Incompleteness of Eigenfunctions of the Two-Dimensional Coulomb Potential

This simulation shows the superposition of eigenfunctions of the two-dimensional Coulomb potential with n = 1, 2, ... 10, all with m = 0. To better study the central area, the grid has a higher resolution than usual (leading to a slower computation of time evolution) and allows to zoom in by changing the central charge.

The default coefficients are chosen to approximate a Gaussian with v0 = 0 and width s = 1.2e-9 m. Start the simulation and compare it carefully with the applet Gaussian wave packet in a two-dimensional Coulomb potential. Small differences are partially due to the higher accuracy.

Another reason is the difference in the start distribution: The superposition doesn't give an exact Gaussian, but shows a dip with an additional peak at the center that can be pronounced by using smaller Z values (e.g. Z = 0.5). This difference is not a result of inaccurate coefficients or missing values for larger values of n, but of the incompleteness of the eigenfunctions. For higher energies (> 0 actually) the Coulomb potential has unbound states - corresponding to the classical hyperbolic orbits -, which can not be approximated using the standard eigenfunctions. The initial Gaussian wave function contains a certain fraction of unbound states which is missing here.