[CHM 331] txtbook #60

[Andrew J. Pounds]

You use the /concept/ of triple integration in spherical polar 
coordinates.  The spherical harmonics, by themselves, form an 
orthonormal set over the unit solid defined in spherical polar 
coordinates (that is, $\theta =0\rightarrow \pi$, $\phi=0\rightarrow 
2\pi$).  Because of this fact, even if you integrated over $R$, 
$\theta$, and $\phi$ coordinates to perform the orthogonality tests you 
would still get zero because of the orthogonality properties of the 
spherical harmonics alone.  Blindly applying triple integration in the 
case of normalization will, however, not work because the integral...

$\int_0^\infty r^2 dr$

will obviously never converge.   Since there is no R coordinate (the 
spherical harmonics are only over angular terms) then you are, in this 
case, safe dropping the R term from the triple integration as you 
mention.   Don't worry, we will be including the R term in the 
integration this week.  If this is not covered to your satisfaction in 
the Alberty and Silbey text, then I recommend that you also take a look 
at the Steiner text.

The lure of /Methmatica/ is great -- beware of it's dangers.  Some of 
these integrals are best done by hand.

On 09/14/13 22:08, Connor Holt wrote:
> Dr. Pounds,
>
> In class you said that we have to use triple integration to verify normality. This works for orthogonality, but when doing triple integrals in mathematica to verify normality I get divergence issues. It will only converge to 1 if I take out the r^2 portion of the Jacobian and turn it into a double integral. Is this how it should work?
>
> Thanks
>
> Connor


-- 
Andrew J. Pounds, Ph.D.  (pounds_aj at mercer.edu)
Professor of Chemistry and Computer Science
Mercer University,  Macon, GA 31207   (478) 301-5627
http://faculty.mercer.edu/pounds_aj

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