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<font face="serif">I have in my notes that I talked briefly about
how to do my problem 3 on HW 3 (the superposition problem), but I
also remember noting that some of you were not present for that
class period.<br>
<br>
Let's say you have wavefunction <img style="vertical-align:
middle" src="cid:part1.02020509.06090909@mercer.edu"
alt="$\psi$"> that represents a superposition of eigenstates (it
is made up of a combination of eigenstates, <img
style="vertical-align: middle"
src="cid:part2.04080308.06000805@mercer.edu"
alt="$\phi_1,\phi_2,\ldots,\phi_n$">.) If you try to measure
the energy of <img style="vertical-align: middle"
src="cid:part1.02020509.06090909@mercer.edu" alt="$\psi$"> what
will actually happen is one of the individual discrete eigenstate
energies, <img style="vertical-align: middle"
src="cid:part4.04090808.02080406@mercer.edu"
alt="$E_1,E_2,\ldots,E_n$"> will be returned. <br>
<br>
In mathematical parlance <img style="vertical-align: middle"
src="cid:part1.02020509.06090909@mercer.edu" alt="$\psi$"> can
be resolved into the basis functions </font><font face="serif"><img
style="vertical-align: middle"
src="cid:part2.04080308.06000805@mercer.edu"
alt="$\phi_1,\phi_2,\ldots,\phi_n$">. As such, the probability
of measuring energy <img style="vertical-align: middle"
src="cid:part7.00010307.03060304@mercer.edu" alt="$E_1$"> can be
determined through integration over all space as will thusly be
demonstrated.<br>
<br>
In this problem the basis functions are particle in a box
wavefunctions, so they have to be integrated from 0 to L. the
probability, for example, of getting an energy from the first
basis function when the energy of <img style="vertical-align:
middle" src="cid:part1.02020509.06090909@mercer.edu"
alt="$\psi$"> was measured would be:<br>
<br>
<img style="vertical-align: middle"
src="cid:part9.07000500.01080902@mercer.edu"
alt="$\left[\int_0^L \phi_1 \psi \right]^2$">.<br>
<br>
Building the first five wavefunction for the 1D PIB should be
pretty straightforward in Mathematica and the integrations should
also be relatively straightforward. You will, however want to set
up your wavefunctions so that they have <img
style="vertical-align: middle"
src="cid:part10.06080306.02080408@mercer.edu" alt="$L$"> as a
parameter and <img style="vertical-align: middle"
src="cid:part11.07000702.02070405@mercer.edu" alt="$x$"> as a
variable. Let me know if you have any questions.<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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