 1]}, {Subscript[m, 3], Subscript[m, 2]}]

Free variables

Subscript[j, 1]

Subscript[j, 2]

Subscript[j, 3]

Subscript[m, 1]

Subscript[m, 2]

Subscript[m, 3]

Symbol info

3j symbol; Example: \Wignerthreejsym@@{j_1}{j_2}{j_3}{m_1}{m_2}{m_3}

Will be translated to: ThreeJSymbol[{$0, $3}, {$1, $4}, {$3, $5}] Relevant links to definitions: DLMF: http://dlmf.nist.gov/34.2#E4 Mathematica: https://reference.wolfram.com/language/ref/ThreeJSymbol.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \Wignerthreejsym [\Wignerthreejsym]

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

(LaTeX -> Maple) No translation possible for given token: Cannot extract information from feature set: \Wignerthreejsym [\Wignerthreejsym]

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$j_{2}$

$j_{3}$

$j_{1}$

$m_{3}$

${\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{2}&j_{3}&j_{1}\\m_{2}&m_{3}&m_{1}\end{pmatrix}}={\begin{pmatrix}j_{3}&j_{1}&j_{2}\\m_{3}&m_{1}&m_{2}\end{pmatrix}}$

$j$

$m_{2}$

$m_{1}$

$m$

Is part of

${\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{2}&j_{3}&j_{1}\\m_{2}&m_{3}&m_{1}\end{pmatrix}}={\begin{pmatrix}j_{3}&j_{1}&j_{2}\\m_{3}&m_{1}&m_{2}\end{pmatrix}}$

Complete translation information:

{
  "id" : "FORMULA_ac35c3efbad138eb4844417fa385c7aa",
  "formula" : "\\begin{pmatrix}\n  j_1 & j_2 & j_3\\\\\n  m_1 & m_2 & m_3\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n  j_2 & j_3 & j_1\\\\\n  m_2 & m_3 & m_1\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n  j_3 & j_1 & j_2\\\\\n  m_3 & m_1 & m_2\n\\end{pmatrix}",
  "semanticFormula" : "\\Wignerthreejsym{j_1}{j_2}{j_3}{m_1}{m_2}{m_3} = \\Wignerthreejsym{j_2}{j_3}{j_1}{m_2}{m_3}{m_1} = \\Wignerthreejsym{j_3}{j_1}{j_2}{m_3}{m_1}{m_2}",
  "confidence" : 0.6657404039393544,
  "translations" : {
    "Mathematica" : {
      "translation" : "ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] == ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}, {Subscript[m, 2], Subscript[m, 1]}] == ThreeJSymbol[{Subscript[j, 3], Subscript[m, 3]}, {Subscript[j, 1], Subscript[m, 1]}, {Subscript[m, 3], Subscript[m, 2]}]",
      "translationInformation" : {
        "subEquations" : [ "ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[m, 1], Subscript[m, 3]}] = ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}, {Subscript[m, 2], Subscript[m, 1]}]", "ThreeJSymbol[{Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}, {Subscript[m, 2], Subscript[m, 1]}] = ThreeJSymbol[{Subscript[j, 3], Subscript[m, 3]}, {Subscript[j, 1], Subscript[m, 1]}, {Subscript[m, 3], Subscript[m, 2]}]" ],
        "freeVariables" : [ "Subscript[j, 1]", "Subscript[j, 2]", "Subscript[j, 3]", "Subscript[m, 1]", "Subscript[m, 2]", "Subscript[m, 3]" ],
        "tokenTranslations" : {
          "\\Wignerthreejsym" : "3j symbol; Example: \\Wignerthreejsym@@{j_1}{j_2}{j_3}{m_1}{m_2}{m_3}\nWill be translated to: ThreeJSymbol[{$0, $3}, {$1, $4}, {$3, $5}]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/34.2#E4\nMathematica:  https://reference.wolfram.com/language/ref/ThreeJSymbol.html"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\Wignerthreejsym [\\Wignerthreejsym]"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Maple) No translation possible for given token: Cannot extract information from feature set: \\Wignerthreejsym [\\Wignerthreejsym]"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "j_{2}", "j_{3}", "j_{1}", "m_{3}", "\\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}", "j", "m_{2}", "m_{1}", "m" ],
  "isPartOf" : [ "\\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}" ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	In [[quantum mechanics]], the '''Wigner 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. \|editor1-last=Wightman \|editor1-first=Arthur S. \|date=1951 \|chapter=On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups \|title=The Collected Works of Eugene Paul Wigner \|volume=3 \|pages=608–654 \|doi=10.1007/978-3-662-02781-3_42 \|isbn=978-3-642-08154-5 }}</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-integer\|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''<sub>3</sub> → −''m''<sub>3</sub>}}: :<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> == 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} = (-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 ([[T-symmetry#Microscopic_phenomena:_time_reversal_invariance\|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>{{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>{{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}}</ref> == Orthogonality relations == A system of two angular momenta with magnitudes {{math\|''j''<sub>1</sub>}} and {{math\|''j''<sub>2</sub>}} can be described either in terms of the uncoupled basis states (labeled by the quantum numbers {{math\|''m''<sub>1</sub>}} and {{math\|''m''<sub>2</sub>}}), or the coupled basis states (labeled by {{math\|''j''<sub>3</sub>}} and {{math\|''m''<sub>3</sub>}}). 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''<sub>1</sub> ''j''<sub>2</sub> ''j''<sub>3</sub>}<!--ignore-->}} is equal to 1 when the triad (''j''<sub>1</sub>, ''j''<sub>2</sub>, ''j''<sub>3</sub>) 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 symbol\|6-''j'']] and [[9-j symbol\|9-''j'']] symbols, all of which are irreducible summations of 3-''jm'' symbols where no {{mvar\|m}} variables remain. ==Relation to spherical harmonics== The 3-''jm'' symbols give the integral of the products of three [[spherical harmonics]]{{Citation needed\|date=October 2020\|reason=It is not clear which reference contains this claim.}} :<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. === 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_D-matrix#Wigner_(small)_d-matrix\|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. == 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> :<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 [[Giulio Racah\|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> ==See also== [[Clebsch–Gordan coefficients]] [[Spherical harmonics]] [[6-j symbol]] [[9-j symbol]] ==References== <references /> <!-- ---------------------------------------------------------- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for a discussion of different citation methods and how to generate footnotes using the<ref>, </ref> and <reference /> tags ----------------------------------------------------------- --> <div class="references"> [[Lawrence Biedenharn\|L. 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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=Comp. Phys. Comm. \|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 }} * [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]]
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