Finite difference Schrodinger equation

[aaaa202]

I am simulating electrons inside a cylindrical well like the one shown on the first figure.
My current work has been on solving the Schrodinger equation numerically for the above potential and then finding corrections to the solution such that it is consistent with Poissons equation.
To do so I need to apply the inverse Laplacian to the electron density as shown in figure 2, which will give me the correction to the potential profile (at least in the first iteration). The problem is however, that applying the inverse Laplacian to the electron density shown gives me a correction as shown on figure 3. As seen it makes sense that this is peaked around the origin, since the electron density is highest at this point. What however doesn't make sense, is that it curves upwards in one direction and doesn't go to zero in a spherically symmetric manner. Everything in my problem has circular symmetry, so I have no idea why it doesn't go to zero with circular symmetry. Does anyone have an idea why this can be? Is it a numerical problem? I doubt so since my mesh size is not very big.

 

[DrClaude]

It's hard to answer without more details of the method you use. It could indeed be due to the propagation of an error. Looking at figure 3, the correction seems to be symmetric along the right axis, but not the left axis. Is your calculation starting from the negative end of the left axis and then proceeds from negative to positive values of the left axis?

 

[aaaa202]

My 2d grid is indexed like the one attached. It starts at one corner and then the value of the wave function at each point on the grid represents an entry in an NxN dimensional vector.

 

[aaaa202]

For a given point the second order derivative is calculated using 4 points in the vicinity as marked by the red cross on the this figure. I guess this causes problems for points on the edge.

 

[aaaa202]

hmm I think I figured it out