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Journal of Modern Physics

Transient Quantum Beat Oscillations in Extreme-Relativistic Diffraction in Time

Salvador Godoy

Published Jan 1, 2021• 1 citation
Publisher

Abstract

In the solution of the Klein-Gordon equation for the shutter problem, we prove that, at internuclear distances, a relativistic beam of Pi-mesons has a probability density which oscillates in time in a similar way to the spatial dependence in optical Fresnel diffraction from a straight edge. However, for an extreme-relativistic beam, the Fresnel oscillations turn into quantum damped beat oscillations. We prove that quantum beat oscillations are the consequence, at extreme-relativistic velocities, of the interference between the initial incident wave function, and the Green’s function in the relativistic shutter problem. This is a pure quantum relativistic phenomenon.

DOI
10.4236/jmp.2021.121001

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In the solution of the Klein-Gordon equation for the shutter problem, we prove that, at internuclear distances, a relativistic beam of Pi-mesons has a probability density which oscillates in time in a similar way to the spatial dependence in optical Fresnel diffraction from a straight edge. However, for an extreme-relativistic beam, the Fresnel oscillations turn into quantum damped beat oscillations. We prove that quantum beat oscillations are the consequence, at extreme-relativistic velocities, of the interference between the initial incident wave function, and the Green’s function in the relativistic shutter problem. This is a pure quantum relativistic phenomenon.
In a previous note a dynamical description of resonance scattering was given, and transient terms appeared in the wave function describing the process. To understand the physical significance of these terms, the transient effects that appear when a shutter is opened are discussed in this paper. For a nonrelativistic beam of particles, the transient current has a close mathematical resemblance with the intensity of light in the Fresnel diffraction by a straight edge. This is the reason for calling the transient phenomena by the name of diffraction in time.The shutter problem is discussed for particles whose wave functions satisfy the Schr\"odinger equation, the ordinary wave equation, and the Klein-Gordon equation. Only for the Schr\"odinger time-dependent equation do the transient wave functions resemble the solutions that appear in Sommerfeld's theory of diffraction. The connection of transient phenomena with the time-energy uncertainty relation, and the interpretation of the transient current in a scattering process, are briefly discussed. The relativistic wave functions for the shutter problem may play an important role in the dynamical description of a relativistic scattering process.
We explore the dynamics of relativistic quantum waves in a potential step by using an exact solution to the Klein-Gordon equation with a point-source initial condition. We show that in both the propagation and Klein-tunneling regimes, the zitterbewegung effect manifests itself as a series of quantum beats of the particle density in the long-time limit. We demonstrate that the beating phenomenon is characterized by the zitterbewegung frequency and that the amplitude of these oscillations decays as ${t}^{\ensuremath{-}3/2}$. We show that the beating effect also manifests itself in the free Klein-Gordon and Dirac equations within a quantum shutter setup, which involve the dynamics of cutoff quantum states. We also find a time domain where the particle density of the point source is governed by the propagation of a main wavefront, exhibiting an oscillating pattern similar to the diffraction-in-time phenomenon observed in nonrelativistic systems. The relative positions of these wavefronts are used to investigate the time delay of quantum waves in the Klein-tunneling regime. We show that depending on the energy difference E\mathcal{E}E between the source and the potential step, the time delay can be positive, negative, or zero. The latter case corresponds to a super-Klein-tunneling configuration, where E\mathcal{E}E equals half the energy of the potential step.
We explore the dynamics of two-dimensional massless Dirac fermions within a quantum shutter approach, which involves the time evolution of an initial cutoff plane wave. We show that the probability density is governed by an interplay between diffraction in time and Zitterbewegung phenomena typical of relativistic quantum shutter systems with nonzero mass. The time diffraction appears as an oscillatory pattern in the probability density, similar to the effect predicted by Moshinsky in 1952 [Phys. Rev. 88, 625 (1952)] for Schr\"odinger free matter waves. The Zitterbewegung manifests itself as high-frequency oscillations embedded in the time-diffraction profile. We found that these two transient effects are induced by the transverse momentum component of the incident wave ky{k}_{y}ky​ that acts as an effective mass of the system. Furthermore, this effective mass can be manipulated by tuning the incidence angle of the initial quantum state, which allows us to control the frequencies of the transients. In particular, we demonstrate that near a normal incidence condition, the Zitterbewegung appears as a series of quantum beats in the probability density, with a beating frequency 2kyvF2{k}_{y}{v}_{F}2ky​vF​, where vF{v}_{F}vF​ is the Fermi velocity.
Relativistic diffraction in time wave functions can be used as a basis for causal scattering waves. We derive such exact wave function for a beam of Dirac and Klein-Gordon particles. The transient Dirac spinors are expressed in terms of integral defined functions which are the relativistic equivalent of the Fresnel integrals. When plotted versus time the exact relativistic densities show transient oscillations which resemble a diffraction pattern. The Dirac and Klein-Gordon time oscillations look different, hence relativistic diffraction in time depends strongly on the particle spin.
The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus.