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
The parabolic potential is unique in that a Schrodinger wave packet period is the same as the period of a classic particle in the same potential. Therefore the accuracy of a given numerical TDSE algorithm can easily be checked in this potential. For the 3 point second derivative one sees that the wave function does not return to its initial position after one classic period in the parabolic potential. For 5 or more points second derivative the wave function seems to return to its initial position. This program runs the algorithm for 2 periods of wave packet and particle oscillation. The variables available are initial wave speed, parabola potential/ V=kP/2*x*x coefficient kP, 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.