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ClebschGordan






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.38.23.0007.01









  


  










Input Form





Sum[ClebschGordan[{Subscript[j, 1], 0}, {Subscript[j, 2] + j, 0}, {Subscript[j, 1] - Subscript[j, 2] + j, 0}]^2/ ((2 Subscript[j, 2] - 1) (2 Subscript[j, 1] - 2 Subscript[j, 2] + 2 j + 1)), {Subscript[j, 2], 0, Subscript[j, 1]}] == -(KroneckerDelta[Subscript[j, 1], 0]/(2 j + 1)) /; Element[j, Integers] && j >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], SubscriptBox["j", "1"]], FractionBox[SuperscriptBox[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "2"], "+", "j"]], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", "j"]], ",", "0"]], "}"]]]], "]"]], "2"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["j", "2"]]], "-", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "-", RowBox[List["2", SubscriptBox["j", "2"]]], "+", RowBox[List["2", "j"]], "+", "1"]], ")"]]]]]]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["j", "1"], ",", "0"]], "]"]], RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]]]]], "/;", RowBox[List[RowBox[List["j", "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[GreaterEqual]", "0"]]]]]]]]










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> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </munderover> <mfrac> <msup> <semantics> <mrow> <mo> &#9001; </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> </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> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8290; </mo> <mtext> &#8287; </mtext> <mn> 0 </mn> </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;, RowBox[List[&quot;j&quot;, &quot;+&quot;, SubscriptBox[&quot;j&quot;, &quot;2&quot;]]], &quot;\[MediumSpace]&quot;, &quot;0&quot;, &quot;\[MediumSpace]&quot;, &quot;0&quot;]], &quot;\[MediumSpace]&quot;, &quot;\[VerticalSeparator]&quot;, &quot;\[MediumSpace]&quot;, RowBox[List[SubscriptBox[&quot;j&quot;, &quot;1&quot;], &quot;\[MediumSpace]&quot;, RowBox[List[&quot;j&quot;, &quot;+&quot;, SubscriptBox[&quot;j&quot;, &quot;2&quot;]]], &quot;\[MediumSpace]&quot;, RowBox[List[&quot;j&quot;, &quot;+&quot;, SubscriptBox[&quot;j&quot;, &quot;1&quot;], &quot;-&quot;, SubscriptBox[&quot;j&quot;, &quot;2&quot;]]], &quot;\[MediumSpace]&quot;, &quot;0&quot;]]]], &quot;\[RightAngleBracket]&quot;]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mn> 2 </mn> </msup> <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> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <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> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mfrac> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> j </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <apply> <ci> ClebschGordan </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </list> <list> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </list> <list> <apply> <plus /> <ci> j </ci> <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> <cn type='integer'> 0 </cn> </list> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <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> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> j </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], SubscriptBox["j", "1"]], FractionBox[SuperscriptBox[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "2"], "+", "j"]], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", "j"]], ",", "0"]], "}"]]]], "]"]], "2"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "-", RowBox[List["2", " ", SubscriptBox["j", "2"]]], "+", RowBox[List["2", " ", "j"]], "+", "1"]], ")"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["j", "1"], ",", "0"]], "]"]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]]], "/;", RowBox[List[RowBox[List["j", "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2001-12-21





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