Quantum Chaos On A Curved Surface

The system studied in the thesis is a particle in a two-dimensional box on the surface of a sphere with constant radius. The different systems have different radii while the box dimension is kept the same, so the curvature of the surface of the box is different for the different systems. In a system with a sphere of a large radius the surface of the box is almost flat. What happens if the radius is decreased and the symmetry is broken? Will the system become chaotic if the radius is small enough…


1 Introduction
1.1 The aim of the thesis
2 Theory
2.1 Quantum mechanics
2.2 Regular systems become chaotic
2.3 Quantum chaos
3 Method
3.1 Finite Difference Method
3.2 The kinetic energy part of the Hamiltonian matrix
3.3 Building the box
4 Test of the method
5 Statistical Properties
5.1 Distribution of the probability function
5.1.1 Results
5.1.2 Pictures
5.2 The Porter-Thomas distribution for different eigenfunctions
5.2.1 Results
5.2.2 Pictures
5.3 Amplitude distribution of the eigenfunction
5.3.1 Results
5.3.2 Pictures
6 Show the curvature
7 Limits and tips for the reader
7.1 Resolution of the FDM
7.2 Better resolution of the box
7.3 The smallest sphere posible
8 Discussion and conclusion
8.1 The eigenfunctions ψ and the probability density |ψ|
8.2 Porter-Thomas
8.3 Amplitude distribution and Gaussian clock curve
8.4 Conclusion

Author: Wärnå, John

Source: Linköping University

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