The Quantum Baker's Map.
We use the Quantum Baker's Map as a case study for investigating the quantum and semiclassical mechanics of systems whose classical analogues undergo chaotic dynamics. A common image for the nature of such a system is that the quantum-semiclassical correspondence allows the quantum mechanics to capture the details of the chaos only up to the time t* when the chaos generates classical structures small on the scale of Planck's constant. After this, the correspondence is expected to be met only statistically, with the essence of the chaos captured by random quantum fluctuations. With the Baker's map, we first perform an exploratory study. A short time semiclassical theory, valid only until t* is derived in the form of a linear wavepacket dynamics. The consequences for the implied heavy eigenfunction scarring and for the trace of the propagator are compared to the quantum results. Unexpectedly, the semiclassical trace remains accurate beyond t*. Some exotic quantum properties are found that cannot be explained by the short time semiclassics. Further study reveals level clustering, large recurrences in the trace of the propagator, and excesses of small eigenvector components. Many of the unusual behaviors have a very strong and peculiar hbar dependence. We next develop a method for computing semiclassically beyond t*. Even when the classical structures are much smaller than Planck's constant, the semiclassical theory (based on these structures!) accurately describes the quantum behavior. Finally, we turn to a study of the breakdown of the semiclassics. Evidence is given that the quantum-semiclassical correspondence may continue out to times of the order needed to resolve individual stationary states. However, the semiclassics are breaking down fast enough that a pure semiclassical theory alone should not be able to describe all the details of the stationary states and spectrum, even as h to 0.
- Pub Date:
- Physics: General; Physics: Atomic; Physics: Molecular