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 ClebschGordan

 http://functions.wolfram.com/07.38.07.0004.01

 Input Form

 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}]

 Standard Form

 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[RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", RowBox[List[SubscriptBox["m", "1"], "+", SubscriptBox["m", "2"]]]]], "]"]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["j", "1"], "+", "j", "-", SubscriptBox["m", "1"], "-", "m"]]], SuperscriptBox["2", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", "1"]]]], " ", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", "j"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", "j", "+", "1"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["m", "1"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["m", "1"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["m", "2"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["m", "2"]]], ")"]], "!"]]]]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t", "2"]]], ")"]], FractionBox[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"]]], "2"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "t"]], RowBox[List["1", "+", "t"]]], ")"]], FractionBox[RowBox[List[SubscriptBox["m", "1"], "-", SubscriptBox["m", "2"]]], "2"]], " ", RowBox[List["WignerD", "[", RowBox[List[RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["j", "1"]]], ",", "m", ",", "j", ",", RowBox[List["ArcCos", "[", "t", "]"]]]], "]"]], " ", RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", 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"]], "}"]]]], "]"]]]]]]

 MathML Form

 j 1 j 2 m 1 m 2 j 1 j 2 j m 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]] δ KroneckerDelta m , m 1 + m 2 ( - 1 ) j 1 + j - m 1 - m 2 j 1 + j 2 + 1 ( j 1 + j 2 - j ) ! ( j 1 + j 2 + j + 1 ) ! 2 j + 1 ( j 1 + m 1 ) ! ( j 1 - m 1 ) ! ( j 2 + m 2 ) ! ( j 2 - m 2 ) ! - 1 1 ( 1 - t 2 ) j 1 + j 2 2 ( 1 - t 1 + t ) m 1 - m 2 2 d j 2 - j 1 , m j ( cos - 1 ( t ) ) t /; 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 ( { j 1 , m 1 } , { j 2 , m 2 } , { j , m } ) Condition 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 -1 Subscript m 1 -1 m 2 Subscript j 1 Subscript j 2 1 -1 Subscript j 1 Subscript j 2 -1 j 1 2 Subscript j 1 Subscript j 2 j 1 1 2 2 j 1 1 2 Subscript j 1 Subscript m 1 1 2 Subscript j 1 -1 Subscript m 1 1 2 Subscript j 2 Subscript m 2 1 2 Subscript j 2 -1 Subscript m 2 1 2 -1 t -1 1 1 -1 t 2 Subscript j 1 Subscript j 2 2 -1 1 -1 t 1 t -1 Subscript m 1 -1 Subscript m 2 2 -1 Subscript d Subscript j 2 -1 Subscript j 1 m j t 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 Subscript j 1 Subscript m 1 Subscript j 2 Subscript m 2 j m [/itex]

 Rule Form

 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]", RowBox[List[FractionBox[RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", RowBox[List[SubscriptBox["mm", "1"], "+", SubscriptBox["mm", "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["j", "1"], "+", "j", "-", SubscriptBox["mm", "1"], "-", "m"]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", "j"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", "j", "+", "1"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]]], ")"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "t"]], RowBox[List["1", "+", "t"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["mm", "1"], "-", SubscriptBox["mm", "2"]]], ")"]]]]], " ", RowBox[List["WignerD", "[", RowBox[List[RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["j", "1"]]], ",", "m", ",", "j", ",", RowBox[List["ArcCos", "[", "t", "]"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List[SuperscriptBox["2", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", "1"]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["mm", "1"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["mm", "1"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["mm", "2"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["mm", "2"]]], ")"]], "!"]]]]], ")"]]]]], "/;", RowBox[List["\[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ]", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["mm", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["mm", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "m"]], "}"]]]], "]"]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29