Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Algebraic Methods and Exact Solutions of Quantum Parametric Oscillators(Izmir Institute of Technology, 2019) Çetindaş, Osman; Atılgan Büyükaşık, Şirin; Pashaev, Oktay; Atılgan Büyükaşık, Şirin; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this thesis, we study different approaches for solving the Schrödinger equation for quantum parametric oscillators. The Wei-Norman algebraic approach, the Lewis- Riesenfeld invariant approach, the Malkin-Manko-Trifonov approach are investigated. For each approach, the wave function solutions of the Schrödinger equation, the propagator and dynamical invariants are found and their relations with each other are shown. In the Wei-Norman Algebraic approach, for constructing wave functions, explicit form of evolution operator is obtained uniquely in terms of two linearly independent classical solutions of the corresponding classical equation of motion. In Lewis-Riesenfeld approach, quadratic invariants are found in terms of the solution of Ermakov-Pinney equation and using the eigenstates of these invariants, wave function solutions are constructed. Setting initial values for Ermakov-Pinney solution, results of Wei-Norman and Lewis- Riesenfeld approaches are compared, then this solution is expressed in terms of same two linearly independent classical solutions. In Malkin-Manko-Trifonov approach, linear invariants which are symmetry operators for the Schrödinger equation, are constructed in terms of complex-valued solutions of the classical equation. Using these invariants, quadratic invariants are constructed and their eigenstates are used to find wave function solutions. Moreover, initial values for complex solutions of classical equation of motion are posed, and comparison of the three approaches is given.Master Thesis Solutions of Initial and Boundary Value Problems for Inhomogeneous Burgers Equations With Time-Variable Coefficients(Izmir Institute of Technology, 2016) Bozacı, Aylin; Atılgan Büyükaşık, Şirin; Atılgan Büyükaşık, Şirin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this thesis, we have investigated initial-boundary value problems on semiinfinite line for inhomogeneous Burgers equation with time-variable coecients. We have formulated the solutions for the cases with Dirichlet and Neumann boundary conditions. We showed that the Dirichlet problem for the variable parametric Burgers equation is solvable in terms of a linear ordinary dierential equation and a linear second kind singular Volterra integral equation. Then, for particular models with special initial and Dirichlet boundary conditions we found a class of exact solutions. Next, we considered the Neumann problem and showed that it reduces to a second order linear ordinary dierential equation and the standard heat equation with initial and nonlinear boundary conditions. Finally, we formulated the Cauchy problem for the variable parametric Burgers equation on the non-characteristic line, and obtained its solution in terms of a linear ODE and the series solution of the corresponding Cauchy problem for the heat equation. We gave examples to illustrate how some well known solutions of the Burgers equation can be recovered by solving a corresponding Cauchy problem.Master Thesis Exactly Solvable Generalized Quantum Harmonic Oscillators Related With the Classical Orthogonal Polynomials(Izmir Institute of Technology, 2016) Çayiç, Zehra; Atılgan Büyükaşık, Şirin; Atılgan Büyükaşık, Şirin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this thesis, we study exactly solvable generalized parametric oscillators related with the classical orthogonal polynomials of Hermite, Laguerre and Jacobi type. These quantum models with specific damping term, frequency and external forces are solved using Wei-Norman Lie algebraic approach. The exact form of the evolution operator is explicitly obtained in terms of two linearly independent homogeneous solutions and a particular solution of the corresponding classical equation of motion. Then, time evolution of wave functions and Glauber coherent states are constructed. Probability densities, expectation values and uncertainty relations are found and their properties are investigated according to the influence of the external forces. Besides, some examples with explicit solutions are given and their plots are constructed for the probability densities and uncertainty relations.