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Title:  Quantization of Chirikov map and quantum KAM theorem 
Author(s):  Shi, KangJie 
Doctoral Committee Chair(s):  Chang, ShauJin 
Department / Program:  Physics 
Discipline:  Physics 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Chirikov map
quantum KAM theorem nonlinear dynamics chaos 
Abstract:  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 walllike 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 nearintegrable 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 quasiperiodically (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 nearintegrable systems the eigenstate wave function in the above band can either have a walllike structure or have a vanishing amplitude. These conclusions agree with the numerical results of the quantum Chirikov map. 
Issue Date:  1987 
Genre:  Dissertation / Thesis 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/25458 
Rights Information:  1987 KangJie Shi 
Date Available in IDEALS:  20110620 
Identifier in Online Catalog:  3476325 
This item appears in the following Collection(s)

Dissertations and Theses  Physics
Dissertations in Physics 
Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois