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%                         Computational Physics
%                             SPRING 2009
%                              FINAL EXAM
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%                     Instructor: Kevin H. Knuth
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%                  Time-Dependent Schrodinger Equation
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%
%  In this problem you will use the method of finite differences to solve
%  for the time evolution of a Gaussian wavepacket in a quantum potential
%  well.  Follow sections 30.4, 30.5, and 30.6 in Landau and Paez 2004.
%
%  You will describe the wavepacket with a Gaussian function with the
%  following parameters:
%   sigma_o = 0.5;
%   ko = 17*pi
%
%  For the finite difference method define arrays for the real and
%  imaginary parts of the wavefunction.  The spatial dimension should be on
%  the order of 750 elements.  You will also need to store both the past
%  and future times, so these arrays could have dimension 750 x 2.
%
%
%  Solve the problem for the infinite potential well letting time run 
%  something like 5000 steps (start with just a few steps until you are 
%  sure that it works).  You should use dx = 0.02 and dt - 1/2 dx^2 
%       - Make a figure every 100 steps or so (memory permitting) and make 
%         a movie of the evolution of the wavepacket.
%       - Make a movie of the probability density as a function of time
%       - Make a 3D plot of probability versus time
%       - Check how well probability is conserved over time
%
%   Repeat the above for the Harmonic Oscillator potential well V = 1/2 x^2
%   and take the initial momentum to be ko = 3*pi and the time and space
%   steps of dt = 0.02 and dt = 1/4 dx^2.
%       Repeat the analysis above for the Harmonic Oscillator
%
%   Why does the wavefunction break up for the square well, but not the
%   Harmonic Oscillator?
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% Problem by: Kevin H. Knuth
% Created on: 4 May 2009
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