[CHM 331] Problem 2h

[Andrew J. Pounds]

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 $r$ 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 $r$ for s orbitals, you can pull out those integrations

$\left< \hat{r} \right> = \int_0^{\pi} \int_0^{ 2 \pi} \sin \theta d\phi 
d\theta \int_0^{\infty} \psi^*_{nlm}r \psi_{nlm} r^2 dr$


and the term out front is just a constant, $4 \pi$,


$\left< \hat{r} \right> = 4 \pi \int_0^{\infty} \psi^*_{nlm}r \psi_{nlm} 
r^2 dr$


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.

Let me know if you have questions.



-- 
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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