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http://functions.wolfram.com/07.38.07.0007.01
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ClebschGordan[{Subscript[j, 1], Subscript[m, 1]},
{Subscript[j, 2], Subscript[m, 2]}, {j, m}] ==
(1/(8 Pi^2)) ((Sqrt[(Subscript[j, 1] + Subscript[j, 2] + j + 1)!]
Sqrt[(Subscript[j, 1] + Subscript[j, 2] - j)!] Sqrt[2 j + 1])/
(Sqrt[(2 Subscript[j, 1])!] Sqrt[(2 Subscript[j, 2])!]))
Integrate[Sin[\[Beta]] WignerD[Subscript[m, 1], Subscript[j, 1],
Subscript[j, 1], \[Alpha], \[Beta], \[Gamma]]
WignerD[Subscript[m, 2], -Subscript[j, 2], Subscript[j, 2], \[Alpha],
\[Beta], \[Gamma]] Conjugate[WignerD[m, Subscript[j, 1] -
Subscript[j, 2], j, \[Alpha], \[Beta], \[Gamma]]],
{\[Alpha], 0, 2 Pi}, {\[Beta], 0, Pi}, {\[Gamma], 0, 2 Pi}] /;
\[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\
\[ScriptA]\[ScriptL]\[ScriptCapitalQ][{Subscript[j, 1], Subscript[m, 1]},
{Subscript[j, 2], Subscript[m, 2]}, {j, m}]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "m"]], "}"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["8", SuperscriptBox["\[Pi]", "2"]]]], FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", "j", "+", "1"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", "j"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["2", SubscriptBox["j", "1"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["2", SubscriptBox["j", "2"]]], ")"]], "!"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Integral]", "0", RowBox[List["2", "\[Pi]"]]], RowBox[List[UnderoverscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[UnderoverscriptBox["\[Integral]", "0", RowBox[List["2", "\[Pi]"]]], RowBox[List[RowBox[List["Sin", "[", "\[Beta]", "]"]], " ", RowBox[List["WignerD", "[", RowBox[List[SubscriptBox["m", "1"], ",", SubscriptBox["j", "1"], ",", SubscriptBox["j", "1"], ",", "\[Alpha]", ",", "\[Beta]", ",", "\[Gamma]"]], "]"]], " ", RowBox[List["WignerD", "[", RowBox[List[SubscriptBox["m", "2"], ",", RowBox[List["-", SubscriptBox["j", "2"]]], ",", SubscriptBox["j", "2"], ",", "\[Alpha]", ",", "\[Beta]", ",", "\[Gamma]"]], "]"]], " ", RowBox[List["Conjugate", "[", RowBox[List["WignerD", "[", RowBox[List["m", ",", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"]]], ",", "j", ",", "\[Alpha]", ",", "\[Beta]", ",", "\[Gamma]"]], "]"]], "]"]], " ", RowBox[List["\[DifferentialD]", "\[Gamma]"]], " ", RowBox[List["\[DifferentialD]", "\[Beta]"]], " ", RowBox[List["\[DifferentialD]", "\[Alpha]"]]]]]]]]]]]]]], "/;", RowBox[List["\[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ]", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "m"]], "}"]]]], "]"]]]]]]
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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> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mo> ❘ </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mi> j </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> π </mi> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msubsup> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> β </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> D </mi> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> α </mi> <mo> , </mo> <mi> β </mi> <mo> , </mo> <mi> γ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> D </mi> <mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> α </mi> <mo> , </mo> <mi> β </mi> <mo> , </mo> <mi> γ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mover> <mrow> <msubsup> <mi> D </mi> <mrow> <mi> m </mi> <mo> , </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mi> j </mi> </msubsup> <mo> ( </mo> <mrow> <mi> α </mi> <mo> , </mo> <mi> β </mi> <mo> , </mo> <mi> γ </mi> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> γ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> β </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> α </mi> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> AngleBracket </ci> <apply> <ci> VerticalSeparator </ci> <apply> <times /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <ci> j </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> γ </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </uplimit> <apply> <int /> <bvar> <ci> β </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <int /> <bvar> <ci> α </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </uplimit> <apply> <times /> <apply> <sin /> <ci> β </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> D </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> α </ci> <ci> β </ci> <ci> γ </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> D </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> α </ci> <ci> β </ci> <ci> γ </ci> </apply> <apply> <ci> OverBar </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> D </ci> <ci> m </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> j </ci> </apply> <ci> α </ci> <ci> β </ci> <ci> γ </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <ci> j </ci> <ci> m </ci> </list> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
