Clebsch‐Gordan coefficients: Summation (formula 07.38.23.0035)

ClebschGordan

$Mathematica Notation$

Hypergeometric Functions

ClebschGordan[{j₁,m₁},{j₂,m₂},{j,m}]

Summation

Finite summation

Involving two Clebsch Gordan coefficients and one 6j symbol

http://functions.wolfram.com/07.38.23.0035.01

Input Form

Sum[(-1)^(c + d + f) Sqrt[2 c + 1] Sqrt[2 e + 1] ClebschGordan[{e, \[Epsilon]}, {a, \[Alpha]}, {f, \[CurlyPhi]}] ClebschGordan[{d, \[Delta]}, {c, \[Gamma]}, {f, \[CurlyPhi]}] SixJSymbol[{b, a, c}, {f, d, e}], {f, Max[Abs[a - e], Abs[c - d]], Min[a + e, c + d]}, {\[CurlyPhi], -f, f}] == ClebschGordan[{a, \[Alpha]}, {b, \[Beta]}, {c, \[Gamma]}] ClebschGordan[{d, \[Delta]}, {b, \[Beta]}, {e, \[Epsilon]}]

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> f </mi> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> a </mi> <mo> - </mo> <mi> e </mi> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> e </mi> </mrow> <mo> , </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> φ </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> f </mi> </mrow> </mrow> <mi> f </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> <mo> + </mo> <mi> f </mi> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> ϵ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> α </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> φ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["e", "\[MediumSpace]", "a", "\[MediumSpace]", "\[Epsilon]", "\[MediumSpace]", "\[Alpha]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["e", "\[MediumSpace]", "a", "\[MediumSpace]", "f", "\[MediumSpace]", "\[CurlyPhi]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> c </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> δ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> γ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> c </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> f </mi> <mo> ⁢ </mo> 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ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> δ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> β </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> e </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> ϵ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["d", "\[MediumSpace]", "b", "\[MediumSpace]", "\[Delta]", "\[MediumSpace]", "\[Beta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["d", "\[MediumSpace]", "b", "\[MediumSpace]", "e", "\[MediumSpace]", "\[Epsilon]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> φ </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> </lowlimit> <uplimit> <ci> f </ci> </uplimit> <apply> <sum /> <bvar> <ci> f </ci> </bvar> <lowlimit> <apply> <max /> <apply> <abs /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> </apply> </apply> <apply> <abs /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <min /> <apply> <plus /> <ci> a </ci> <ci> e </ci> </apply> <apply> <plus /> <ci> c </ci> <ci> d </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> c </ci> <ci> d </ci> <ci> f </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> e </ci> <ci> ϵ </ci> </list> <list> <ci> a </ci> <ci> α </ci> </list> <list> <ci> f </ci> <ci> φ </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> d </ci> <ci> δ </ci> </list> <list> <ci> c </ci> <ci> γ </ci> </list> <list> <ci> f </ci> <ci> φ </ci> </list> </apply> <apply> <ci> SixJSymbol </ci> <list> <ci> b </ci> <ci> a </ci> <ci> c </ci> </list> <list> <ci> f </ci> <ci> d </ci> <ci> e </ci> </list> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> ClebschGordan </ci> <list> <ci> a </ci> <ci> α </ci> </list> <list> <ci> b </ci> <ci> β </ci> </list> <list> <ci> c </ci> <ci> γ </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> d </ci> <ci> δ </ci> </list> <list> <ci> b </ci> <ci> β </ci> </list> <list> <ci> e </ci> <ci> ϵ </ci> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-12-21