![](/common/images/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
http://functions.wolfram.com/07.38.23.0041.01
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
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]}]
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["c", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["a", "-", "e"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["b", "-", "f"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["a", "+", "e"]], ",", RowBox[List["b", "+", "f"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Gamma]", "=", RowBox[List["-", "c"]]]], "c"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", "e"]]], FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", "a"]], "+", "1"]]], RowBox[List["(", RowBox[List[RowBox[List["2", "c"]], "+", "1"]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["2", "b"]], "+", "1"]]]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]], RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["c", ",", "f", ",", "b"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "e", ",", "a"]], "}"]]]], "]"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]]]], "]"]]]]]]]]
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
<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> c </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> b </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> a </mi> <mo> + </mo> <mi> e </mi> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> γ </mi> <mo> = </mo> <mrow> <mo> - </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> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <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> 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> <mi> e </mi> </mtd> <mtd> <mi> a </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> 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> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> φ </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> δ </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> d </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> a </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> α </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["f", "\[MediumSpace]", "d", "\[MediumSpace]", "\[CurlyPhi]", "\[MediumSpace]", "\[Delta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["f", "\[MediumSpace]", "d", "\[MediumSpace]", "a", "\[MediumSpace]", "\[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> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> f </ci> <ci> φ </ci> </list> <list> <ci> c </ci> <ci> γ </ci> </list> <list> <ci> b </ci> <ci> β </ci> </list> </apply> <apply> <ci> ClebschGordan </ci> <list> <ci> c </ci> <ci> γ </ci> </list> <list> <ci> a </ci> <ci> α </ci> </list> <list> <ci> e </ci> <ci> ϵ </ci> </list> </apply> <apply> <ci> SixJSymbol </ci> <list> <ci> c </ci> <ci> f </ci> <ci> b </ci> </list> <list> <ci> d </ci> <ci> e </ci> <ci> a </ci> </list> </apply> </apply> </apply> </apply> <apply> <times /> <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> <ci> ClebschGordan </ci> <list> <ci> f </ci> <ci> φ </ci> </list> <list> <ci> d </ci> <ci> δ </ci> </list> <list> <ci> a </ci> <ci> α </ci> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/clear.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| | ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["c_", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["a_", "-", "e_"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["b_", "-", "f_"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["a_", "+", "e_"]], ",", RowBox[List["b_", "+", "f_"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Gamma]_", "=", RowBox[List["-", "c_"]]]], "c_"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", " ", "e_"]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "a_"]], "+", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c_"]], "+", "1"]], ")"]]]], ")"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["f_", ",", "\[CurlyPhi]_"]], "}"]], ",", RowBox[List["{", RowBox[List["c_", ",", "\[Gamma]_"]], "}"]], ",", RowBox[List["{", RowBox[List["b_", ",", "\[Beta]_"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["c_", ",", "\[Gamma]_"]], "}"]], ",", RowBox[List["{", RowBox[List["a_", ",", "\[Alpha]_"]], "}"]], ",", RowBox[List["{", RowBox[List["e_", ",", "\[Epsilon]_"]], "}"]]]], "]"]], " ", RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["c_", ",", "f_", ",", "b_"]], "}"]], ",", RowBox[List["{", RowBox[List["d_", ",", "e_", ",", "a_"]], "}"]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["2", " ", "b_"]], "+", "1"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]]]], "]"]]]]]]]] |
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Date Added to functions.wolfram.com (modification date)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/common/images/spacer.gif) |
|
|