First of all, it is clear that one cannot define chaos in quantum mechanics the same way that it is defined in classical mechanics. The classical definition of chaos relies upon the ideas of sensitive dependence on initial conditions and exponentially diverging trajectories. This type of definition is simply not possible in quantum mechanics. To begin with, the Uncertainty Principle prevents us from talking about initial conditions in quantum mechanics since we cannot know the position and velocity of a particle simulataneously. Furthermore, the same principle prevents us from talking about trajectories in quantum mechanics because a trajectory is nothing more than a complete description of the position and velocity of a particle at all times. With these restrictions, one cannot talk about sensitive dependence on initial conditions or expoenentially diverging trajectories in quantum mechanics. Those ideas simply don't make sense in the context of the quantum theory. Thus, it is impossible for a quantum system to exhibit chaotic behavior if one uses the classical definition of chaos.
The probabilistic nature of quantum mechanics also makes it difficult to talk about chaos in the context of the quantum theory. Chaos is all about unpredictability, but unpredictability is built right into the foundation of quantum mechanics. One can never predict the exact behavior of a quantum system. One can only predict the probability that a quantum particle will behave in a certain way. So we are faced with a difficult choice. We can define chaos as it is defined in classical mechanics, in which case there is no chaos in quantum mechanics. Or we can define chaos in a vague way as "unpredictability", in which case all of quantum mechanics is chaotic. Neither of these choices are very appealing. Fortunately, there are other alternatives.
One approach is to study how classical mechanics arises from quantum mechanics. Since all tiny particles must follow the rules of quantum mechanics, and since all matter is thought to be composed of tiny particles, the rules of classical mechanics that are followed by large objects must somehow arise from the rules of quantum mechanics. There is some very interesting research being done in this area, particularly in regard to the theory of decoherence. It may be that decoherence explains how chaotic classical motion can arise from non-chaotic quantum behavior. However, this does not give us a way to define chaos in quantum mechanics. It simply tells us that we need not be concerned if there turns out to be no chaos in quantum mechanics. We can still have chaos in classical mechanics if that is the case.
Another approach to the problem of quantum chaos is to formulate a probabilistic description of classical mechanics and attempt to find a definition of chaos that works in this new context. Instead of looking at individual trajectories in classical mechanics, one can look at a collection of a very large number of trajectories. One then examines how this collection evolves as time goes by. Collections of trajectories in chaotic classical systems do evolve in a much more complicated way than do collections of trajectories in regular classical systems. It turns out that the probability distributions in a quantum system evolve in very similar ways. However, the Uncertainty Principle prevents the quantum distributions from exactly following the classical collection of trajectories. Some progress has been made in understanding the relationship between this probabalistic description of classical mechanics and quantum mechanics, but so far no one has come up with a good way to define chaos in quantum mechanics.
A third approach to the problem of quantum chaos is to ignore the question of defining chaos in quantum mechanics and instead concentrate on identifying features of a quantum system that correspond to chaos in a classical system. In this approach one does not worry about whether or not the quantum system is "chaotic". Instead, one tries to determine whether or not quantum versions of classically chaotic systems behave differently from quantum versions of classically regular systems. A great deal of progress has been made along these lines in recent years. For example, it has been discovered that the eigenvalues of a chaotic quantum system (the quantum version of a classically chaotic system) have different statistical properties than do the eigenvalues of a regular quantum system (the quantum version of a classically regular system). Another interesting development is that the distribution of eigenvalues in a chaotic quantum system can be determined using information about periodic orbits (trajectories that repeat the same path over and over) of the classical system.
Our research at the Viking Center for Computational Physics centers on this third approach to studying quantum chaos. In particular, we investigate properties of the eigenstates (as opposed to the eigenvalues) of a quantum system that relate to chaos in the classical version of the system. More details about our work can be found in the next section.
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