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Fourth Year Project Poster - Coggle Diagram
Fourth Year Project Poster
Theoretical Background
Adiabatic Quantum Mechanics
Adiabatic Theorem
Hypothesis, and conclusion.
Plausibility for use in our problem
The interest in studying this problem
Dynamical Systems notions
Definitions
Instantaneous Eigenstates
Group 1 and group 2 elements: the sigma function
Level Number
Random Dynamical System
Set-up of Problem
Choice of Hamiltonian
Spectral property
Off-diagonal Elements Vanish
Self-adjointness: a dense domain for which it is self-adjoint exists
Choice of Hilbert Space
Possible interpretation as a qubit and SHO model, where the potential is different for spin up or spin down, and the spin of the qubit does not change during the separation period
Motivation for Hamiltonian
Simplest model. It is easy to write down permutations following the set up.
Further Possible Directions for Study
Questions concerning Ergodicity
Different Choice of Hamiltonian
Results (Theoretical)
Evolution Formula and Algorithm
Asymptotic Behaviour, and error estimate
Result on difference of each iterate
Group 2 is increasing, group 1 is decreasing
Finding some fixed points
Distribution of the group one and group two elements
Relative density is known
Evolution of Expectation assuming random walk model
Results (Numerical)