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Mechanisms and machine science

Numerical Simulation of One-Dimensional Barrier Penetration

Xingrong Zheng, Yujie Li, Kaiqiang Xie

Published Jan 1, 2024• 0 citations
Publisher
DOI
10.1007/978-3-031-62664-7_25

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A standard way to solve a Schrödinger equation is to discreteize the radial coordinates and apply a numerical method for a differential equation, such as the Runge-Kutta method or the Numerov method. Here I employ a discrete basis formalism based on a finite mesh method as a simpler alternative, with which the numerical computation can be easily implemented by ordinary linear algebra operations. I compare the numerical convergence of the Numerov integration method to the finite mesh method for calculating penetrabilities of a one-dimensional potential barrier, and show that the latter approach has better convergence properties.
Elementary numerical methods are used to evaluate solutions to Schrödinger’s equation for a variety of one-dimensional potential barriers and to determine the transmission coefficients of free particles. These techniques are used to study the oscillatory variations in transmission coefficients for both continuous and discontinuous potentials. The amplitude of the variations is found to be strongly increased by discontinuities in the potential function but small variations exist even for a smooth, continuous potential function. The techniques described are suitable for intermediate-level undergraduate physics students.