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American Journal of Physics

Variational R -matrix method for quantum tunneling problems

Hervé Le Rouzo

Published Feb 13, 2003• 8 citations
Publisher

Abstract

A variational R-matrix method for solving the one-dimensional time-independent Schrödinger equation is presented. The theory is simple and general and applies to an arbitrary potential. As an example, the method is applied to the pedagogically important problem of quantum tunneling through a barrier. Accurate transmission and reflection coefficients are obtained by the variational R-matrix method which compare favorably with the known analytical solutions.

DOI
10.1119/1.1519231

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A variational R-matrix method for solving the one-dimensional time-independent Schrödinger equation is presented. The theory is simple and general and applies to an arbitrary potential. As an example, the method is applied to the pedagogically important problem of quantum tunneling through a barrier. Accurate transmission and reflection coefficients are obtained by the variational R-matrix method which compare favorably with the known analytical solutions.
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.
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.
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.
Abstract We discuss the procedure of solving the time-dependent Schrödinger equation via a separation ansatz that leads to the time-independent Schrödinger equation. The resulting equation is in the form of an eigenvalue problem for the Hamiltonian, whose solutions are the stationary states. We derive and discuss the bound states and corresponding energy levels for several one-dimensional potentials of interest, including the finite and infinite potential well, the Dirac-delta potential, and the square double-well modelling the spatial configuration of the ammonia molecule. We then turn to the scattered solutions and discuss reflection from and transmission through a finite potential barrier, thus encountering the tunnel effect, which we in turn formulate more generally using the Gamow factor. This leads us to the interesting phenomenon of tunnelling resonances
A new numerical algorithm for tunneling is presented. This algorithm can be used to analyse an onedimensional\npotential barrier with an arbitrary profile. It is able to calculate not only the transmission coefficient\naccurately, but also the phases of the transmitted and reflected waves. Using the Schrodinger equation and plane waves,\nwe introduce a tunneling matrix to relate the transmitted and incident wave functions. Thus the transmission coefficient\nand phase can be calculated from the tunneling matrix.To test the accuracy of this algorithm,we calculate the transmission\ncoefficients and phases for a double -barrier (resonant tunneling) and a triangular barrier (field emission), and\nmake comparisous with corresponding analytical values. It shows that this algorithm is fast, accurate and efficient.
The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.