Re: Product of Spherical Harmonics

*To*: mathgroup at smc.vnet.net*Subject*: [mg4526] Re: Product of Spherical Harmonics*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Wed, 7 Aug 1996 04:17:35 -0400*Organization*: University of Western Australia*Sender*: owner-wri-mathgroup at wolfram.com

Vandemoortele CC Group R&D Center wrote: > it may seem silly, but I can't find the expansion factors for > decomposing a > product of spherical harmonics into a sum of spherical harmonics: > > Y(a,b) Y(c,d) = Sum[ coefficient[a,b,c,d,l,m=-b-d] Y(l,m=-b-d) > ,{l,lower,upper} ] > > I hope to do it faster and smarter with the ClebschGordan /or/ > ThreeJSymbols. That is however where I got stuck. They seem to work >'the other way round' somehow. Actually they work 'both ways round'. Because of the orthonormality of the (complex) spherical harmonics, you can compute the integrals in terms of 3-j symbols or linearize a product of spherical harmonics with 3-j coefficients as the expansion coefficients. The formulae you need is given in a number of places; perhaps the most popular reference is Edmonds, A. R.: "Angular momentum in quantum mechanics", Princeton University Press, 1974 You want Edmonds (4.6.5): [a b c ] * Y(a,ma) Y(b,mb) = Sum[C[ ] Y (c,mc),{c,|a-b|,a+b}] [ma mb mc] where * denotes the complex conjugate, ma+mb+mc == 0, and [a b c ] C[ ] = Sqrt[(2a+1)(2b+1)(2c+1)/(4Pi)] * [ma mb mc] ThreeJSymbol[{a,ma},{b,mb},{c,mc}] * ThreeJSymbol[{a,0},{b,0},{c,0}] * m Note that Y (c,mc) = (-1) Y(c,-mc) so you can get the expansion you require. For more information on the Mathematica Clebsch-Gordan Coefficient package, you should note that it is included with the Mathematica distribution (in Packages`StartUp`) Cheers, Paul _________________________________________________________________ Paul Abbott Department of Physics Phone: +61-9-380-2734 The University of Western Australia Fax: +61-9-380-1014 Nedlands WA 6907 paul at physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/Paul Black holes are where God divided by zero _________________________________________________________________ ==== [MESSAGE SEPARATOR] ====