<html>
<head>
<meta content="text/html; charset=ISO-8859-1"
http-equiv="Content-Type">
</head>
<body bgcolor="#FFFFFF" text="#000000">
<div class="moz-cite-prefix">You use the <i>concept</i> 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, <img
style="vertical-align: middle"
src="cid:part1.03000200.08050608@mercer.edu" alt="$\theta
=0\rightarrow \pi$">, <img style="vertical-align: middle"
src="cid:part2.02070206.04080005@mercer.edu"
alt="$\phi=0\rightarrow 2\pi$">). Because of this fact, even if
you integrated over <img style="vertical-align: middle"
src="cid:part3.06050201.02090908@mercer.edu" alt="$R$">, <img
style="vertical-align: middle"
src="cid:part4.08010702.02030901@mercer.edu" alt="$\theta$">,
and <img style="vertical-align: middle"
src="cid:part5.03030007.09030909@mercer.edu" alt="$\phi$">
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...
<br>
<br>
<img style="vertical-align: middle"
src="cid:part6.07070509.02030701@mercer.edu" alt="$\int_0^\infty
r^2 dr$"><br>
<br>
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.<br>
<br>
The lure of <i>Methmatica</i> is great -- beware of it's
dangers. Some of these integrals are best done by hand. <br>
<br>
On 09/14/13 22:08, Connor Holt wrote:<br>
</div>
<blockquote
cite="mid:C40B2F181831EF44A88CD73525827803026514AC75@MERCERMAIL.MercerU.local"
type="cite">
<pre wrap="">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</pre>
</blockquote>
<br>
<br>
<pre class="moz-signature" cols="72">--
Andrew J. Pounds, Ph.D. (<a class="moz-txt-link-abbreviated" href="mailto:pounds_aj@mercer.edu">pounds_aj@mercer.edu</a>)
Professor of Chemistry and Computer Science
Mercer University, Macon, GA 31207 (478) 301-5627
<a class="moz-txt-link-freetext" href="http://faculty.mercer.edu/pounds_aj">http://faculty.mercer.edu/pounds_aj</a>
</pre>
</body>
</html>