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Química Nova

TRANSMISSION AND REFLECTION COEFFICIENTS BY THE VARIABLE AMPLITUDE METHOD

Éderson D’M. Costa, Luciano Cordeiro, Nelson H. T. Lemes, J. P. Braga

Published Jan 1, 2016• 2 citations
Publisher

Abstract

AMPLITUDE METHOD.In this work, a simple derivation of the variable amplitude method using the variation of parameters to solve a differential equation is presented.The variable amplitude method was originally devised by Tikochinsky in 1977, using the quantum theory of scattering.The method is applied to two model potentials, the rectangular potential barrier and the Eckart potential, both with analytical solutions for the reflection coefficient.Numerical results will be compared with the exact values for several energies.The problem of calculating the reflection coefficient, usually involving extensive algebra as described in several textbooks, is reduced to solving a first order differential equation with initial condition.The method is very simple to apply, representing an attractive tool for teaching introductory quantum mechanics.A simple computer code is available from which reflection coefficients for the Eckart potential can be calculated.

DOI
10.5935/0100-4042.20160061

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AMPLITUDE METHOD.In this work, a simple derivation of the variable amplitude method using the variation of parameters to solve a differential equation is presented.The variable amplitude method was originally devised by Tikochinsky in 1977, using the quantum theory of scattering.The method is applied to two model potentials, the rectangular potential barrier and the Eckart potential, both with analytical solutions for the reflection coefficient.Numerical results will be compared with the exact values for several energies.The problem of calculating the reflection coefficient, usually involving extensive algebra as described in several textbooks, is reduced to solving a first order differential equation with initial condition.The method is very simple to apply, representing an attractive tool for teaching introductory quantum mechanics.A simple computer code is available from which reflection coefficients for the Eckart potential can be calculated.
People are familiar with quantum mechanical reflection and transmission coefficient. In all those cases corresponding potentials are usually assumed as of constant height and depth. For the cases of varying potential, corresponding reflection and transmission coefficients can be found out using WKB approximation method. But due to change of barrier height, reflection and transmission coefficients should be changed from point to point. Here we show the analytical expressions of the instantaneous reflection and transmission coefficients. Here as if we apply the WKB approximation at each point, so we call it as Instantaneous WKB method or IWKB method. Once we know the forward and backward wave amplitudes we can find out corresponding wave function by calculating Eiconal. For the case of analytically complicated potential corresponding differential equation seems to be unsolvable analytically. If the potentials are well behaved then obviously these could be replaced by functions of simple expression of same behaviour which can be integrated analytically and the equation is now possible to solve analytically.
A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.
Transmission and reflection of a quantal particle by a single-hump potential barrier are analysed by means of an amplitude-phase decomposition of the wavefunction on both sides of the barrier. The amplitude-phase analysis of the wavefunction provides a particular invariant of the Ermakov–Lewis type, which originates in the matching process. The transmission and reflection coefficients turn out to be simple functions of this invariant. Numerical calculations of the invariant for an Eckart–Epstein potential barrier provide very accurate results.
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.
In this paper, we calculate the transmission and reflection amplitudes of wave functions for different potentials such as the delta function, the rectangular barrier, the Eckart potential, and the Hulthen potential. We describe the relationship between these amplitudes and compute the reflection resonances between each potential. We describe the transmission and reflection probabilities using the WKB formula and compare the results with ones obtained from matching the boundary conditions. Furthermore, we use a two by two transfer matrix to calculate a rigorous bound on the transmission and reflection probabilities.