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 ClebschGordan

 http://functions.wolfram.com/07.38.03.0024.01

 Input Form

 ClebschGordan[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 1] + 1, Subscript[m, 1]}, {j, 2 Subscript[m, 1]}] == (-1)^((j - 2 Subscript[j, 1] - 1)/2) (((2 Subscript[j, 1] + 2) ((2 Subscript[j, 1] + j + 1)/2)! Sqrt[(j + 2 Subscript[m, 1])!] Sqrt[(j - 2 Subscript[m, 1])!] Sqrt[2 j + 1] Sqrt[(2 Subscript[j, 1] - j + 1)!])/ (((2 Subscript[j, 1] - j + 1)/2)! ((j + 2 Subscript[m, 1] - 1)/2)! ((j - 2 Subscript[m, 1] - 1)/2)! Sqrt[Subscript[j, 1] + Subscript[m, 1] + 1] Sqrt[Subscript[j, 1] - Subscript[m, 1] + 1] Sqrt[j] Sqrt[j + 1] Sqrt[(2 Subscript[j, 1] + j + 2)!])) /; Element[(2 Subscript[j, 1] + j - 1)/2, Integers] && \[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\ \[ScriptA]\[ScriptL]\[ScriptCapitalQ][{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 1] + 1, Subscript[m, 1]}, {j, 2 Subscript[m, 1]}]

 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[RowBox[List[SubscriptBox["j", "1"], "+", "1"]], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", RowBox[List["2", SubscriptBox["m", "1"]]]]], "}"]]]], "]"]], " ", "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], FractionBox[RowBox[List["j", "-", RowBox[List["2", SubscriptBox["j", "1"]]], "-", "1"]], "2"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "+", "2"]], ")"]], RowBox[List[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "+", "j", "+", "1"]], "2"], ")"]], "!"]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["j", "+", RowBox[List["2", SubscriptBox["m", "1"]]]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["j", "-", RowBox[List["2", SubscriptBox["m", "1"]]]]], ")"]], "!"]]], SqrtBox[RowBox[List[RowBox[List["2", "j"]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "-", "j", "+", "1"]], ")"]], "!"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "-", "j", "+", "1"]], "2"], ")"]], "!"]], RowBox[List[RowBox[List["(", FractionBox[RowBox[List["j", "+", RowBox[List["2", SubscriptBox["m", "1"]]], "-", "1"]], "2"], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", FractionBox[RowBox[List["j", "-", RowBox[List["2", SubscriptBox["m", "1"]]], "-", "1"]], "2"], ")"]], "!"]], SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["m", "1"], "+", "1"]]], " ", SqrtBox[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["m", "1"], "+", "1"]]], SqrtBox["j"], " ", SqrtBox[RowBox[List["j", "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "+", "j", "+", "2"]], ")"]], "!"]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], "+", "j", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", 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[RowBox[List[SubscriptBox["j", "1"], "+", "1"]], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", RowBox[List["2", SubscriptBox["m", "1"]]]]], "}"]]]], "]"]]]]]]]]

 MathML Form

 j 1 j 1 + 1 m 1 m 1 j 1 j 1 + 1 j 2 m 1 TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "\[MediumSpace]", RowBox[List[SubscriptBox["j", "1"], "+", "1"]], "\[MediumSpace]", SubscriptBox["m", "1"], "\[MediumSpace]", SubscriptBox["m", "1"]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["j", "1"], "\[MediumSpace]", RowBox[List[SubscriptBox["j", "1"], "+", "1"]], "\[MediumSpace]", "j", "\[MediumSpace]", RowBox[List["2", " ", SubscriptBox["m", "1"]]]]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] ( - 1 ) j - 2 j 1 - 1 2 ( 2 j 1 + 2 ) ( 2 j 1 + j + 1 2 ) ! ( j + 2 m 1 ) ! ( j - 2 m 1 ) ! 2 j + 1 ( 2 j 1 - j + 1 ) ! ( 2 j 1 - j + 1 2 ) ! ( j + 2 m 1 - 1 2 ) ! ( j - 2 m 1 - 1 2 ) ! j 1 + m 1 + 1 j 1 - m 1 + 1 j j + 1 ( 2 j 1 + j + 2 ) ! /; 2 j 1 + j - 1 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 ( { j 1 , m 1 } , { j 1 + 1 , m 1 } , { j , 2 m 1 } ) Condition ClebschGordan Subscript j 1 Subscript m 1 Subscript j 1 1 Subscript m 1 j 2 Subscript m 1 -1 j -1 2 Subscript j 1 -1 2 -1 2 Subscript j 1 2 2 Subscript j 1 j 1 2 -1 j 2 Subscript m 1 1 2 j -1 2 Subscript m 1 1 2 2 j 1 1 2 2 Subscript j 1 -1 j 1 1 2 2 Subscript j 1 -1 j 1 2 -1 j 2 Subscript m 1 -1 2 -1 j -1 2 Subscript m 1 -1 2 -1 Subscript j 1 Subscript m 1 1 1 2 Subscript j 1 -1 Subscript m 1 1 1 2 j 1 2 j 1 1 2 2 Subscript j 1 j 2 1 2 -1 2 Subscript j 1 j -1 2 -1 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 Subscript j 1 Subscript m 1 Subscript j 1 1 Subscript m 1 j 2 Subscript m 1 [/itex]

 Rule Form

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

 2001-10-29