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MathGroup Archive 1996

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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
_________________________________________________________________

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