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Semiclassics, adiabatic decoupling and perturbed periodic structures

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129 Seiten
Veröffentlichung
2009-02-12
Authors
Sträng, Eric
Dissertation


Faculties
Fakultät für Naturwissenschaften
Abstract
This thesis deals with the consequences of periodic structures in quantum mechanics in different semiclassical regimes. The first chapter introduces the Floquet theory. It is applied on the Hill equation. As a special case, the solutions to the Mathieu equation are studied exemplifying the properties of the spectrum of periodic Schrödinger operators. Special attention is brought to the properties of the spectrum and the corresponding Floquet exponent of solutions. We present new results concerning the relation between the eigenvalues and the Floquet exponent. We introduce the Weyl quantization in the second chapter. In particular, we introduce semiclassical propagation results which, up to some known errors, describes the propagation of an initial Gaußian under quantum evolution (Combescure, Robert 1997). This approximation breaks down at what is commonly known as the Ehrenfest time. We use these results to investigate the localization properties of an initial Gaußian. We can show localization of the state up to the Ehrenfest time under the assumption that the flow differential of the corresponding classical motion admits Floquet solutions with purely imaginary Floquet exponents. In this case, we also show the existence of what we call classical revivals. The last part is a study of the spectral properties of perturbed periodic Schrödinger operators. This is done in the limit of weak perturbation epsilon . As the structure of the band spectrum of the unperturbed operator remains unchanged under small enough perturbations, the effective dynamics can be described adiabatically under certain prerequisites. We consider the construction of WKB ansätze (Littlejohn, Flynn, 1991, Emmrich, Weinstein 1996). Our results include Bohr-Sommerfeld quantization conditions modulo O(epsilon²). This is made possible by including geometric terms. The Wannier-Stark system is studied as an example.
Date created
2008
Subject headings
[GND]: Wannier-Funktion
[LCSH]: Floquet theory | Mathematical physics | Solid state physics
[Free subject headings]: Berry phase | Geometric phase | Semiclassics | Wannier-Stark resonance ladder
[DDC subject group]: DDC 530 / Physics
License
Standard (Fassung vom 01.10.2008)
https://oparu.uni-ulm.de/xmlui/license_v2

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DOI & citation

Please use this identifier to cite or link to this item: http://dx.doi.org/10.18725/OPARU-1195

Sträng, Eric (2009): Semiclassics, adiabatic decoupling and perturbed periodic structures. Open Access Repositorium der Universität Ulm und Technischen Hochschule Ulm. Dissertation. http://dx.doi.org/10.18725/OPARU-1195
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