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http://functions.wolfram.com/07.38.07.0004.01
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ClebschGordan[{Subscript[j, 1], Subscript[m, 1]},
{Subscript[j, 2], Subscript[m, 2]}, {j, m}] ==
KroneckerDelta[m, Subscript[m, 1] + Subscript[m, 2]]
((-1)^(Subscript[j, 1] + j - Subscript[m, 1] - m)/
2^(Subscript[j, 1] + Subscript[j, 2] + 1))
((Sqrt[(Subscript[j, 1] + Subscript[j, 2] - j)!]
Sqrt[(Subscript[j, 1] + Subscript[j, 2] + j + 1)!] Sqrt[2 j + 1])/
(Sqrt[(Subscript[j, 1] + Subscript[m, 1])!]
Sqrt[(Subscript[j, 1] - Subscript[m, 1])!]
Sqrt[(Subscript[j, 2] + Subscript[m, 2])!]
Sqrt[(Subscript[j, 2] - Subscript[m, 2])!]))
Integrate[(1 - t^2)^((Subscript[j, 1] + Subscript[j, 2])/2)
((1 - t)/(1 + t))^((Subscript[m, 1] - Subscript[m, 2])/2)
WignerD[Subscript[j, 2] - Subscript[j, 1], m, j, ArcCos[t]],
{t, -1, 1}] /; \[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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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> <mrow> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> , </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </msub> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> j </mi> <mo> - </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> m </mi> </mrow> </msup> <msup> <mn> 2 </mn> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </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> </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> <mo> + </mo> <mn> 1 </mn> </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> <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> </msqrt> <mo> ⁢ </mo> <msqrt> <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> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <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> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <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> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mfrac> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> d </mi> <mrow> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mi> m </mi> </mrow> <mi> j </mi> </msubsup> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </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> 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> <times /> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> <apply> <plus /> <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> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <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> <cn type='integer'> 1 </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> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </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> <ci> j </ci> <cn type='integer'> 1 </cn> </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> <plus /> <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> </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> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </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'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </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'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <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> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> t </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <ci> j </ci> </apply> <apply> <arccos /> <ci> t </ci> </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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