Quantum analysis of states near a separatrix

Abstract

Quantum modifications of classical chaos have been of great interest in recent years. This interest ls due in part to a basic contradiction. In certain classical systems the existence of chaos, as defined, for example, by the lack of existence of isolating integrals, can be proven. In contrast, such chaos does not exist in bounded quantum systems, in which quastpertodictty is easily shown. This may not mean that quantum systems are not chaotic, but only that the correct definition of quantum chaos is not yet known. Indeed, the search for the correct definition of quantum chaos is the thrust of much of recent research. As examples we mention the work on the relationship of the sensitivity t,2 and distribution of eigenvalues to nonseparabiltty 3-5. Along these lines, the idea ts that a quantum analysis of classical ly chaotic systems should reveal the nature of quantum chaos. It is furthermore expected that such a connection wil l most likely be revealed tn the study of states of large quantum number, i.e., In the semiclassical limit. The semlclasslcal limit by itself has also been an area of much recent study. Examples of such work include that of Berry concerning the relation of the quantum adiabatic phase 6 to the previously known classical adiabatic phase 7 and the consequence of tunneling 8 on the quantum adiabatic theorem 9,10. We note also the burgeoning literature on wavepacket evolution and coherent states I !-!6. Finally, we mention the use of adiabatic switching coupled with classical calculations to obtain quantum spectra 17, i 8. In classical systems, chaos f irst appears near separatrtces of integrable systems as they are perturbed. In part this is due to the exponential separation of ne~'-separatrtx orbits. Given the previous introduction, we are, therefore, motivated to study the quantum mechanics of states near a separatrix in the semiclassical limit. Our result is that quantum mechanics drastically modifies the system even in the limit of very large quantum number, N ~10 7. The reason is that the classical excitation frequency, the orbit frequency, vanishes for orbits with energy E equal to the separatrtx energy E x. In contrast, the quantum excitation frequency, the frequency separation