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 (-1)^S}

${\displaystyle (-1)^{S}}$

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

Semantic latex: (-1)^S

Confidence: 0

Mathematica

Translation: (- 1)^(S)

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Sub Equations

(- 1)^(S)

Free variables

S

Tests

Symbolic

Numeric

SymPy

Translation: (- 1)**(S)

Information

Sub Equations

(- 1)**(S)

Free variables

S

Tests

Symbolic

Numeric

Maple

Translation: (- 1)^(S)

Information

Sub Equations

(- 1)^(S)

Free variables

S

Tests

Symbolic

Numeric

Dependency Graph Information

Is part of

${\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=(-1)^{S}{\begin{Bmatrix}j_{4}&j_{5}&j_{6}\\j_{1}&j_{2}&j_{3}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=(-1)^{S}{\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\j_{5}&j_{4}&j_{6}\\j_{8}&j_{7}&j_{9}\end{Bmatrix}}$

Description

phase factor

example

odd permutation of row

column

Complete translation information:

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Specify your own input

Formula input
Wikitext context	[[File:Jucys diagram for Wigner 9-j symbol.svg\|thumb\|[[Jucys diagram]] for the Wigner 9-j symbol. The diagram describes a summation over six [[3-jm symbol]]s. Plus signs on each nodes indicate an anticlockwise reading of the lines for the 3-jm symbol, whereas minus signs indicate clockwise. Due to its symmetries, there are many ways in which the diagram can be drawn.]] In physics, Wigner's '''9-''j'' symbols''' were introduced by [[Eugene Paul Wigner]] in 1937. They are related to [[angular momentum coupling\|recoupling coefficients]] in [[quantum mechanics]] involving four angular momenta :<math> \sqrt{(2j_3+1)(2j_6+1)(2j_7+1)(2j_8+1)} \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} = \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 \| ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle. </math> ==Recoupling of four angular momentum vectors== Coupling of two angular momenta <math>\mathbf{j}_1</math> and <math>\mathbf{j}_2</math> is the construction of simultaneous eigenfunctions of <math>\mathbf{J}^2</math> and <math>J_z</math>, where <math>\mathbf{J}=\mathbf{j}_1+\mathbf{j}_2</math>, as explained in the article on [[Clebsch–Gordan coefficients]]. Coupling of three angular momenta can be done in several ways, as explained in the article on [[Racah W-coefficient]]s. Using the notation and techniques of that article, total angular momentum states that arise from coupling the angular momentum vectors <math>\mathbf{j}_1</math>, <math>\mathbf{j}_2</math>, <math>\mathbf{j}_4</math>, and <math>\mathbf{j}_5</math> may be written as :<math> \| ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle. </math> Alternatively, one may first couple <math>\mathbf{j}_1</math> and <math>\mathbf{j}_4</math> to <math>\mathbf{j}_7</math> and <math>\mathbf{j}_2</math> and <math>\mathbf{j}_5</math> to <math>\mathbf{j}_8</math>, before coupling <math>\mathbf{j}_7</math> and <math>\mathbf{j}_8</math> to <math>\mathbf{j}_9</math>: :<math> \|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle. </math> Both sets of functions provide a complete, orthonormal basis for the space with dimension <math>(2j_1+1)(2j_2+1)(2j_4+1)(2j_5+1)</math> spanned by :<math> \|j_1 m_1\rangle \|j_2 m_2\rangle \|j_4 m_4\rangle \|j_5 m_5\rangle, \;\; m_1=-j_1,\ldots,j_1;\;\; m_2=-j_2,\ldots,j_2;\;\; m_4=-j_4,\ldots,j_4;\;\;m_5=-j_5,\ldots,j_5. </math> Hence, the transformation between the two sets is unitary and the matrix elements of the transformation are given by the scalar products of the functions. As in the case of the [[Racah W-coefficient]]s the matrix elements are independent of the total angular momentum projection quantum number (<math>m_9</math>): :<math> \|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle = \sum_{j_3}\sum_{j_6} \| ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 \| ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle. </math> ==Symmetry relations== A 9-''j'' symbol is invariant under reflection about either diagonal as well as [[even permutation]]s of its rows or columns: :<math> \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} = \begin{Bmatrix} j_1 & j_4 & j_7\\ j_2 & j_5 & j_8\\ j_3 & j_6 & j_9 \end{Bmatrix} = \begin{Bmatrix} j_9 & j_6 & j_3\\ j_8 & j_5 & j_2\\ j_7 & j_4 & j_1 \end{Bmatrix} = \begin{Bmatrix} j_7 & j_4 & j_1\\ j_9 & j_6 & j_3\\ j_8 & j_5 & j_2 \end{Bmatrix}. </math> An odd permutation of rows or columns yields a phase factor <math>(-1)^S</math>, where :<math>S=\sum_{i=1}^9 j_i. </math> For example: :<math> \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} = (-1)^S \begin{Bmatrix} j_4 & j_5 & j_6\\ j_1 & j_2 & j_3\\ j_7 & j_8 & j_9 \end{Bmatrix} = (-1)^S \begin{Bmatrix} j_2 & j_1 & j_3\\ j_5 & j_4 & j_6\\ j_8 & j_7 & j_9 \end{Bmatrix}. </math> ==Reduction to 6j symbols== The 9-''j'' symbols can be calculated as sums over triple-products of 6-''j'' symbols where the summation extends over all {{math\|''x''}} admitted by the triangle conditions in the factors: :<math> \begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \\ j_7 & j_8 & j_9 \end{Bmatrix} = \sum_x (-1)^{2 x}(2 x + 1) \begin{Bmatrix} j_1 & j_4 & j_7 \\ j_8 & j_9 & x \end{Bmatrix} \begin{Bmatrix} j_2 & j_5 & j_8 \\ j_4 & x & j_6 \end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \\ x & j_1 & j_2 \end{Bmatrix} </math>. ==Special case== When <math>j_9=0</math> the 9-''j'' symbol is proportional to a [[6-j symbol]]: :<math> \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & 0 \end{Bmatrix} = \frac{\delta_{j_3,j_6} \delta_{j_7,j_8}}{\sqrt{(2j_3+1)(2j_7+1)}} (-1)^{j_2+j_3+j_4+j_7} \begin{Bmatrix} j_1 & j_2 & j_3\\ j_5 & j_4 & j_7 \end{Bmatrix}. </math> ==Orthogonality relation== The 9-''j'' symbols satisfy this orthogonality relation: :<math> \sum_{j_7 j_8} (2j_7+1)(2j_8+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end{Bmatrix} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}} {(2j_3+1)(2j_6+1)}. </math> The [[3-jm symbol#Orthogonality relations\|''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 zero otherwise. ==3''n''-j symbols== The [[6-j symbol]] is the first representative, {{math\|''n'' {{=}} 2}}, of {{math\|3''n''}}-''j'' symbols that are defined as sums of products of {{math\|''n''}} of Wigner's 3-''jm'' coefficients. The sums are over all combinations of {{math\|m}} that the {{math\|3''n''}}-''j'' coefficients admit, i.e., which lead to non-vanishing contributions. If each 3-''jm'' factor is represented by a vertex and each j by an edge, these {{math\|3''n''}}-''j'' symbols can be mapped on certain [[Table of simple cubic graphs\|3-regular graphs]] with {{math\|3''n''}} vertices and {{math\|2''n''}} nodes. The 6-''j'' symbol is associated with the [[Complete graph\|K<sub>4</sub>]] graph on 4 vertices, the 9-''j'' symbol with the [[Gas water electricity\|utility graph]] on 6 vertices (''K''<sub>3,3</sub>), and the two distinct (non-isomorphic) 12-''j'' symbols with the [[Hypercube graph\|''Q''<sub>3</sub>]] and [[Wagner graph]]s on 8 vertices. Symmetry relations are generally representative of the automorphism group of these graphs. ==See also== * [[Clebsch–Gordan coefficients]] * [[3-j symbol]], also called 3-jm symbol * [[Racah W-coefficient]] * [[6-j symbol]] * [[9-j symbol]] ==References== {{no footnotes\|date=September 2010}} * {{cite book \|last1= Biedenharn \|first1= L. C. \|last2= van Dam \|first2= H. \|title= Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers \|year= 1965 \|publisher= [[Academic Press]] \|location= New York \|isbn= 0120960567}} * {{cite book \|last= Edmonds \|first= A. R. \|title= Angular Momentum in Quantum Mechanics \|year= 1957 \|publisher= [[Princeton University Press]] \|location= Princeton, New Jersey \|isbn= 0-691-07912-9 \|url-access= registration \|url= https://archive.org/details/angularmomentumi0000edmo }} * {{cite book \|last= Condon \|first= Edward U. \|author2= Shortley, 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= 0-521-09209-4 \|chapter= Chapter 3}} {{dlmf\|id=34 \|title=3j,6j,9j Symbols\|first=Leonard C.\|last= Maximon}} {{cite book \|last= Messiah \|first= Albert \|title= Quantum Mechanics (Volume II) \|year= 1981 \| edition= 12th \|publisher= [[Elsevier\|North Holland Publishing]] \|location= New York \|isbn= 0-7204-0045-7}} * {{cite book \|last= Brink \|first= D. M. \|author2= Satchler, G. R. \|title= Angular Momentum \|year= 1993 \|edition= 3rd \|publisher= [[Clarendon Press]] \|location= Oxford \|isbn= 0-19-851759-9 \|chapter= Chapter 2 \|url-access= registration \|url= https://archive.org/details/angularmomentum0000brin }} * {{cite book \|last= Zare \|first= Richard N. \|title= Angular Momentum \|year=1988 \|publisher= [[John Wiley & Sons\|John Wiley]] \|location= New York \|isbn= 0-471-85892-7 \|chapter= Chapter 2}} * {{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= 0201135078 }} * {{cite book \|last= Varshalovich \|first= D. A. \|author2= Moskalev, A. N.\|author3= Khersonskii, V. K. \|title= Quantum Theory of Angular Momentum \| year= 1988 \|publisher= [[World Scientific]] \|location= Singapore \|isbn= 9971-50-107-4 }} {{cite journal \|first1=H. A. \|last1=Jahn \|first2=J. \|last2=Hope \|title=Symmetry properties of the Wigner 9j symbol \|journal=Physical Review \|year=1954 \|volume=93 \|issue=2 \|page=318 \|doi=10.1103/PhysRev.93.318 \|bibcode=1954PhRv...93..318J }} ==External links== {{cite web \|first1=Anthony \|last1=Stone \|url=http://www-stone.ch.cam.ac.uk/wigner.shtml \|title=Wigner coefficient calculator }} (Gives answer in exact fractions) * {{ cite web \|url=http://plasma-gate.weizmann.ac.il/369j.html \|title=369j-symbol calculator \|author=Plasma Laboratory of Weizmann Institute of Science }} (Answer as floating point numbers) * {{cite web \|url=http://caagt.ugent.be/yutsis/GYutsisApplet.caagt \|title=GYutsis Applet \|first1=Veerl \|last1=Fack \|first2=Dries \|last2=van Dyck }} * {{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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