[CHM 331] Superposition Problem

[Andrew J. Pounds]

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.

Let's say you have wavefunction $\psi$ that represents a superposition 
of eigenstates (it is made up of a combination of eigenstates, 
$\phi_1,\phi_2,\ldots,\phi_n$.)  If you try to measure the energy of 
$\psi$ what will actually happen is one of the individual discrete 
eigenstate energies, $E_1,E_2,\ldots,E_n$ will be returned.

In mathematical parlance $\psi$ can be resolved into the basis functions 
$\phi_1,\phi_2,\ldots,\phi_n$.  As such, the probability of measuring 
energy $E_1$ can be determined through integration over all space as 
will thusly be demonstrated.

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 $\psi$ was measured would be:

$\left[\int_0^L \phi_1 \psi \right]^2$.

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 $L$ as a parameter and $x$ as a 
variable.  Let me know if you have any 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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