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 ClebschGordan

 http://functions.wolfram.com/07.38.23.0033.01

 Input Form

 Sum[(-1)^(c - \[Gamma] + e - \[Epsilon]) ClebschGordan[{a, \[Alpha]}, {b, \[Beta]}, {c, \[Gamma]}] ClebschGordan[{c, \[Gamma]}, {f, \[CurlyPhi]}, {j, \[Mu]}] ClebschGordan[{e, \[Epsilon]}, {b, \[Beta]}, {g, \[Eta]}] ClebschGordan[{d, \[Delta]}, {f, \[CurlyPhi]}, {e, \[Epsilon]}], {\[Beta], -b, b}, {\[Gamma], -c, c}, {\[Epsilon], -e, e}, {\[CurlyPhi], -f, f}] == (-1)^(a + d - \[Alpha] - \[Delta]) Sqrt[2 c + 1] Sqrt[2 e + 1] Sqrt[2 g + 1] Sqrt[2 j + 1] Sum[ClebschGordan[{g, \[Eta]}, {j, -\[Mu]}, {k, \[Kappa]}] ClebschGordan[{d, \[Delta]}, {a, -\[Alpha]}, {k, \[Kappa]}] NineJSymbol[{c, b, a}, {f, e, d}, {j, g, k}], {k, Max[Abs[g - j], Abs[a - d]], Min[g + j, a + d]}, {\[Kappa], -k, k}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Beta]", "=", RowBox[List["-", "b"]]]], "b"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Gamma]", "=", RowBox[List["-", "c"]]]], "c"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Epsilon]", "=", RowBox[List["-", "e"]]]], "e"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[CurlyPhi]", "=", RowBox[List["-", "f"]]]], "f"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["c", "-", "\[Gamma]", "+", "e", "-", "\[Epsilon]"]]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["c", ",", "\[Gamma]"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "\[Mu]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "\[Beta]"]], "}"]], ",", RowBox[List["{", RowBox[List["g", ",", "\[Eta]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "\[CurlyPhi]"]], "}"]], ",", RowBox[List["{", RowBox[List["e", ",", "\[Epsilon]"]], "}"]]]], "]"]]]]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "d", "-", "\[Alpha]", "-", "\[Delta]"]]], SqrtBox[RowBox[List[RowBox[List["2", "c"]], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["2", "e"]], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["2", "g"]], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["2", "j"]], "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["g", "-", "j"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["a", "-", "d"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["g", "+", "j"]], ",", RowBox[List["a", "+", "d"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Kappa]", "=", RowBox[List["-", "k"]]]], "k"], RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["g", ",", "\[Eta]"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", RowBox[List["-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "\[Kappa]"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", RowBox[List["-", "\[Alpha]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "\[Kappa]"]], "}"]]]], "]"]], RowBox[List["NineJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["c", ",", "b", ",", "a"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "e", ",", "d"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "g", ",", "k"]], "}"]]]], "]"]]]]]]]]]]]]]]

 MathML Form

 β = - b b γ = - c c ϵ = - e e φ = - f f ( - 1 ) c + e - γ - ϵ a b α β a b c γ TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["a", "\[MediumSpace]", "b", "\[MediumSpace]", "\[Alpha]", "\[MediumSpace]", "\[Beta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["a", "\[MediumSpace]", "b", "\[MediumSpace]", "c", "\[MediumSpace]", "\[Gamma]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] c f γ φ c f j μ TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["c", "\[MediumSpace]", "f", "\[MediumSpace]", "\[Gamma]", "\[MediumSpace]", "\[CurlyPhi]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["c", "\[MediumSpace]", "f", "\[MediumSpace]", "j", "\[MediumSpace]", "\[Mu]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] e b ϵ β e b g η TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["e", "\[MediumSpace]", "b", "\[MediumSpace]", "\[Epsilon]", "\[MediumSpace]", "\[Beta]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["e", "\[MediumSpace]", "b", "\[MediumSpace]", "g", "\[MediumSpace]", "\[Eta]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] d f δ φ d f e ϵ TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["d", "\[MediumSpace]", "f", "\[MediumSpace]", "\[Delta]", "\[MediumSpace]", "\[CurlyPhi]"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["d", "\[MediumSpace]", "f", "\[MediumSpace]", "e", "\[MediumSpace]", "\[Epsilon]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] ( - 1 ) a + d - α - δ 2 c + 1 2 e + 1 2 g + 1 2 j + 1 k = max ( "\[LeftBracketingBar]" g - j "\[RightBracketingBar]" , "\[LeftBracketingBar]" a - d "\[RightBracketingBar]" ) min ( g + j , a + d ) κ = - k k g j η - μ g j k κ TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["g", "\[MediumSpace]", "j", "\[MediumSpace]", "\[Eta]", "\[MediumSpace]", RowBox[List["-", "\[Mu]"]]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["g", "\[MediumSpace]", "j", "\[MediumSpace]", "k", "\[MediumSpace]", "\[Kappa]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] d a δ - α d a k κ TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List["d", "\[MediumSpace]", "a", "\[MediumSpace]", "\[Delta]", "\[MediumSpace]", RowBox[List["-", "\[Alpha]"]]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List["d", "\[MediumSpace]", "a", "\[MediumSpace]", "k", "\[MediumSpace]", "\[Kappa]"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] { c b a f e d j g k } φ -1 f f ϵ -1 e e γ -1 c c β -1 b b -1 c e -1 γ -1 ϵ ClebschGordan a α b β c γ ClebschGordan c γ f φ j μ ClebschGordan e ϵ b β g η ClebschGordan d δ f φ e ϵ -1 a d -1 α -1 δ 2 c 1 1 2 2 e 1 1 2 2 g 1 1 2 2 j 1 1 2 κ -1 k k k g -1 j a -1 d g j a d ClebschGordan g η j -1 μ k κ ClebschGordan d δ a -1 α k κ c b a f e d j g k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Beta]_", "=", RowBox[List["-", "b_"]]]], "b_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Gamma]_", "=", RowBox[List["-", "c_"]]]], "c_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Epsilon]_", "=", RowBox[List["-", "e_"]]]], "e_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[CurlyPhi]_", "=", RowBox[List["-", "f_"]]]], "f_"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["c_", "-", "\[Gamma]_", "+", "e_", "-", "\[Epsilon]_"]]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "\[Alpha]_"]], "}"]], ",", RowBox[List["{", RowBox[List["b_", ",", "\[Beta]_"]], "}"]], ",", RowBox[List["{", RowBox[List["c_", ",", "\[Gamma]_"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["c_", ",", "\[Gamma]_"]], "}"]], ",", RowBox[List["{", RowBox[List["f_", ",", "\[CurlyPhi]_"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "\[Mu]_"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["e_", ",", "\[Epsilon]_"]], "}"]], ",", RowBox[List["{", RowBox[List["b_", ",", "\[Beta]_"]], "}"]], ",", RowBox[List["{", RowBox[List["g_", ",", "\[Eta]_"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d_", ",", "\[Delta]_"]], "}"]], ",", RowBox[List["{", RowBox[List["f_", ",", "\[CurlyPhi]_"]], "}"]], ",", RowBox[List["{", RowBox[List["e_", ",", "\[Epsilon]_"]], "}"]]]], "]"]]]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "d", "-", "\[Alpha]", "-", "\[Delta]"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "e"]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "g"]], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["g", "-", "j"]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List["a", "-", "d"]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List["g", "+", "j"]], ",", RowBox[List["a", "+", "d"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["\[Kappa]", "=", RowBox[List["-", "k"]]]], "k"], RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["g", ",", "\[Eta]"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", RowBox[List["-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "\[Kappa]"]], "}"]]]], "]"]], " ", RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List["d", ",", "\[Delta]"]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", RowBox[List["-", "\[Alpha]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "\[Kappa]"]], "}"]]]], "]"]], " ", RowBox[List["NineJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["c", ",", "b", ",", "a"]], "}"]], ",", RowBox[List["{", RowBox[List["f", ",", "e", ",", "d"]], "}"]], ",", RowBox[List["{", RowBox[List["j", ",", "g", ",", "k"]], "}"]]]], "]"]]]]]]]]]]]]]]

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

 2001-12-21