For the harmonic oscillator the Gaussian wave function has a lot of interesting properties, but for the two-dimensional Coulomb potential it is not very illuminative: Starting the applet with its default values shows a very complicated and seemingly chaotic behaviour.

Looking more closely one finds features that reveal connections between the classical and quantum description of the system:

- While the wave function oscillates wildly in the central area, it changes much more slowly at greater distance from the center. This is a reminiscence of the decrease of the orbital velocities with distance that is due to Kepler's third law. From a quantum point of view one would instead stress that the inner regions correspond mainly to eigenstates with low energy quantum number n, i.e. high (absolute) value of the binding energy, which oscillate faster.
- Observing the behaviour for a longer time, one finds a kind of oscillating behaviour between states that are concentrated on the left side and mirror images on the right side. Measure the period of such oscillations and compare with the period of a corresponding classical elongated ellipse (beware: this is not trivial!).

Another interesting phenomenon is the decrease of the total probability P with time, which reaches a steady value < 1 after some time. Study this decrease for different values of s and v0. Can you find a qualitative explanation of your findings?