Re: Integrating SphericalHarmonicY

*To*: mathgroup at smc.vnet.net*Subject*: [mg73297] Re: Integrating SphericalHarmonicY*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Fri, 9 Feb 2007 23:42:42 -0500 (EST)*Organization*: The University of Western Australia*References*: <eq9c4v$n37$1@smc.vnet.net> <eqh6rm$hqg$1@smc.vnet.net>

In article <eqh6rm$hqg$1 at smc.vnet.net>, wgempel at yahoo.com wrote: > Sure, I had to calculate some expectation values for non-relativistic > hydrogen wavefunctions for a homework assignment. That was fine and I > have already submitted that work. When I entered the formulas for the > wavefunctions, I wanted to check that they were normalized for general > n,l,m (just to double check that I had the right normalization > constants). I was unable to figure out how to do this for the general > case using mathematica. Instead I ended just trying several test case > (e.g. n = 18, l = 11, m = -4) until I felt confident the > normalization was correct. There are other portions of the integral > that fail to simplify, but I was struck by my inability to coax > mathematica into simplifying this basic integral. Such integrals are _not_ trivial for general values of n, l, m. > Mainly, I am just trying to learn Mathematica. I came across this > simplification that I think of as a basic pattern, and I was unable to > figure out the right way to represent it in the system. It seems to > me that Mathematica should be able to recognize simple integrals of > orthogonal polynomials (if the parameters are correctly constrained), > so I assume that I am missing some technique. Since projecting > functions onto orthogonal polynomials is quite common (in physics), I > want to know the correct way to work with these functions in > Mathematica. The "correct" approach is essentially the approach you use in physics (and mathematics) if you are doing the calculation by hand. To compute integrals such as Integrate[LegendreP[m, x] x^n, {x, -1, 1}] (where it is implicit that n and m are non-negative integers), or to show that Integrate[LegendreP[m, x] LegendreP[n, x], {x, -1, 1}] vanishes for n != m, one can use the generating function (1 - 2 x t + t^2)^(1/2) == Sum[t^n LegendreP[n, x], {n, 0, Infinity}] and interchange the order of summation and integration. As an exercise, show that Integrate[(LegendreP[n + 1, x] - LegendreP[n - 1, x])/(1 - x)^(3/2), {x, -1, 1}] == -4 Sqrt[2] for n = 1, 2, 3, ... (see The Mathematica Journal 7 (2)). 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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**Re: Integrating SphericalHarmonicY**

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