The use of better estimates of the second derivative in the Schrodinger equation with a parabolic potential showed that the iterated solution had essentially perfect periodicity at the classic particle period. So I thought I'd try the improved estimate on the swaged potential which usually does not have the perfect period. This program runs the iterative algorithm for 2 periods of wave packet and particle oscillation. Unfortunately the improved estimate of the second derivative does not result in perfect periodicity of the wave packet for the swaged potential. The variables available are initial wave speed, maximum potential, swage width, packet width, and time increment per step. All of these variables are limited to values which generally do not cause the algorithm to become unstable. Some interesting printouts Vs time are packet kinetic energy, potential energy, and packet*packet integral which should and does remain quite constant. After the run is complete the value, xTAMin/xTA, for the closest packet centroid distance to the classic turnaround is printed. To demonstrate the difference between classic and wave packet motion, the packet centroid position- particle position is plotted (blue). Another plot (green)shows how the width of the packet changes as time progresses. For periodic behavior one expects the width to return to its starting value after one period and that is definitely not the case. The following link is to a document that shows the algebra of computing the second derivative used in the Schrodinger equation.
Improved Second Derivative Approximation For 3, 5, 7 and 9 points the coefficients are:
cA3=[1,-2,1]/dx^2
cA5=[-1/12,4/3,-5/2,4/3,-1/12]/dx^2
cA7=[1/90,-3/20,3/2,-49/18,3/2,-3/20,1/90]/dx^2
cA9=[-(1/560), 8/315, -(1/5), 8/5, -(205/72), 8/5, -(1/5), 8/315, -(1/560)]/dx^2