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http://functions.wolfram.com/07.38.23.0007.01
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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
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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"]]]]]]]]
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<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> ∑ </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> 〈 </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁢ </mo> <mtext>   </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> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "\[MediumSpace]", RowBox[List["j", "+", SubscriptBox["j", "2"]]], "\[MediumSpace]", "0", "\[MediumSpace]", "0"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["j", "1"], "\[MediumSpace]", RowBox[List["j", "+", SubscriptBox["j", "2"]]], "\[MediumSpace]", RowBox[List["j", "+", SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"]]], "\[MediumSpace]", "0"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mn> 2 </mn> </msup> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mfrac> <msub> <semantics> <mi> δ </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> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> j </mi> <mo> ∈ </mo> <mi> ℕ </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> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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