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
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