Clebsch‐Gordan coefficients: Summation (formula 07.38.23.0040)

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/07.38.23.0040.01

Input Form

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

Standard Form

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MathML Form

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</mo> <mi> c </mi> </mrow> </mrow> <mi> c </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> d </mi> <mo> + </mo> <mi> e </mi> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> α </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> β </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> c </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> γ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["a", "\[MediumSpace]", "b", "\[MediumSpace]", "\[Alpha]", "\[MediumSpace]", "\[Beta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["a", "\[MediumSpace]", "b", "\[MediumSpace]", "c", "\[MediumSpace]", "\[Gamma]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> c </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> φ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> γ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> c </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["f", "\[MediumSpace]", "c", "\[MediumSpace]", "\[CurlyPhi]", "\[MediumSpace]", "\[Gamma]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["f", "\[MediumSpace]", "c", "\[MediumSpace]", "e", "\[MediumSpace]", "\[Epsilon]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <semantics> <mo> { </mo> <annotation encoding='Mathematica'> TagBox[StyleBox["{", Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> <mtext>   </mtext> <mtable> <mtr> <mtd> <mi> a </mi> </mtd> <mtd> <mi> b </mi> </mtd> <mtd> <mi> c </mi> </mtd> </mtr> <mtr> <mtd> <mi> e </mi> </mtd> <mtd> <mi> f </mi> </mtd> <mtd> <mi> d </mi> </mtd> </mtr> </mtable> <mtext>   </mtext> <semantics> <mo> } </mo> <annotation encoding='Mathematica'> TagBox[StyleBox["}", Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> e </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mo> - 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Rule Form

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

2001-12-21