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Topic: Re: Product of Spherical Harmonics
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Paul Abbott

Posts: 1,437
Registered: 12/7/04
Re: Product of Spherical Harmonics
Posted: Jul 30, 1996 2:56 AM
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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@physics.uwa.edu.au
AUSTRALIA http://www.pd.uwa.edu.au/Paul

Black holes are where God divided by zero
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