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 d^l_{mn}}

${\displaystyle d_{mn}^{l}}$

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

Semantic latex: d_{mn}^l

Confidence: 0

Mathematica

Translation: (Subscript[d, m, n])^(l)

Information

Sub Equations

(Subscript[d, m, n])^(l)

Free variables

Subscript[d, m, n]

l

m

n

Tests

Symbolic

Numeric

SymPy

Translation: (Symbol('{d}_{m, n}'))**(l)

Information

Sub Equations

(Symbol('{d}_{m, n}'))**(l)

Free variables

Symbol('{d}_{m, n}')

l

m

n

Tests

Symbolic

Numeric

Maple

Translation: (d[m, n])^(l)

Information

Sub Equations

(d[m, n])^(l)

Free variables

d[m, n]

l

m

n

Tests

Symbolic

Numeric

Dependency Graph Information

Is part of

${\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {2l_{3}+1}}}$

${\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {l_{2}+l_{3}+1}}}$

Description

Wigner function

symbol

better approximation

Regge symmetry

Complete translation information:

{
  "id" : "FORMULA_dd35bc6f2bbfaf5a4389fd8ab3b9d1cc",
  "formula" : "d^l_{mn}",
  "semanticFormula" : "d_{mn}^l",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "(Subscript[d, m, n])^(l)",
      "translationInformation" : {
        "subEquations" : [ "(Subscript[d, m, n])^(l)" ],
        "freeVariables" : [ "Subscript[d, m, n]", "l", "m", "n" ]
      },
      "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" : "(Symbol('{d}_{m, n}'))**(l)",
      "translationInformation" : {
        "subEquations" : [ "(Symbol('{d}_{m, n}'))**(l)" ],
        "freeVariables" : [ "Symbol('{d}_{m, n}')", "l", "m", "n" ]
      },
      "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" : "(d[m, n])^(l)",
      "translationInformation" : {
        "subEquations" : [ "(d[m, n])^(l)" ],
        "freeVariables" : [ "d[m, n]", "l", "m", "n" ]
      },
      "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" : [ {
    "section" : 9,
    "sentence" : 0,
    "word" : 15
  } ],
  "includes" : [ ],
  "isPartOf" : [ "\\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}}", "\\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}}" ],
  "definiens" : [ {
    "definition" : "Wigner function",
    "score" : 0.722
  }, {
    "definition" : "symbol",
    "score" : 0.7125985104912714
  }, {
    "definition" : "better approximation",
    "score" : 0.660423639753057
  }, {
    "definition" : "Regge symmetry",
    "score" : 0.573332519662682
  } ]
}

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. 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=Marcos \|last1=Moshinsky \|title=Wigner coefficients for the SU<sub>3</sub> 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 }} {{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 }} {{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=http://cds.cern.ch/record/345262 }} {{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 SU<sub>n</sub> \|journal=J. Math. Phys. \|volume=5 \|issue=12 \|year=1964 \|page=1723 \|doi=10.1063/1.1704095 \|bibcode = 1964JMP.....5.1723B }} {{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=S. J. \|last1=P. McNamee \|first2=Frank \|last2=Chilton \|title=Tables of Clebsch-Gordan coefficients of SU<sub>3</sub> \|journal=Rev. Mod. Phys. \|volume=36 \|year=1964 \|page=1005 \|doi=10.1103/RevModPhys.36.1005 \|issue=4 \|bibcode=1964RvMP...36.1005M }} {{Cite journal \|first1=K. T. \|last1=Hecht \|title=SU<sub>3</sub> 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 }} {{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. 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 }} {{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=J. P. \|last1=Draayer \|first2=Yoshimi \|last2=Akiyama \|title=Wigner and Racah coefficients for SU<sub>3</sub> \|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 }} {{Cite journal \|first1=Yoshimi \|last1=Akiyama \|first2=J. P. \|last2=Draayer \|title=A users' guide to fortran programs for Wigner and Racah coefficients of SU<sub>3</sub> \|journal=Comp. Phys. Comm. \|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 }} {{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=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 }} {{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=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 SU<sub>6</sub> and SU<sub>3</sub> \|journal=J. Phys. A \|volume=15 \|issue=4 \|year=1982 \|page=1087 \|doi=10.1088/0305-4470/15/4/014 \|bibcode=1982JPhA...15.1087B }} {{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=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]]
	Purge cached results.

LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

Specify your own input

Navigation menu

Search