[Date Index]
[Thread Index]
[Author Index]
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
Prev by Date:
**FindRoot Precision Problem**
Next by Date:
**Re: record intermediate steps**
Previous by thread:
**Re: Integrating SphericalHarmonicY**
Next by thread:
**dot minus operator**
| |