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http://functions.wolfram.com/07.38.23.0034.01
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Sum[(-1)^(2 e) Sqrt[2 c + 1] Sqrt[2 d + 1] ClebschGordan[{b, \[Beta]},
{d, \[Delta]}, {e, \[Epsilon]}] ClebschGordan[{f, \[CurlyPhi]},
{c, \[Gamma]}, {e, \[Epsilon]}] SixJSymbol[{a, b, c}, {e, f, d}],
{e, Max[Abs[b - d], Abs[c - f]], Min[b + d, c + f]},
{\[Epsilon], -e, e}] == ClebschGordan[{a, \[Alpha]}, {b, \[Beta]},
{c, \[Gamma]}] ClebschGordan[{a, \[Alpha]}, {f, \[CurlyPhi]},
{d, \[Delta]}]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["e", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "d"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["c", "-", "f"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["b", "+", "d"]], ",", RowBox[List["c", "+", "f"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Epsilon]", "=", RowBox[List["-", "e"]]]], "e"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", "e"]]], SqrtBox[RowBox[List[RowBox[List["2", "c"]], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["2", "d"]], "+", "1"]]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]], RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", "c"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "f", ",", "d"]], "}"]]]], "]"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]]]], "]"]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> e </mi> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> b </mi> <mo> - </mo> <mi> d </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> f </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> b </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> , </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> ϵ </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> e </mi> </mrow> </mrow> <mi> e </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </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> d </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> β </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> δ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> d </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["b", "\[MediumSpace]", "d", "\[MediumSpace]", "\[Beta]", "\[MediumSpace]", "\[Delta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["b", "\[MediumSpace]", "d", "\[MediumSpace]", "e", "\[MediumSpace]", "\[Epsilon]"]]]], "\[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> 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> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> α </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> φ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> δ </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["a", "\[MediumSpace]", "f", "\[MediumSpace]", "\[Alpha]", "\[MediumSpace]", "\[CurlyPhi]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["a", "\[MediumSpace]", "f", "\[MediumSpace]", "d", "\[MediumSpace]", "\[Delta]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> ϵ </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> </lowlimit> <uplimit> <ci> e </ci> </uplimit> <apply> <sum /> <bvar> <ci> e </ci> </bvar> <lowlimit> <apply> <max /> <apply> <abs /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> <apply> <abs /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <min /> <apply> <plus /> <ci> b </ci> <ci> d </ci> </apply> <apply> <plus /> <ci> c </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> </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> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> b </ci> <ci> β </ci> </list> <list> <ci> d </ci> <ci> δ </ci> </list> <list> <ci> e </ci> <ci> ϵ </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> f </ci> <ci> φ </ci> </list> <list> <ci> c </ci> <ci> γ </ci> </list> <list> <ci> e </ci> <ci> ϵ </ci> </list> </apply> <apply> <ci> SixJSymbol </ci> <list> <ci> a </ci> <ci> b </ci> <ci> c </ci> </list> <list> <ci> e </ci> <ci> f </ci> <ci> d </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> a </ci> <ci> α </ci> </list> <list> <ci> f </ci> <ci> φ </ci> </list> <list> <ci> d </ci> <ci> δ </ci> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["e_", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b_", "-", "d_"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["c_", "-", "f_"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["b_", "+", "d_"]], ",", RowBox[List["c_", "+", "f_"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Epsilon]_", "=", RowBox[List["-", "e_"]]]], "e_"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", " ", "e_"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "c_"]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "d_"]], "+", "1"]]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", "\[Beta]_"]], "}"]], ",", RowBox[List["{", RowBox[List["d_", ",", "\[Delta]_"]], "}"]], ",", RowBox[List["{", RowBox[List["e_", ",", "\[Epsilon]_"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["f_", ",", "\[CurlyPhi]_"]], "}"]], ",", RowBox[List["{", RowBox[List["c_", ",", "\[Gamma]_"]], "}"]], ",", RowBox[List["{", RowBox[List["e_", ",", "\[Epsilon]_"]], "}"]]]], "]"]], " ", RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "b_", ",", "c_"]], "}"]], ",", RowBox[List["{", RowBox[List["e_", ",", "f_", ",", "d_"]], "}"]]]], "]"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
