Transient Quantum Beat Oscillations in Extreme-Relativistic Diffraction in Time

Salvador Godoy

doi:10.4236/jmp.2021.121001

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.

Highlights

Quantum beat oscillations are a common subject in Atomic and Molecular Spectroscopy
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
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

Summary

Introduction

Quantum beat oscillations are a common subject in Atomic and Molecular Spectroscopy. In Atomic physics, the term quantum beat refers to a superposed oscillatory behavior in the light intensity emitted by some suddenly excited atomic systems in their subsequent decay [1]. In Molecular Spectroscopy, quantum beat spectroscopy is a Doppler-free time domain method based on the creation of molecular coherences with a laser pulse and the measurement of their subsequent time evolution [2] In this same context is the work of Villavicencio et al [3], where transient phenomena of phase-modulated cutoff wave packets were explored by deriving an exact general solution to Schrödinger’s equation for finite-range potentials involving arbitrary initial quantum states. They show that the dynamical features of the probability density are governed by a virtual two-level system. For an extreme-relativistic beam, the Fresnel oscillations turn into quantum damped beats!

The Klein-Gordon Shutter Problem

Time-Dependent Density

Relativistic Diffraction in Time for Pi-Mesons

Damped Beats in Extreme-Relativistic Diffraction in Time

Conclusions

The Klein-Gordon Solution for the 1D Shutter Problem