# Quantum Free Particle Superoscillations in 1+1 Dimensions: A First-Quantised Approach

Herklots, Jack (2019) Quantum Free Particle Superoscillations in 1+1 Dimensions: A First-Quantised Approach. Doctor of Philosophy (PhD) thesis, University of Kent,. (KAR id:80770)

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## Abstract

The principal differences between non-relativistic and relativistic quantum mechanics are: the existence of negative energy states, and the natural emergence of spin within a relativistic framework. We consider both these effects in the creation and evolution of superoscillating wavepackets.

In this thesis, we expand this work into a relativistic framework by applying it to the Klein-Gordon and Dirac equations, to do this we require a thorough examination of 1+1 dimensional relativistic propagators. Although initially derived interms of Bessel functions, these have two key limits which allow for simpler calculation: the light-cone limit $$(x→x0+ct)$$ and the WKB limit $$(h→0)$$.

The wall effect is also noticeably different in relativistic and non-relativistic contexts. The walls appear at different points in space and, despite the relativistic wall effect being evident from $$t=$$0, within the Schrödinger equation, it doesn’t appear until a time of $$t=$$1/32. This contrast in the wall effects of both cases is the leading cause of the inequality of the disappearance times.

Where positive and negative energy wavefunctions appear as $$h→$$0, mixed energy superoscillations appear at the light-cone. Mixed energy superoscillations do not exhibit the wall effect and neither do they exist in a non-relativistic description. However, they do have a disappearance time. In contrast to the positive and negative energy states,$$t_d→$$0 as a superoscillatory parameter is increased. It is within a mixed energy construct that the effect of spin on the evolution of relativistic superoscillations appears; one of the components of the Dirac equation does not superoscillate. This is caused by this term existing at the WKB limit as opposed to the light-cone.

Item Type: Thesis (Doctor of Philosophy (PhD)) Q Science Faculties > Sciences > School of Physical Sciences System Moodle System Moodle 09 Apr 2020 09:07 UTC 14 Apr 2020 08:23 UTC https://kar.kent.ac.uk/id/eprint/80770 (The current URI for this page, for reference purposes)