After few months I came back to my grant project in quantum simulations. For now the main goal is to simulate the following equation in three dimensions:

The biggest issue is the singularity in 1/R. There is something more – I have never simulated such complicated equation, so I started from something much much easier – the equations for free quantum particle in space.

Â After this (having very simple evolution) I checked if version with separated variables ( psi = f + i g) gives the same solution:

Now I’m trying again to obtain same results with another change of coordinate system: psi = R exp(i Theta). There are some problems, unfortunately…

Because of desperation, I changed the system into one dimensional space, where I have bigger control. Still, the last pair of equations are difficult to simulate. I suspect that it can be partially of troublesome behaviour of theta variable – it is something I call periodical variable, which is given with the error of 2 pi. That generates wrong behaviour in getting the derivative – sometimes small change of argument generates great change of function value. I have a method to get rid of it, in changing a definition of theta derivative:

This way I can remove a discontinuity in theta.