Quantization of Chirikov map and quantum KAM theorem

KAM theorem is one of the most important theorems in classical nonlinear dynamics and chaos. To extend KAM theorem to the regime of quantum mechanics, we first study the quantum Chirikov map, whose classical counterpart provides a good example of RAM theorem. Under resonance condition 2n11=1/N, we obtain the eigenstates of the evolution operator of this system. We find that the wave functions in the coherent state representation (CSR) are very similar to the classical trajectories. In particular, some of these wave functions have wall-like structure at the locations of classical KAM curves. We also find that a local average is necessary for a Wigner function to approach its classical limit in the phase space. We then study the general problem theoretically. Under similar conditions for establishing the classical KAM theorem, we obtain a quantum extension of RAM theorem. By constructing successive unitary transformations, we can greatly reduce the perturbation part of a near-integrable Hamiltonian system in a region associated with a Diophantine number W0. This reduction is restricted only by the magnitude of n. We can summarize our results as follows: In the CSR of a nearly integrable quantum system, associated with a Diophantine number W0 , there is a band near the corresponding KAM torus of the classical limit of the system. In this band, a Gaussian wave packet moves quasi-periodically (and remain close to the KAM torus) for a long time, with possible diffusion in both the size and the shape of its wave packet. The upper bound of the tunnelling rate out of this band for the wave packet can be made much smaller than any given power of n if the original perturbation is sufficiently small (but independent of n). When n-->0, we reproduce the classical KAM theorem. For most near--integrable systems the eigenstate wave function in the above band can either have a wall-like structure or have a vanishing amplitude. These conclusions agree with the numerical results of the quantum Chirikov map.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Year dc:date

2011

Author dc:creator

• Shi, Kang-Jie

Contributors dc:contributor

• Chang, Shau-Jin

• Chirikov map
• quantum KAM theorem
• nonlinear dynamics
• chaos

Statement dc:rights

• 1987 Kang-Jie Shi

Language dc:language

en

Identifier

3476325

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/25458