Quantum Chaos

It is well-known that starting from a quantum-mechanical description of some phenomena one can arrive at a classical description of the same phenomena by letting hbar go to zero. This limit is very satisifying as it provides a bridge between two very different descriptions. A very important difference between the two is whereas the Schroedinger equation, for example, many classical counterparts are nonlinear. It has become clear that it is this classical nonlinearity which leads to chaos and that linear systems do not exhibit chaos. Because one strongly believes the the transition from classical to quantum is obtained via a limiting process, one suspects that one should find renmants of chaos in the corresponding quantum mechanical system. Quantum chaos is basically the search for such remnants.

Work by Berry and others has shown that it is interesting to examine the spacing between energy levels in quantum mechanical systems. In Classical and quantum analysis for chaos in the Discrete Self-Trapping equation we examine the three degree of freedom discrete self-trapping model. The paper shows similar agreement in the level spacings for much of the parameter range studied, but there is a small N limit a significant deviation is discovered. The origins of this deviations remain unknown and could form the topic of further study.

A valuable tool in this study is the existance of systems for which one can exactly solve the quantum mechanical problem of determining the energy-level spectrum, thus allowing for a calculation of the spacings. As there are not many such systems, any addition is welcom. In my unpublished work on the two degree-of-freedom discrete self-trapping equation , I carry out the nontrivial task of explicitly calculating the action angle variables for two DOF DST. The work remains unpublished as while I have finished the quantum mechanical calculations, I have not carried out the numerical diagonalisation of the associated matrices so that the classical/quantum comparison can be made.