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http://functions.wolfram.com/07.38.17.0048.01
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ClebschGordan[{Subscript[j, 1], Subscript[m, 1]},
{Subscript[j, 2], Subscript[m, 2]}, {j, m}] ==
(1/Sqrt[1 + j + Subscript[j, 1] - Subscript[m, 2]])
(Sqrt[1 + 2 j] ClebschGordan[
{(1/2) (j + Subscript[j, 1] + Subscript[m, 2]),
(1/2) (j - Subscript[j, 1] + Subscript[m, 1] + m)},
{Subscript[j, 2], -j + Subscript[j, 1]},
{(1/2) (j + Subscript[j, 1] - Subscript[m, 2]),
(1/2) (-j + Subscript[j, 1] + Subscript[m, 1] + m)}])
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Cell[BoxData[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", SqrtBox[RowBox[List["1", "+", "j", "+", SubscriptBox["j", "1"], "-", SubscriptBox["m", "2"]]]]], RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", "j"]]]]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "+", SubscriptBox["j", "1"], "+", SubscriptBox["m", "2"]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "-", SubscriptBox["j", "1"], "+", SubscriptBox["m", "1"], "+", "m"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", RowBox[List[RowBox[List["-", "j"]], "+", SubscriptBox["j", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "+", SubscriptBox["j", "1"], "-", SubscriptBox["m", "2"]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", SubscriptBox["j", "1"], "+", SubscriptBox["m", "1"], "+", "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> <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> <mfrac> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> j </mi> </mrow> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> m </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "+", SubscriptBox["j", "1"], "+", SubscriptBox["m", "2"]]], ")"]]]], "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "+", "m", "-", SubscriptBox["j", "1"], "+", SubscriptBox["m", "1"]]], ")"]]]], "\[MediumSpace]", RowBox[List[SubscriptBox["j", "1"], "-", "j"]]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "+", SubscriptBox["j", "1"], "+", SubscriptBox["m", "2"]]], ")"]]]], "\[MediumSpace]", SubscriptBox["j", "2"], "\[MediumSpace]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "+", SubscriptBox["j", "1"], "-", SubscriptBox["m", "2"]]], ")"]]]], "\[MediumSpace]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "m", "+", SubscriptBox["j", "1"], "+", SubscriptBox["m", "1"]]], ")"]]]]]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <times /> <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> <power /> <apply> <plus /> <ci> j </ci> <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'> 2 </cn> </apply> </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> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> j </ci> <ci> m </ci> <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> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </list> <list> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> j </ci> <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'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> m </ci> <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> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", 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_"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", "j"]]]]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "+", SubscriptBox["j", "1"], "+", SubscriptBox["mm", "2"]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "-", SubscriptBox["j", "1"], "+", SubscriptBox["mm", "1"], "+", "m"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", RowBox[List[RowBox[List["-", "j"]], "+", SubscriptBox["j", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "+", SubscriptBox["j", "1"], "-", SubscriptBox["mm", "2"]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", SubscriptBox["j", "1"], "+", SubscriptBox["mm", "1"], "+", "m"]], ")"]]]]]], "}"]]]], "]"]]]], SqrtBox[RowBox[List["1", "+", "j", "+", SubscriptBox["j", "1"], "-", SubscriptBox["mm", "2"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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