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- All Subjects: Schrödinger equation
- Creators: Castillo-Chavez, Carlos
- Creators: Ferry, David
Description
The Quantum Harmonic Oscillator is one of the most important models in Quantum Mechanics. Analogous to the classical mass vibrating back and forth on a spring, the quantum oscillator system has attracted substantial attention over the years because of its importance in many advanced and difficult quantum problems. This dissertation deals with solving generalized models of the time-dependent Schrodinger equation which are called generalized quantum harmonic oscillators, and these are characterized by an arbitrary quadratic Hamiltonian of linear momentum and position operators. The primary challenge in this work is that most quantum models with timedependence are not solvable explicitly, yet this challenge became the driving motivation for this work. In this dissertation, the methods used to solve the time-dependent Schrodinger equation are the fundamental singularity (or Green's function) and the Fourier (eigenfunction expansion) methods. Certain Riccati- and Ermakov-type systems arise, and these systems are highlighted and investigated. The overall aims of this dissertation are to show that quadratic Hamiltonian systems are completely integrable systems, and to provide explicit approaches to solving the time-dependent Schr¨odinger equation governed by an arbitrary quadratic Hamiltonian operator. The methods and results established in the dissertation are not yet well recognized in the literature, yet hold for high promise for further future research. Finally, the most recent results in the dissertation correspond to the harmonic oscillator group and its symmetries. A simple derivation of the maximum kinematical invariance groups of the free particle and quantum harmonic oscillator is constructed from the view point of the Riccati- and Ermakov-type systems, which shows an alternative to the traditional Lie Algebra approach. To conclude, a missing class of solutions of the time-dependent Schr¨odinger equation for the simple harmonic oscillator in one dimension is constructed. Probability distributions of the particle linear position and momentum, are emphasized with Mathematica animations. The eigenfunctions qualitatively differ from the traditional standing waves of the one-dimensional Schrodinger equation. The physical relevance of these dynamic states is still questionable, and in order to investigate their physical meaning, animations could also be created for the squeezed coherent states. This will be addressed in future work.
ContributorsLopez, Raquel (Author) / Suslov, Sergei K (Thesis advisor) / Radunskaya, Ami (Committee member) / Castillo-Chavez, Carlos (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2012
Description
In the traditional setting of quantum mechanics, the Hamiltonian operator does not depend on time. While some Schrödinger equations with time-dependent Hamiltonians have been solved, explicitly solvable cases are typically scarce. This thesis is a collection of papers in which this first author along with Suslov, Suazo, and Lopez, has worked on solving a series of Schrödinger equations with a time-dependent quadratic Hamiltonian that has applications in problems of quantum electrodynamics, lasers, quantum devices such as quantum dots, and external varying fields. In particular the author discusses a new completely integrable case of the time-dependent Schrödinger equation in R^n with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. The author also considers several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schrödinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator. Finally, the author constructs integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrödinger equation with variable quadratic Hamiltonians. An extension of the Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application.
ContributorsCordero-Soto, Ricardo J (Author) / Suslov, Sergei (Thesis advisor) / Castillo-Chavez, Carlos (Thesis advisor) / Engman, Martin (Committee member) / Herrera-Valdez, Marco (Committee member) / Arizona State University (Publisher)
Created2011
Description
Moore's law has been the most important driving force for the tremendous progress of semiconductor industry. With time the transistors which form the fundamental building block of any integrated circuit have been shrinking in size leading to smaller and faster electronic devices.As the devices scale down thermal effects and the short channel effects become the important deciding factors in determining transistor architecture.SOI (Silicon on Insulator) devices have been excellent alternative to planar MOSFET for ultimate CMOS scaling since they mitigate short channel effects. Hence as a part of thesis we tried to study the benefits of the SOI technology especially for lower technology nodes when the channel thickness reduces down to sub 10nm regime. This work tries to explore the effects of structural confinement due to reduced channel thickness on the electrostatic behavior of DG SOI MOSFET. DG SOI MOSFET form the Qfinfet which is an alternative to existing Finfet structure. Qfinfet was proposed and patented by the Finscale Inc for sub 10nm technology nodes.
As part of MS Thesis we developed electrostatic simulator for DG SOI devices by implementing the self consistent full band Schrodinger Poisson solver. We used the Empirical Pseudopotential method in conjunction with supercell approach to solve the Schrodinger Equation. EPM was chosen because it has few empirical parameters which give us good accuracy for experimental results. Also EPM is computationally less expensive as compared to the atomistic methods like DFT(Density functional theory) and NEGF (Non-equilibrium Green's function). In our workwe considered two crystallographic orientations of Si,namely [100] and [110].
As part of MS Thesis we developed electrostatic simulator for DG SOI devices by implementing the self consistent full band Schrodinger Poisson solver. We used the Empirical Pseudopotential method in conjunction with supercell approach to solve the Schrodinger Equation. EPM was chosen because it has few empirical parameters which give us good accuracy for experimental results. Also EPM is computationally less expensive as compared to the atomistic methods like DFT(Density functional theory) and NEGF (Non-equilibrium Green's function). In our workwe considered two crystallographic orientations of Si,namely [100] and [110].
ContributorsLaturia, Akash (Author) / Vasileska, Dragica (Thesis advisor) / Ferry, David (Committee member) / Goodnick, Stephen (Committee member) / Arizona State University (Publisher)
Created2016