Tunneling in energy eigenstates and complex quantum trajectories

TL;DR

This paper introduces a new method for calculating quantum reflection probabilities using complex trajectories and an extended probability density, showing high accuracy across various potential barriers and revealing slight deviations at moderate energies.

Contribution

It proposes a novel ansatz for reflection probability based on complex quantum trajectories and tests its validity across multiple potential barriers, confirming its accuracy.

Findings

High agreement with standard results for rectangular barriers

Very good agreement for Eckart and Morse barriers

Minor deviations at moderate energies, up to 0.1% of standard values

Abstract

Complex quantum trajectory approach, which arose from a modified de Broglie-Bohm interpretation of quantum mechanics, has attracted much attention in recent years. The exact complex trajectories for the Eckart potential barrier and the soft potential step, plotted in a previous work, show that more trajectories link the left and right regions of the barrier, when the energy is increased. In this paper, we evaluate the reflection probability using a new ansatz based on these observations, as the ratio between the total probabilities of reflected and incident trajectories. While doing this, we also put to test the complex-extended probability density previously postulated for these quantum trajectories. The new ansatz is preferred since the evaluation is solely done with the help of the complex-extended probability density along the imaginary direction and the trajectory pattern itself. The calculations are performed for a rectangular potential barrier, symmetric Eckart and Morse barriers, and a soft potential step. The predictions are in perfect agreement with the standard results for potentials such as the rectangular potential barrier. For the other potentials, there is very good agreement with standard results, but it is exact only for low and high energies. For moderate energies, there are slight deviations. These deviations result from the periodicity of the trajectory pattern along the imaginary axis and have a maximum value only as much as $0.1 \%$ of the standard value. Measurement of such deviation shall provide an opportunity to falsify the ansatz.