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Re: Product of Spherical Harmonics
Posted:
Jul 30, 1996 2:56 AM


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=bd] Y(l,m=bd) > ,{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 3j symbols or linearize a product of spherical harmonics with 3j 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,ab,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 ClebschGordan Coefficient package, you should note that it is included with the Mathematica distribution (in Packages`StartUp`)
Cheers, Paul
_________________________________________________________________ Paul Abbott Department of Physics Phone: +6193802734 The University of Western Australia Fax: +6193801014 Nedlands WA 6907 paul@physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/Paul
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