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

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.

Superoscillations are a phenomena in which a function can oscillate faster than it’s fastest Fourier component. As a trade-off they suffer by being exponentially smaller than conventional oscillations. Previous studies of superoscillations within the construct of the Schrödinger equation have identified two key features: disappearance after a time, \(t_d\), and an asymmetrical evolution either side of a line in space (the wall effect).

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 \((h→0)\).

Positive and negative energy superoscillations are best described in the WKBlimit. Through application of asymptotic integration, we find that both positiveand negative energy superoscillations possess a disappearance time and wall effect. For both Klein Gordon and Dirac equations, \(t_d\) is equal. This implies, in terms of disappearance spin has no effect on the evolution of a superoscillatory, relativistic wavefunction of positive or negative energy. However, relativistic superoscillations disappear faster than non-relativistic superoscillations, and in each case, the disappearance time tends to a finite value as a superoscillatory parameter is increased. For non-relativistic superoscillations \(t_d→\)1/2 compared to \(t_d→\)0.3715 for the relativistic case.

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.

The asymmetries between the wall effects are caused by differences in the behaviour of saddles (complex momenta) found when evaluating the wavefunction as an integral over the propagator. Again, there is no disparity between the walls of the Klein-Gordon or Dirac equation; spin does not affect the evolution of positive ornegative energy superoscillations.

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.