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http://functions.wolfram.com/07.38.23.0012.01
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Sum[ClebschGordan[{j, m}, {Subscript[j, 2], Subscript[m, 2]},
{Subscript[j, 1], Subscript[m, 1]}] ClebschGordan[
{Derivative[1][j], -Derivative[1][m]}, {Subscript[j, 1],
Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}],
{Subscript[m, 1], -Subscript[j, 1], Subscript[j, 1]},
{Subscript[m, 2], -Subscript[j, 2], Subscript[j, 2]}] ==
(-1)^(Subscript[j, 2] - Subscript[j, 1] - j)
((Sqrt[2 Subscript[j, 1] + 1] Sqrt[2 Subscript[j, 2] + 1])/(2 j + 1))
KroneckerDelta[j, Derivative[1][j]] KroneckerDelta[m, Derivative[1][m]] /;
\[ScriptCapitalT]\[ScriptR]\[ScriptI]\[ScriptA]\[ScriptN]\[ScriptG]\
\[ScriptU]\[ScriptL]\[ScriptA]\[ScriptR]\[ScriptCapitalQ][Subscript[j, 1],
Subscript[j, 2], j] && Element[j - m, Integers] && -j <= m <= j
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["m", "1"], "=", RowBox[List["-", SubscriptBox["j", "1"]]]]], SubscriptBox["j", "1"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["m", "2"], "=", RowBox[List["-", SubscriptBox["j", "2"]]]]], SubscriptBox["j", "2"]], RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "m"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SuperscriptBox["j", "\[Prime]"], ",", RowBox[List["-", SuperscriptBox["m", "\[Prime]"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]]]], "]"]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["j", "1"], "-", "j"]]], FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "2"]]], "+", "1"]]]]], RowBox[List[RowBox[List["2", "j"]], "+", "1"]]], RowBox[List["KroneckerDelta", "[", RowBox[List["j", ",", SuperscriptBox["j", "\[Prime]"]]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", SuperscriptBox["m", "\[Prime]"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["\[ScriptCapitalT]\[ScriptR]\[ScriptI]\[ScriptA]\[ScriptN]\[ScriptG]\[ScriptU]\[ScriptL]\[ScriptA]\[ScriptR]\[ScriptCapitalQ]", "[", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "j"]], "]"]], "\[And]", RowBox[List[RowBox[List["j", "-", "m"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "j"]], "\[LessEqual]", "m", "\[LessEqual]", "j"]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </munderover> <mrow> <munderover> <mo> ∑ </mo> <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> </munderover> <mrow> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mi> j </mi> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mi> m </mi> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mi> j </mi> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["j", "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", "m", "\[MediumSpace]", SubscriptBox["m", "2"]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["j", "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", SubscriptBox["j", "1"], "\[MediumSpace]", SubscriptBox["m", "1"]]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msup> <mi> j </mi> <mo> ′ </mo> </msup> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mo> - </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msup> <mi> j </mi> <mo> ′ </mo> </msup> <mo> ⁢ </mo> <mtext>   </mtext> <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> 2 </mn> </msub> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SuperscriptBox["j", "\[Prime]", Rule[MultilineFunction, None]], "\[MediumSpace]", SubscriptBox["j", "1"], "\[MediumSpace]", RowBox[List["-", SuperscriptBox["m", "\[Prime]", Rule[MultilineFunction, None]]]], "\[MediumSpace]", SubscriptBox["m", "1"]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SuperscriptBox["j", "\[Prime]", Rule[MultilineFunction, None]], "\[MediumSpace]", SubscriptBox["j", "1"], "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", SubscriptBox["m", "2"]]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> j </mi> <mo> , </mo> <msup> <mi> j </mi> <mo> ′ </mo> </msup> </mrow> </msub> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> 𝒯𝓇𝒾𝒶𝓃ℊ𝓊ℓ𝒶𝓇𝒬 </mi> <mo> ⁡ </mo> <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> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ≤ </mo> <mi> m </mi> <mo> ≤ </mo> <mi> j </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </lowlimit> <uplimit> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </lowlimit> <uplimit> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> ClebschGordan </ci> <list> <ci> j </ci> <ci> m </ci> </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> <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> </apply> <apply> <ci> ClebschGordan </ci> <list> <apply> <partialdiff /> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <partialdiff /> <ci> m </ci> </apply> </apply> </list> <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> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </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> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </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> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> j </ci> <apply> <partialdiff /> <ci> j </ci> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> <apply> <partialdiff /> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> 𝒯𝓇𝒾𝒶𝓃ℊ𝓊ℓ𝒶𝓇𝒬 </ci> <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> </apply> <apply> <in /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <integers /> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> m </ci> <ci> j </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["m_", "1"], "=", RowBox[List["-", SubscriptBox["j", "1"]]]]], SubscriptBox["j", "1"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["m_", "2"], "=", RowBox[List["-", SubscriptBox["j", "2"]]]]], SubscriptBox["j", "2"]], RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "m_"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m_", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m_", "1"]]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SuperscriptBox["j", "\[Prime]", Rule[MultilineFunction, None]], ",", RowBox[List["-", SuperscriptBox["m_", "\[Prime]", Rule[MultilineFunction, None]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m_", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m_", "2"]]], "}"]]]], "]"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["j", "1"], "-", "j"]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], "+", "1"]]]]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["j", ",", SuperscriptBox["j", "\[Prime]", Rule[MultilineFunction, None]]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", SuperscriptBox["m", "\[Prime]", Rule[MultilineFunction, None]]]], "]"]]]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]], "/;", RowBox[List[RowBox[List["\[ScriptCapitalT]\[ScriptR]\[ScriptI]\[ScriptA]\[ScriptN]\[ScriptG]\[ScriptU]\[ScriptL]\[ScriptA]\[ScriptR]\[ScriptCapitalQ]", "[", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "j"]], "]"]], "&&", RowBox[List[RowBox[List["j", "-", "m"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "j"]], "\[LessEqual]", "m", "\[LessEqual]", "j"]]]]]]]]]] |
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