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http://functions.wolfram.com/07.38.23.0039.01
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Sum[(-1)^(2 c) Sqrt[2 e + 1] Sqrt[2 f + 1] ClebschGordan[{b, \[Beta]},
{d, \[Delta]}, {f, \[CurlyPhi]}] ClebschGordan[{a, \[Alpha]},
{f, \[CurlyPhi]}, {c, \[Gamma]}] SixJSymbol[{a, b, e}, {d, c, f}],
{f, Max[Abs[a - c], Abs[b - d]], Min[a + c, b + d]},
{\[CurlyPhi], -f, f}] == ClebschGordan[{b, \[Beta]}, {a, \[Alpha]},
{e, \[Epsilon]}] ClebschGordan[{d, \[Delta]}, {e, \[Epsilon]},
{c, \[Gamma]}]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["f", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["a", "-", "c"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["b", "-", "d"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["a", "+", "c"]], ",", RowBox[List["b", "+", "d"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[CurlyPhi]", "=", RowBox[List["-", "f"]]]], "f"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", "c"]]], SqrtBox[RowBox[List[RowBox[List["2", "e"]], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["2", "f"]], "+", "1"]]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]]]], "]"]], RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", "e"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "c", ",", "f"]], "}"]]]], "]"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]]]], "]"]]]]]]]]
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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> 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> c </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> b </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> c </mi> </mrow> <mo> , </mo> <mrow> <mi> b </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> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> f </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> 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["b", "\[MediumSpace]", "d", "\[MediumSpace]", "\[Beta]", "\[MediumSpace]", "\[Delta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["b", "\[MediumSpace]", "d", "\[MediumSpace]", "f", "\[MediumSpace]", "\[CurlyPhi]"]]]], "\[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> 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]", "f", "\[MediumSpace]", "\[Alpha]", "\[MediumSpace]", "\[CurlyPhi]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["a", "\[MediumSpace]", "f", "\[MediumSpace]", "c", "\[MediumSpace]", "\[Gamma]"]]]], "\[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> e </mi> </mtd> </mtr> <mtr> <mtd> <mi> d </mi> </mtd> <mtd> <mi> c </mi> </mtd> <mtd> <mi> f </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> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> β </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> α </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> b </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["b", "\[MediumSpace]", "a", "\[MediumSpace]", "\[Beta]", "\[MediumSpace]", "\[Alpha]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["b", "\[MediumSpace]", "a", "\[MediumSpace]", "e", "\[MediumSpace]", "\[Epsilon]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> e </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> δ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> ϵ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> e </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["d", "\[MediumSpace]", "e", "\[MediumSpace]", "\[Delta]", "\[MediumSpace]", "\[Epsilon]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["d", "\[MediumSpace]", "e", "\[MediumSpace]", "c", "\[MediumSpace]", "\[Gamma]"]]]], "\[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> c </ci> </apply> </apply> </apply> <apply> <abs /> <apply> <plus /> <ci> b </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> c </ci> </apply> <apply> <plus /> <ci> b </ci> <ci> d </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </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> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> f </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> f </ci> <ci> φ </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> c </ci> <ci> γ </ci> </list> </apply> <apply> <ci> SixJSymbol </ci> <list> <ci> a </ci> <ci> b </ci> <ci> e </ci> </list> <list> <ci> d </ci> <ci> c </ci> <ci> f </ci> </list> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> ClebschGordan </ci> <list> <ci> b </ci> <ci> β </ci> </list> <list> <ci> a </ci> <ci> α </ci> </list> <list> <ci> e </ci> <ci> ϵ </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> d </ci> <ci> δ </ci> </list> <list> <ci> e </ci> <ci> ϵ </ci> </list> <list> <ci> c </ci> <ci> γ </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["f_", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["a_", "-", "c_"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["b_", "-", "d_"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["a_", "+", "c_"]], ",", RowBox[List["b_", "+", "d_"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[CurlyPhi]_", "=", RowBox[List["-", "f_"]]]], "f_"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", " ", "c_"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "e_"]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "f_"]], "+", "1"]]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", "\[Beta]_"]], "}"]], ",", RowBox[List["{", RowBox[List["d_", ",", "\[Delta]_"]], "}"]], ",", RowBox[List["{", RowBox[List["f_", ",", "\[CurlyPhi]_"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "\[Alpha]_"]], "}"]], ",", RowBox[List["{", RowBox[List["f_", ",", "\[CurlyPhi]_"]], "}"]], ",", RowBox[List["{", RowBox[List["c_", ",", "\[Gamma]_"]], "}"]]]], "]"]], " ", RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "b_", ",", "e_"]], "}"]], ",", RowBox[List["{", RowBox[List["d_", ",", "c_", ",", "f_"]], "}"]]]], "]"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
