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http://functions.wolfram.com/07.38.07.0008.01
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ClebschGordan[{Subscript[j, 1], Subscript[m, 1]},
{Subscript[j, 2], Subscript[m, 2]}, {j, m}]
ClebschGordan[{Subscript[j, 1], Subscript[n, 1]},
{Subscript[j, 2], Subscript[n, 2]}, {j, n}] ==
((2 j + 1)/(8 Pi^2)) Integrate[Sin[\[Beta]] WignerD[Subscript[m, 1],
Subscript[n, 1], Subscript[j, 1], \[Alpha], \[Beta], \[Gamma]]
WignerD[Subscript[m, 2], Subscript[n, 2], Subscript[j, 2], \[Alpha],
\[Beta], \[Gamma]] Conjugate[WignerD[m, n, 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}] && \[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\
\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ][
{Subscript[j, 1], Subscript[n, 1]}, {Subscript[j, 2], Subscript[n, 2]},
{j, n}]
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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> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mi> j </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> m </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", SubscriptBox["m", "1"], "\[MediumSpace]", SubscriptBox["m", "2"]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["j", "1"], "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", "j", "\[MediumSpace]", "m"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mi> j </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> n </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", SubscriptBox["n", "1"], "\[MediumSpace]", SubscriptBox["n", "2"]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["j", "1"], "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", "j", "\[MediumSpace]", "n"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </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> n </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> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </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> <mi> n </mi> </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> <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> <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> n </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> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> ClebschGordan </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> <ci> ClebschGordan </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </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> n </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <ci> j </ci> <ci> n </ci> </list> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <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> <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> n </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> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </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> <ci> n </ci> </apply> <ci> j </ci> </apply> <ci> α </ci> <ci> β </ci> <ci> γ </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <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> <ci> 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </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> n </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <ci> j </ci> <ci> n </ci> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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){kind=small}
