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arXiv:quant-ph/0407249

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

Joseph Kimeu, Mai, Roland, Kingshuk Majumdar

Published Jul 29, 2004• 3 citations
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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.

DOI
10.48550/arxiv.quant-ph/0407249

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Application of the variational RRR-matrix method to one-dimensional quantum tunneling
We have applied the variational RRR-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 RRR-matrix method is a simple, non-iterative, and effective method to solve one-dimensional quantum tunneling problems.
This article presents a simple, yet general and exact analytic-numerical method of solving one-dimensional quantum mechanics problems. It is based on the concept of dividing an arbitrary potential into convenient segments and analyzing the multiple reflection of a plane matter wave between neighboring potential walls. Amplitude reflection and transmission coefficients of each segment are the basis for the analysis. A simple iterative technique for calculating them by a square barrier approximation is given. This method is applicable to various potential barriers, wells, and periodic structures including continuous variations of potential energy and particle effective mass.
A simple, yet accurate, method of solving the Schrödinger equation across an arbitrary potential barrier is described, based on the analytic transfer-matrix technique. This paper gives the final analytical expression for the transmission probability. Numerical calculations for four typical potential barrier structures are compared with the traditional WKB, modified WKB and the modified Airy function (MAF) methods. Excellent agreement with an exact example is demonstrated. Most importantly, in cases where other potential-barrier methods fail, our method performs well.
A simle numerical method of calculating transmission coefficients across arbitrary potentials is presented by using the combination of a transfer matrix procedure and the solution of the wave functions in arbitrary potentials.This method is applied to a parabolic potential and a double barrier potential in an electric field.The transmission coefficients calculated are in agreement with those obtained using a multistep potential approximation.The wave functions at the resonant states are also given for the two examples.
Based on the similarity between time-independent Schrodinger equation and Helmholtz equation, the analytical transfer matrix method was utilized to analyze macroscopic quantum tunneling effect, and the potential filed and eigenvalue were corresponded to the refractive index and propagation constant, respectively.Without solving the Schrodinger equation and any approximation,the exact analytic expressions of reflection coefficient and reflection probability of the particle were obtained.Initial results show the equations obtained here are exact and have clear physical meanings.It is expected to be applied in the researches of many basic quantum phenomena,such as quantum tunneling effect,quantum reflection, quantum particles and tunneling resonance.