Application of the variational $R$-matrix method to one-dimensional quantum tunneling

TL;DR

This paper demonstrates that the variational R-matrix method effectively calculates quantum tunneling probabilities in one-dimensional barriers, showing good agreement with exact and other numerical methods across various potential profiles.

Contribution

The paper introduces the application of the variational R-matrix method to compute tunneling probabilities for multiple potential profiles, highlighting its simplicity and accuracy.

Findings

Results agree well with exact analytical solutions.

The method is non-iterative and computationally efficient.

Applicable to various potential barrier shapes.

Abstract

We have applied the variational $R$-matrix method to calculate the reflection and tunneling probabilities of particles tunneling through one-dimensional potential barriers for five different types of potential profiles -- truncated linear step, truncated exponential step, truncated parabolic, bell-shaped, and Eckart. Our variational results for the transmission and reflection coefficients are compared with exact analytical results and results obtained from other numerical methods. We find that our results are in good agreement with them. We conclude that the variational $R$-matrix method is a simple, non-iterative, and effective method to solve one-dimensional quantum tunneling problems.