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Re: spherical harmonics

  • To: mathgroup at smc.vnet.net
  • Subject: [mg129097] Re: spherical harmonics
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Sat, 15 Dec 2012 05:48:31 -0500 (EST)
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  • References: <kaem8f$5qe$1@smc.vnet.net>

Am 14.12.2012 08:59, schrieb agdijkstra at gmail.com:
> I'm trying to do some Mathematica calculations with spherical harmonics. However, even testing the simple orthogonality relation does not work.
> code:
> Integrate[
>   SphericalHarmonicY[l, m, theta, phi] SphericalHarmonicY[lp, mp,
>     theta, phi] Sin[theta], {theta, 0, Pi}, {phi, 0, 2 Pi}]
>
> adding the assumptions that l, m, lp and mp are integers does not help.
>

The second factor has to be taken conjugate complex and the m and l 
havee to be Integers.

But that does not help much for m==mp.

$Assumptions =
    In: Inequality[0, LessEqual, \[Theta], LessEqual,
     Pi] && 0 <= \[Phi] <= 2*Pi &&
    {l, L} >= 0 &&
    -l <= m <= l &&
    -L <= M <= L &&
    Element[{l, m, L, M},Integers]

     The complex Hilbert space product is

     <f,g> = int_V  dV ( Conjugate[f] * g )

     Avoiding the problems with Conjugate we take the conjugate by
     making one "m" negative

    In: integrand =
     FullSimplify[
     Sin[\[Theta]]*
     FunctionExpand[SphericalHarmonicY[l, m, \[Theta], \[Phi]]]*
     FunctionExpand[SphericalHarmonicY[L, -M, \[Theta], \[Phi]]]]


Out:(1/(4*Pi))*E^(I*(m - M)*\[Phi])*
   Sqrt[((1 + 2*l)*(1 + 2*L)*(l - m)!*
      (L + M)!)/((l + m)!*(L - M)!)]*
   Hypergeometric2F1Regularized[-l, 1 + l,
    1 - m, Sin[\[Theta]/2]^2]*
   Hypergeometric2F1Regularized[-L, 1 + L,
    1 + M, Sin[\[Theta]/2]^2]*Sin[\[Theta]]*
   Tan[\[Theta]/2]^(-m + M)

   The integral with respect to  \[Phi] is zero if m!=M.

   About the orthogonality in \[Theta] for the rest for m==M,
   namely the families of associated Legendre polynomials  of the same 
  upper index m


In: Reduce[Integrate[integrand /. M -> m,
     {\[Theta], 0, Pi}] == 0]



Out: Integrate[(1/(4*Pi))*
     Sqrt[((1 + 2*l)*(1 + 2*L)*(l - m)!*
        (L + m)!)/((L - m)!*(l + m)!)]*
     Hypergeometric2F1Regularized[-l,
      1 + l, 1 - m, Sin[\[Theta]/2]^2]*
     Hypergeometric2F1Regularized[-L,
      1 + L, 1 + m, Sin[\[Theta]/2]^2]*Sin[\[Theta]],
    {\[Theta], 0, Pi}] == 0

No result.


Since, axial angular momentum m given,
with the effective Hamiltonian on the unit sphere00, with fixed 
m-centrifugal potential,
for all integer eigenvalues  l >= m the Schrödinger
equation is

(-d/du (1 - u^2) d/du + m^2/(1 - u^2) ) P_lm = l (l + 1) P_lm

with u= Cos[\[Theta]], -Sin[\[Theta]] d\[Theta] = du

Theorem: Eigenfunctions corresponding to different eigenvalues
of an symmetric operator with respect to a given metric are orthogonal.


In the general parameter case, Mathematica does not seem to provide
any information
about orthogonality relations in the families of orthogonal
polynomials,
ie Legendre, Chebycheff, Gegenbauer, Hermite, Laguerre and Jacobi in
contradistinction to the eigenfunctions of d^2/dx^2 with different 
boundary conditions at 0 and 2 Pi.

Given the assumptions from above

In: Integrate[Sin[l x] Sin[L x], {x, 0, 2 \[Pi]}]
     Integrate[Sin[(l + 1/2) x] Sin[(L + 1/2) x], {x, 0, 2 \[Pi]}]
     Integrate[LegendreP[l, x] LegendreP[L, x], {x, -1, 1}]

Out: 0
      0
      Integrate[LegendreP[l, x] LegendreP[L, x], {x, -1, 1}]

-- 

Roland Franzius





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