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,
td, 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,
td 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
td→1/2 compared to
td→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,
td→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.
Abstract A superoscillatory function is one in which the function oscillates faster than its fastest Fourier component. Superoscillations are now a well-established feature of certain types of wavepacket. Here we discuss quantum superoscillations in a wavepacket evolving according to the harmonic oscillator hamiltonian. The evolution of the wavepacket is investigated using an expansion in terms of eigenfunctions and in terms of the propagator using both an exact integration, and a saddle point approximation. The creation and decay of superoscillations as well as the barrier between them and normal oscillations is shown to depend on the behaviour of the saddle points. The work reported here is the result of a student project and we argue that super oscillations makes an excellent topic with which to introduce students to theoretical research.
Abstract We investigate potential scattering and tunneling dynamics of a particle wavepacket evolving according to the relativistic Schrödinger equation (also known as the Salpeter equation). The tunneling properties of the Salpeter equation differ from those of the standard relativistic wave equations (such as the Klein–Gordon or Dirac equations). In particular, the tunneling solutions must be found by working in momentum space, given that the equation in configuration space contains a pseudo-differential operator. The resulting integral equations are derived and solved numerically for wavepackets scattering on model potential barriers. The solutions are characterized by the absence of Klein tunneling and an effect of the potential on the fraction of the transmitted wavepacket that propagates outside the light cone, a feature that has in the past been well-studied only for free propagation.
We investigate wavepacket dynamics for a relativistic particle in a box evolving according to the relativistic Schrödinger (also known as the Salpeter) equation. We derive the solutions for an infinite well -- which contrary to the standard relativistic wave equations (such as the Klein-Gordon or Dirac equations) -- are well defined, and use these solutions to construct wavepackets. We obtain expressions for the wavepacket revival times and explore the corresponding quantum carpets (the space-time probability density plots) for different dynamical regimes. We further analyze level spacing statistics as the dynamics goes from the non-relativistic regime to the ultra-relativistic limit.
A superoscillatory function—that is, a band-limited function f(x) oscillating faster than its fastest Fourier component—is taken to be the initial state of a freely-evolving quantum wavefunction ψ. The superoscillations persist for unexpectedly long times, but eventually disappear through the interaction of contributions to ψ with complex momenta that are exponentially disparate in magnitude; this is established by applying the asymptotics of integrals, supported by numerics. f(x) can alternatively be regarded as the wave generated by a diffraction grating, propagating paraxially and without evanescence as ψ in the space beyond. The persistence of superoscillations is then interpreted as the propagation of sub-wavelength structure farther into the field than the more familiar evanescent waves.
We investigate relativistic wavepacket dynamics for an electron tunneling through a potential barrier employing space-time resolved solutions to relativistic quantum field theory (QFT) equations. We prove by linking the QFT property of micro-causality to the wavepacket behavior that the tunneling dynamics is fully causal, precluding instantaneous or superluminal effects that have recently been reported in the literature. We illustrate these results by performing numerical computations for an electron tunneling through a potential barrier for standard tunneling as well for Klein tunneling. In all cases (Klein tunneling \ or regular tunneling across a standard or a supercritical potential) the transmitted wavepacket remains in the causal envelope of the propagator, even when its average position lies ahead of the average position of the corresponding freely propagated wavepacket.