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<font face="serif">So some of you have asked about problem 2h on HW
set 4. Specifically when some of you are integrating to get the
expectation value of <img style="vertical-align: middle"
src="cid:part1.06060906.02080605@mercer.edu" alt="$r$"> you are
getting a value that is too small. Here is what is most likely
going on. Since there is no angular dependence on the expectation
value of <img style="vertical-align: middle"
src="cid:part1.06060906.02080605@mercer.edu" alt="$r$"> for s
orbitals, you can pull out those integrations<br>
<br>
<img style="vertical-align: middle"
src="cid:part3.08060506.03010303@mercer.edu" alt="$\left<
\hat{r} \right> = \int_0^{\pi} \int_0^{ 2 \pi} \sin \theta
d\phi d\theta \int_0^{\infty} \psi^*_{nlm}r \psi_{nlm} r^2 dr$"><br>
<br>
<br>
and the term out front is just a constant, <img
style="vertical-align: middle"
src="cid:part4.02090201.02060905@mercer.edu" alt="$4 \pi$">,<br>
<br>
</font><br>
<font face="serif"><img style="vertical-align: middle"
src="cid:part5.06020508.01000809@mercer.edu" alt="$\left<
\hat{r} \right> = 4 \pi \int_0^{\infty} \psi^*_{nlm}r
\psi_{nlm} r^2 dr$"><br>
<br>
<br>
Once you include that it should bring your expectation values in
line with what one expects based on your graphs. Understand, ever
though there is no angular dependence we are still required to do
the integration over the 3D volume elements because we made the
change to spherical polar coordinates. This is an error that
trips up many. I thought I had mentioned it in class, but I was
on so many different cold medicines this past week that I really
don't remember.<br>
<br>
Let me know if you have questions.<br>
<br>
<br>
<br>
</font>
<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>
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