{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/25458"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/25458","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Quantization of Chirikov map and quantum KAM theorem","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 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.","abstract_html":"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.","abstract_has_math":false,"creators":["Shi, Kang-Jie"],"institution":null,"degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Physics","degree_department":null,"school":null,"contributors":["Chang, Shau-Jin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-06-20T15:55:09Z","date_published":"2011-06-20T15:55:09Z","updated_at":"2026-07-22T22:25:24Z","subjects":["Chirikov map","quantum KAM theorem","nonlinear dynamics","chaos"],"languages":["en"],"rights":["1987 Kang-Jie Shi"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["3476325"],"render_values":[{"text":"3476325","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/25458","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Chang, Shau-Jin"]},{"key":"dc:creator","label":"Author","values":["Shi, Kang-Jie"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-06-20T15:55:09Z","10000-01-01","1987"]},{"key":"dc:type","label":"Dc Type","values":["text","Dissertation / Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Physics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Chirikov map","quantum KAM theorem","nonlinear dynamics","chaos"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["1987 Kang-Jie Shi"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["3476325","http://hdl.handle.net/2142/25458"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Submitted by Carolyn Mead (cmead2@illinois.edu) on 2011-06-20T15:55:09Z No. of bitstreams: 1 1987_shi.pdf: 3881637 bytes, checksum: 0652f091822b4e8290c47de5a54ba2e9 (MD5)","Made available in DSpace on 2011-06-20T15:55:09Z (GMT). No. of bitstreams: 1 1987_shi.pdf: 3881637 bytes, checksum: 0652f091822b4e8290c47de5a54ba2e9 (MD5) Previous issue date: 1987","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Carolyn Mead (cmead2@illinois.edu) on 2011-06-20T15:55:09Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:15:21-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: Thesis","Thesis","U of I Only"]},{"key":"dc:title","label":"Title","values":["Quantization of Chirikov map and quantum KAM theorem"]}]}],"canonical_facts":{"dc:contributor":["Chang, Shau-Jin"],"dc:creator":["Shi, Kang-Jie"],"dc:date":["2011-06-20T15:55:09Z","10000-01-01","1987"],"dc:description":["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.","Submitted by Carolyn Mead (cmead2@illinois.edu) on 2011-06-20T15:55:09Z No. of bitstreams: 1 1987_shi.pdf: 3881637 bytes, checksum: 0652f091822b4e8290c47de5a54ba2e9 (MD5)","Made available in DSpace on 2011-06-20T15:55:09Z (GMT). No. of bitstreams: 1 1987_shi.pdf: 3881637 bytes, checksum: 0652f091822b4e8290c47de5a54ba2e9 (MD5) Previous issue date: 1987","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Carolyn Mead (cmead2@illinois.edu) on 2011-06-20T15:55:09Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:15:21-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: Thesis","Thesis","U of I Only"],"dc:identifier":["3476325","http://hdl.handle.net/2142/25458"],"dc:language":["en"],"dc:rights":["1987 Kang-Jie Shi"],"dc:subject":["Chirikov map","quantum KAM theorem","nonlinear dynamics","chaos"],"dc:title":["Quantization of Chirikov map and quantum KAM theorem"],"dc:type":["text","Dissertation / Thesis"],"thesis:degree_discipline":["Physics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-22T22:25:24Z"}