Spherical harmonic function: Summation (formula 07.37.23.0008)

SphericalHarmonicY

$Mathematica Notation$

Hypergeometric Functions

SphericalHarmonicY[lambda,mu,theta,phi]

Summation

Finite summation

Involving Clebsch-Gordan functions

http://functions.wolfram.com/07.37.23.0008.01

Input Form

Sum[ClebschGordan[{Subscript[n, 1], k}, {Subscript[n, 2], M - k}, {L, M}] SphericalHarmonicY[Subscript[n, 1], k, \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 2], M - k, \[CurlyTheta], \[CurlyPhi]], {k, Max[M - Subscript[n, 2], -Subscript[n, 1]], Min[M + Subscript[n, 2], Subscript[n, 1]]}] == Sqrt[((2 Subscript[n, 1] + 1) (2 Subscript[n, 2] + 1))/(4 Pi (2 L + 1))] ClebschGordan[{Subscript[n, 1], 0}, {Subscript[n, 2], 0}, {L, 0}] SphericalHarmonicY[L, M, \[CurlyTheta], \[CurlyPhi]] /; Element[Subscript[n, 1], Integers] && Subscript[n, 1] >= 0 && Element[Subscript[n, 2], Integers] && Subscript[n, 2] >= 0 && Element[L, Integers] && Element[M, Integers] && Abs[Subscript[n, 1] - Subscript[n, 2]] <= L <= Subscript[n, 1] + Subscript[n, 2] && -L <= M <= L

Standard Form

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MathML Form

<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> <mi> k </mi> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> M </mi> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> M </mi> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mi> k </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> M </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mi> L </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mi> M </mi> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["n", "1"], "\[MediumSpace]", SubscriptBox["n", "2"], "\[MediumSpace]", "k", "\[MediumSpace]", "M"]], "-", "k"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["n", "1"], "\[MediumSpace]", SubscriptBox["n", "2"], "\[MediumSpace]", "L", "\[MediumSpace]", "M"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mi> k </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mrow> <mi> M </mi> <mo> - </mo> <mi> k </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> L </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mi> L </mi> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "\[MediumSpace]", SubscriptBox["n", "2"], "\[MediumSpace]", "0", "\[MediumSpace]", "0"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["n", "1"], "\[MediumSpace]", SubscriptBox["n", "2"], "\[MediumSpace]", "L", "\[MediumSpace]", "0"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> L </mi> <mi> M </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> L </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> M </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ≤ </mo> <mi> L </mi> <mo> ≤ </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mi> L </mi> </mrow> <mo> ≤ </mo> <mi> M </mi> <mo> ≤ </mo> <mi> L </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <max /> <apply> <plus /> <ci> M </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <min /> <apply> <plus /> <ci> M </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> ClebschGordan </ci> <apply> <ci> AngleBracket </ci> <apply> <ci> VerticalSeparator </ci> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> k </ci> <ci> M </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> L </ci> <ci> M </ci> </apply> </apply> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> M </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> L </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> ClebschGordan </ci> <list> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </list> <list> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 0 </cn> </list> <list> <ci> L </ci> <cn type='integer'> 0 </cn> </list> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> L </ci> <ci> M </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> ℕ </ci> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> ℕ </ci> </apply> <apply> <in /> <ci> L </ci> <ci> ℤ </ci> </apply> <apply> <in /> <ci> M </ci> <integers /> </apply> <apply> <leq /> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> L </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> L </ci> </apply> <ci> M </ci> <ci> L </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

SphericalHarmonicY[n,m,theta,phi]