So far I have talked only about quantum systems in which an S electron is trapped or spatially confined. Chaotic effects are also present in atomic systems where an electron can roam freely, as it does when it is scattered from the atoms in a molecule. Here energy is no longer quantized, and the electron can take on any value, but the effectiveness of the scattering depends on the energy.
Chaos shows up in quantum scattering as variations in the amount of time the electron is temporarily caught inside the molecule during the scattering process. For simplicity the problem can be examined in two dimensions. To the electron, a molecule consisting of four atoms looks like a small maze. When the electron approaches one of the atoms, it has two choices: it can turn left or right. Each possible trajectory of the electron through the molecule can be recorded as a series of left and right turns around the atom until the particle finally emerges. All of the trajectories are unstable: even a minute change in the energy or the initial direction of the approach will cause a large change in the direction in which the electron eventually leaves molecule.
The chaos in the scattering process comes from the fact that the number of trajectories increases rapidly with path length. Only an interpretation From the quantum mechanical point of view gives reasonable results; a purely classical calculation yields nonsensical results. In quantum mechanics each classical trajectory is used to define a little wavelet that finds its way through the molecule. The quantum mechanical result follows from simply adding up all such wavelets.
Recently I have done a calculation of the scattering process for a special case in which the sum of the wavelets is exact An electron of known momentum hits a and emerges with the same momentum. The arrival time for the electron to reach a fixed monitoring station varies as a function of the momentum and the way in which it varies is so fascinating about this problem. The arrival time fluctuates over small changes in the momentum but over large changes a chaotic imprint emerges which never settles down to any simple pattern.
A particularly tantalizing aspect of the chaotic scattering process is that it may connect the mysteries of quantum chaos with the mysteries of number theory. The calculation of the time delay leads straight into what is probably the most enigmatic object in mathematics, Riemann's zeta function. Actually it was first employed by Leonhard Euler in the middle of the 18th century to show the existence of an infinite number of prime numbers (integers that cannot be divided by any smaller integer other than one). About a century later Bernhard Riemann, one of the founders of modem mathematics, employed the function to delve into the distribution of the primes. In his only paper on the subject, he called the function by the Greek letter zeta.
The zeta function is a function of two variables, x and y which exist in the complex plane). To understand the distribution of prime numbers, Riemann needed to know when the zeta function has the value of zero. Without giving a valid argument, he stated that it is zero only when x is set equal to 1/2. Vast calculations have shown that he was right without exception for the first billion zeros, but no mathematician has come even close to providing a proof. If Riemann's conjecture is correct, all kinds of interesting properties of prime numbers could be proved.
The values of y for which the zeta function is zero form a set of numbers that is much like the spectrum of energies of an atom. Just as one can study the distribution of energy levels in the spectrum so can one study the distribution of zeros for the zeta function. Here the prime numbers play the same role as the classical closed orbits of the hydrogen atom in a magnetic field: the primes indicate some of the hidden correlations among the zeros of the zeta function.