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MathGroup Archive 2007

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Re: Integrating SphericalHarmonicY

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73240] Re: Integrating SphericalHarmonicY
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 8 Feb 2007 03:43:58 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <eq9c4v$n37$1@smc.vnet.net>

In article <eq9c4v$n37$1 at smc.vnet.net>, wgempel at yahoo.com wrote:

> I would like
> 
> Integrate[
>     Conjugate[SphericalHarmonicY[l,m,theta,phi]]
>     SphericalHarmonicY[l,m,theta,phi] Sin[theta],
>     {theta, 0, Pi}, {phi, 0, 2 Pi}]
> 
> to evaluate to 1 (without having to force it through a rule every
> time).

You need to "help" Conjugate (for l and m integral)

  Conjugate[SphericalHarmonicY[l,m,theta,phi]] :>
    SphericalHarmonicY[l,m,theta,-phi]

or 

  Conjugate[SphericalHarmonicY[l,m,theta,phi]] :>
    (-1)^m SphericalHarmonicY[l,-m,theta,phi]


 Table[{l, m, (-1)^ m Integrate[
    SphericalHarmonicY[l,-m,theta,phi]
    SphericalHarmonicY[l,m,theta,phi] Sin[theta],
    {theta, 0, Pi}, {phi, 0, 2 Pi}]}, {l,0,4},{m,-l,l}]

More generally, you can compute integrals of triple-products of 
SphericalHarmonicY functions via ThreeJSymbol:

 Table[{l, m, Sqrt[2l + 1] (-1)^(l - m)*
    ThreeJSymbol[{l, m}, {l, -m}, {0, 0}]}, {l,0,4},{m,-l,l}]

See also, 

  http://physics.uwa.edu.au/pub/Computational/CP2/2.Schroedinger.nb

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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