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Journal of Physical Studies

Reformulation of transmission and reflection problems in terms of quantum wave impedance function

Orest Hryhorchak

Published Jan 1, 2021• 1 citation
Publisher

Abstract

On the basis of 1D Schr odinger equation, a non-linear rst-order dierential equation (Riccati type) for a quantum wave impedance function was derived.The advantages of this approach were discussed and demonstrated for the case of a single rectangular barrier.Both the scattering and the bound states problems were reformulated in terms of quantum wave impedance.The application of the method for solving these problems was considered.The expressions for the reection and transmission coecients were found on the basis of the quantum wave impedance approach.

DOI
10.30970/jps.25.4001

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On the basis of 1D Schr odinger equation, a non-linear rst-order dierential equation (Riccati type) for a quantum wave impedance function was derived.The advantages of this approach were discussed and demonstrated for the case of a single rectangular barrier.Both the scattering and the bound states problems were reformulated in terms of quantum wave impedance.The application of the method for solving these problems was considered.The expressions for the reection and transmission coecients were found on the basis of the quantum wave impedance approach.
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A method for determining the quantum wave impedance function for a quantum-mechanical system with an arbitrary piecewise constant potential has been developed.Using this method, both the well-known iterative formula [A.N. Khondker, M. R. Khan, A. F. M. Anwar, J. Appl.Phys.63, 5191 (1988)] and an alternative approach for calculating quantum wave impedance were derived.This approach was applied to both scattering and bound state cases.The general form of the wave functions for bound states, as well as the transmission and reection coecients in the scattering case, were obtained for systems with a potential consisting of an arbitrary number of rectangular barriers and wells.The application of the developed method was demonstrated by describing symmetric rectangular double-well and double-barrier quantum-mechanical systems.
An application of the quantum wave impedance method for the analytical study of quantum-mechanical systems containing a set of delta-wells and delta-barriers is considered in this study.The algorithm for determining both transmission and reflection coefficients for a set of delta-barriers was proposed.Similarly, the scheme for analytically deriving the expressions for energy levels and wave functions of bound states for a set of delta-wells was developed.In both cases, the solution for the quantum wave impedance function in the region with a constant potential [1] and the matching conditions at the origins of deltafunctions were used.As an illustration of the application of the proposed approach, the quantum-mechanical systems with symmetric double-barrier potential and double-well potential were investigated.In the case of a single delta-well, the eigenenergy value as well as the expression for the eigenfunction were obtained directly using the non-linear differential equation for the quantum wave impedance function.The results demonstrate the effectiveness of the quantum wave impedance method in handling quantum-mechanical systems with deltalike potentials, offering a robust framework for further studies in this area.
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An application of a quantum wave impedance method for a study of quantum-mechanical systems which con\-tain singular zero-range potentials is considered. It was shown how to reformulate the problem of an investigation of mentioned systems in terms of a quantum wave impedance. As a result both the scattering and bound states problems are solved for systems of single δδδ, double δδδ and single δ−δ′δ-δ'δ−δ′ potentials. The formalization of solving systems with an arbitrary combination of a piesewise constant potential and a δδδ-potentials with the help of a quantum wave impedance approach is described.
We show that application of the immersing and transfer-matrix methods to one-dimensional problems of particles scattering leads to the system of two linear equations for the functions F and Φ expressed by means of the transmission and reflection amplitudes. The expressions of these functions are derived. The offered method is illustrated by the finding of transmission and reflection coefficients for the potential barrier with a constant height. The developed method can be applied in solving the quasi-one-dimensional and two-dimensional problems of scattering.