Bound states of the one-dimensional Dirac equation for scalar and vector double square-well potentials

TL;DR

This paper analytically investigates bound states of the one-dimensional Dirac equation with scalar and vector double square-well potentials, providing detailed numerical results and exploring quantum tunneling phenomena.

Contribution

It introduces a transfer-matrix method to analyze bound states in the Dirac equation with scalar and vector DSPs, including cases with equal magnitudes, and compares results with Schrödinger equation.

Findings

Numerical eigenvalues, wave functions, and densities for three DSP cases.

Differences and similarities between Dirac and Schrödinger results.

Wave packet dynamics illustrating quantum tunneling.

Abstract

We have analytically studied bound states of the one-dimensional Dirac equation for scalar and vector double square-well potentials (DSPs), by using the transfer-matrix method. Detailed numerical calculations of the eigenvalue, wave function and density probability have been performed for the three cases: (1) vector DSP only, (2) scalar DSP only, and (3) scalar and vector DSPs with equal magnitudes. We discuss the difference and similarity among results of the cases (1)-(3) in the Dirac equation and that in the Schr\"{o}dinger equation. Motion of a wave packet is calculated for a study on quantum tunneling through the central barrier in the DSP.