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Quantile Motion and Tunneling

S. Brandt, Hans Dieter Dahmen, Erion Gjonaj, Tilo Stroh

Published Jan 19, 1995• 9 citations
OpenAlex

Abstract

The concepts of quantile position, trajectory, and velocity are defined. For a tunneling quantum mechanical wave packet, it is proved that its quantile position always stays behind that of a free wave packet with the same initial parameters. In quantum mechanics the quantile trajectories are mathematically identical to Bohm's trajectories. A generalization to three dimensions is given.

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The concepts of quantile position, trajectory, and velocity are defined. For a tunneling quantum mechanical wave packet, it is proved that its quantile position always stays behind that of a free wave packet with the same initial parameters. In quantum mechanics the quantile trajectories are mathematically identical to Bohm's trajectories. A generalization to three dimensions is given.
The de Broglie--Bohm causual (hydrodynamic) formulation of quantum mechanics is computationally implemented in the Lagrangian (moving with the fluid) viewpoint. The quantum potential and force are accurately evaluated with a moving weighted least squares algorithm. The quantum trajectory method is then applied to barrier tunneling on smooth potential surfaces. Analysis of the tunneling mechanism leads to a novel and accurate approximation: shortly after the wave packet is launched, completely neglect all quantum terms in the dynamical equations for motion along the tunneling coordinate.
We discuss applications of the quantile concept of trajectories and velocities to the propagation of electromagnetic signals in wave guides of varying cross section. Quantile motion is a general description of the transport properties of measurable quantities in quantum mechanics as well as in classical electrodynamics. In the latter case we consider the quantile motion of electromagnetic energy as the direct result of a physical measurement. In this sense the quantile velocity corresponds to the electromagnetic signal velocity. We show that this signal velocity is always smaller than the speed of light in vacuum.
Abstract We discuss applications of the quantile concept of trajectories and velocities to the propagation of electromagnetic signals in wave guides of varying cross section. Quantile motion is a general description of the transport properties of measurable quantities in quantum mechanics as well as in classical electrodynamics. In the latter case we consider the quantile motion of electromagnetic energy as the direct result of a physical measurement. In this sense the quantile velocity corresponds to the electromagnetic signal velocity. We show that this signal velocity is always smaller than the speed of light in vacuum.
We discuss applications of the quantile concept of trajectories and velocities to the propagation of electromagnetic signals in wave guides of varying cross section. Quantile motion is a general description of the transport properties of measurable conserved quantities in quantum mechanics as well as in classical electrodynamics. In the latter case we consider the quantile motion of electromagnetic energy as the direct result of a physical measurement. In this sense the quantile velocity corresponds to the electromagnetic signal velocity also in the presence of barriers and inhomogeneities in the medium of propagation. We show that this signal velocity is always smaller than the speed of light in vacuum. Using numerical examples we demonstrate how typical wave phenomena can be described in terms of the quantile motion.
According to Standard Quantum Mechanics (SQM), known as the Copenhagen Interpretation, the complete description of a system of particles is provided by its wave function. However, in the de Broglie-Bohm theory of Bohmian Quantum Mechanics (BQM), the additional element which is introduced apart from the wave function is the particle position, conceived in the classical sense as pursuing a definite continuous track in space-time. In BQM formulation, depending on the configuration of the potential barrier and the energy of the packet, the particle trajectories have been shown to take distinct paths. We will consider several barrier heights and show that in a Bohmian interpretation of the problem, there is no such thing as Quantum Tunnelling.