Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ClebschGordan






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ClebschGordan[{j1,m1},{j2,m2},{j,m}] > Summation > Finite summation > Involving two Clebsch Gordan coefficients





http://functions.wolfram.com/07.38.23.0013.01









  


  










Input Form





Sum[(2 j + 1) ClebschGordan[{Subscript[j, 1], Subscript[m, 1]}, {j, m}, {Subscript[j, 2], Subscript[m, 2]}] ClebschGordan[ {Subscript[j, 1], Derivative[1][Subscript[m, 1]]}, {j, m}, {Subscript[j, 2], Derivative[1][Subscript[m, 2]]}], {j, Abs[Subscript[j, 1] - Subscript[j, 2]], Subscript[j, 1] + Subscript[j, 2]}, {m, -j, j}] == (2 Subscript[j, 2] + 1) KroneckerDelta[Subscript[m, 1], Derivative[1][Subscript[m, 1]]] KroneckerDelta[Subscript[m, 2], Derivative[1][Subscript[m, 2]]] /; \[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\ \[ScriptA]\[ScriptL]\[ScriptCapitalQ][{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 1] + Subscript[j, 2], Subscript[m, 1] + Subscript[m, 2]}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"]]], "]"]]]], RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "j"]]]], "j"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "j"]], "+", "1"]], ")"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "m"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubsuperscriptBox["m", "1", "\[Prime]"]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "m"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubsuperscriptBox["m", "2", "\[Prime]"]]], "}"]]]], "]"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["j", "2"]]], "+", "1"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["m", "1"], ",", SubsuperscriptBox["m", "1", "\[Prime]"]]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["m", "2"], ",", SubsuperscriptBox["m", "2", "\[Prime]"]]], "]"]]]]]], "/;", RowBox[List["\[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ]", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"]]], ",", RowBox[List[SubscriptBox["m", "1"], "+", SubscriptBox["m", "2"]]]]], "}"]]]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> </mrow> <mi> j </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mo> &#9001; </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mi> j </mi> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mi> m </mi> </mrow> <mtext> &#8287; </mtext> <mo> &#10072; </mo> <mtext> &#8287; </mtext> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mi> j </mi> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> &#9002; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftAngleBracket]&quot;, RowBox[List[RowBox[List[SubscriptBox[&quot;j&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, &quot;j&quot;, &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, &quot;m&quot;]], &quot;\[MediumSpace]&quot;, &quot;\[VerticalSeparator]&quot;, &quot;\[MediumSpace]&quot;, RowBox[List[SubscriptBox[&quot;j&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, &quot;j&quot;, &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;j&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;m&quot;, &quot;2&quot;]]]]], &quot;\[RightAngleBracket]&quot;]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> &#9001; </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mi> j </mi> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msubsup> <mi> m </mi> <mn> 1 </mn> <mo> &#8242; </mo> </msubsup> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mi> m </mi> </mrow> <mtext> &#8287; </mtext> <mo> &#10072; </mo> <mtext> &#8287; </mtext> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mi> j </mi> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <msubsup> <mi> m </mi> <mn> 2 </mn> <mo> &#8242; </mo> </msubsup> </mrow> </mrow> <mo> &#9002; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftAngleBracket]&quot;, RowBox[List[RowBox[List[SubscriptBox[&quot;j&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, &quot;j&quot;, &quot;\[MediumSpace]&quot;, SubsuperscriptBox[&quot;m&quot;, &quot;1&quot;, &quot;\[Prime]&quot;, Rule[MultilineFunction, None]], &quot;\[MediumSpace]&quot;, &quot;m&quot;]], &quot;\[MediumSpace]&quot;, &quot;\[VerticalSeparator]&quot;, &quot;\[MediumSpace]&quot;, RowBox[List[SubscriptBox[&quot;j&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, &quot;j&quot;, &quot;\[MediumSpace]&quot;, SubscriptBox[&quot;j&quot;, &quot;2&quot;], &quot;\[MediumSpace]&quot;, SubsuperscriptBox[&quot;m&quot;, &quot;2&quot;, &quot;\[Prime]&quot;, Rule[MultilineFunction, None]]]]]], &quot;\[RightAngleBracket]&quot;]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msubsup> <mi> m </mi> <mn> 1 </mn> <mo> &#8242; </mo> </msubsup> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msubsup> <mi> m </mi> <mn> 2 </mn> <mo> &#8242; </mo> </msubsup> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#119979;&#119997;&#120014;&#120008;&#119998;&#119992;&#119990;&#8467;&#119980; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </lowlimit> <uplimit> <ci> j </ci> </uplimit> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> ClebschGordan </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <ci> j </ci> <ci> m </ci> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <partialdiff /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <ci> j </ci> <ci> m </ci> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <partialdiff /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </list> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <partialdiff /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <partialdiff /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> &#119979;&#119997;&#120014;&#120008;&#119998;&#119992;&#119990;&#8467;&#119980; </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </list> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"]]], "]"]]]], RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m_", "=", RowBox[List["-", "j"]]]], "j"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m_", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "m_"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m_", "2"]]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SuperscriptBox[SubscriptBox["m_", "1"], "\[Prime]", Rule[MultilineFunction, None]]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "m_"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SuperscriptBox[SubscriptBox["m_", "2"], "\[Prime]", Rule[MultilineFunction, None]]]], "}"]]]], "]"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], "+", "1"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["mm", "1"], ",", SuperscriptBox[SubscriptBox["mm", "1"], "\[Prime]", Rule[MultilineFunction, None]]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["mm", "2"], ",", SuperscriptBox[SubscriptBox["mm", "2"], "\[Prime]", Rule[MultilineFunction, None]]]], "]"]]]], "/;", RowBox[List["\[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ]", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["mm", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["mm", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"]]], ",", RowBox[List[SubscriptBox["mm", "1"], "+", SubscriptBox["mm", "2"]]]]], "}"]]]], "]"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-12-21





© 1998- Wolfram Research, Inc.