the potential is rotationally invariant and so is the initial wave function

thus the symmetry of the initial wave function will be preserved in time

Instead of a 2D equation, i now only need to integrate a 1d equation about the radial part of the wave function.

However, in doing so i encounter some difficulties. There are terms like 1/r \partial_r in the equation.

near the origin, this term will diverge!

I guess this is a well-solved problem. Could anyone give me some help?