Rydberg wave packets

It has only been within the past decade that it has been possible to make and probe Rydberg wave packets on a fairly routine basis. I have made a movie of l=0 electron's radial wave packets in H and in Li centered at the same energy near the n=40 state of H. A large number of radial eigenstates participate in the packet. The initial conditions on the packet are such that the packets are localized to small r at t=0. The packets are constructed so that initially the electron wave packets of H and Li are as similar as possible. It is only near the end of the movie that strong differences between the two electronic functions can be seen.

The bottom box plots the electron's radial probability distribution in Li and the upper box is in H. The arrow on the top is a clock that gives the time in units of the classical period of motion; at t=0 the arrow points up. The classical period at the energy of an n=40 state is 9.7 picoseconds. Every time the arrow points up the time is a multiple of the period whereas when it points down the time is a "halfinteger" period. Since the electron starts near the nucleus you should find that every time the arrow is up the electron probability is near the nucleus whereas when the arrow points down the probability is at large distances (as in the image above).
There are a number of interesting features that deserve comment. One of the things to notice is that the relatively high speed of the electron in the radial range 0-1500 atomic units compared to the range 1500-3000 means that whenever the probability enters the small r region the height of the packet decreases and the radial width increases. Also, whenever there is interference between inward and outward moving pieces of the packet, the interference pattern has a much smaller wave length in the small r region than in the large r region. Also, notice that the two wave packets disperse very quickly, after only 2-3 radial periods the wave packets cover almost the whole classically allowed region. Dispersion is a wave phenomenon that arises when the group velocity of a wave changes with the frequency of the wave. The dispersion can also be related to properties of the classical motion: if the classical period is energy dependent then the wave packet will disperse if you excite more than two eigenstates. For the Coulomb potential, the period of motion increases with the energy (by analogy remember that the year for Mercury is shorter than that for Earth is shorter than that for Jupiter is shorter than that for Pluto). Thus, these radial waves are dispersing by having their higher energy components move to the back of the packet.
One of the final interesting features is the similarity between the H and Li results. This might seem somewhat surprising when you realize that the core electrons in Li cause the l=0 energy levels in Li to shift so much they are nearly halfway between the energy levels in H; thus making them almost as dissimilar as possible. The reason for the similarity is that to lowest order the energy levels are all shifted by the same amount. It is only the change in this shift with the number of nodes that will be apparent in the wave packet. Thus it is necessary to have the packets evolve to somewhat long times before the differences become manifested.