UBC Theses and Dissertations
Exploring the Dirac equation Mackovic, Brie
In this thesis low energy excitations of perfectly dimerized trans-polyacetylene are modelled using the one-dimensional Dirac equation. The system is solved on both the half-line and segment, and the solutions are used to explore quantum phenomena. It is discovered that the zero mode of the half-line is a Majorana fermion quasiparticle. It is also found that dominate zero mode coupling to an electron on a scanning tunnelling microscope is achieved with a sufficiently large mass gap of the quantum wire. This allows scattering state excitations to be ignored in calculations in this thesis. It is also shown that the zero mode can facilitate entanglement of two electrons, each in proximity to opposite ends of a long segment of trans-polyacetylene. An algorithm is also developed which teleports the spin state of an electron on a segment of trans-polyacetylene. The quantum measurement used in this algorithm conserves fermion parity symmetry, however charge superselection is violated for three-fourths of the measurement operators. In the thermally isolated system teleportation is successful for all of the measurement operators on the ground state. However, decoherence occurs in the non-thermally isolated system due to thermal mixing of nearly degenerate states, leading to teleportation being successful for only half of the measurement operations on the thermal state.
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