Wave Transport and Chaos in Two-Dimensional Cavities

This thesis focuses on chaotic stationary waves, both quantum mechanical and classical. In particular we study different statistical properties regarding thesewaves, such as energy transport, intensity (or density) and stress tensor components. Also, the methods used to model these waves are investigated, and somelimitations and specialities are pointed out.


1 Introduction
1.1 Chaos
1.2 Origins of Wave Chaos
1.3 Wave Chaos Today
1.4 The Aim of this Thesis
2 Theory
2.1 Quantum Mechanics
2.1.1 The Schr¨odinger and Helmholtz Equations
2.1.2 Normalization
2.1.3 Energy Transport
2.1.4 Quantum Mechanical Stress Tensor
2.1.5 Continuity Equation
2.1.6 Bracket Notation
3 Finite Difference Method
3.1 Discretization
3.1.1 The Schr¨odinger Equation in FDM
3.2 Boundary Conditions
3.2.1 Dirichlet Boundary Condition
3.2.2 Neumann Boundary Condition
3.2.3 Similarities between NBC and DBC
3.2.4 Mixed Boundary Conditions
3.3 The Matrix Eigenvalue Problem
3.3.1 Setting Up the Matrix
3.3.2 Normalization
4 Imaginary Potential
4.1 Currents
4.2 Conservation of Probability Density
4.3 Perturbation Theory
4.4 Balanced Imaginary Potential
4.4.1 Verification of the Balanced Potential Method
4.5 Interaction between Wave Functions
5 Accuracy
5.1 Theoretical Solution
5.2 Numerically Obtained Eigenvalues
5.3 Numerically Obtained Eigenfunctions
5.4 Perturbation Theory
6 Complexity
6.1 Data Complexity
6.2 Time Complexity
6.3 Exclusion of High Potential Grid Points
6.4 Using Symmetry
7 Statistical Distributions
7.1 Density Distributions
7.1.1 Numerically Obtained Density Distributions
7.2 Current Distributions
7.2.1 Numerically Obtained Current Distributions
7.3 Stress Tensor Distributions
7.3.1 Numerically Obtained Stress Tensor Distributions
8 Conclusions and Future Work
8.1 The Method
8.2 The Statistics
8.3 Future Work
A Plots
A.1 Sample Plots
A.2 Density Distributions
A.3 Current Distributions
A.4 Stress Tensor Distributions
B Derivations
B.1 Helmholtz Equation in a Quasi-Two-Dimensional Cavity
B.2 Derivation of the Continuity Equation

Author: Wahlstrand, Bjorn

Source: Linkoping University

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