Clebsch‐Gordan coefficients: Summation (formula 07.38.23.0041)

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/07.38.23.0041.01

Input Form

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

Standard Form

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MathML Form

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</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> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> β </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]", "b", "\[MediumSpace]", "\[Beta]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> γ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> α </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </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["c", "\[MediumSpace]", "a", "\[MediumSpace]", "\[Gamma]", "\[MediumSpace]", "\[Alpha]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["c", "\[MediumSpace]", "a", "\[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> c </mi> </mtd> <mtd> <mi> f </mi> </mtd> <mtd> <mi> b </mi> </mtd> </mtr> <mtr> <mtd> <mi> d </mi> </mtd> <mtd> 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"\[Alpha]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> γ </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </lowlimit> <uplimit> <ci> c </ci> </uplimit> <apply> <sum /> <bvar> <ci> c </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> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <min /> <apply> <plus /> <ci> a </ci> <ci> e </ci> </apply> <apply> <plus /> <ci> b </ci> <ci> f </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> 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Rule Form

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

2001-12-21