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Journal of Physical Studies

Quantum wave impedance calculation for an arbitrary piesewise constant potential

Orest Hryhorchak

Published Jan 1, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.30970/jps.28.4704

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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.
The method of a determination of a quantum wave impedance for an arbitrary piecewise constant potential was developed. On the base of this method both the well-known iterative formula \cite{Khondker_Khan_Anwar:1988} and alternative ways for a quantum wave impedance calculation were derived. A scattering and a bound state case were considered. The general form of a wave function of bound states for an arbitrary piesewise constant potential was obtained as well as transmission and reflection coefficients in a scattering case. The applying of method was demonstarted on the system of double well/barrier.
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.
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.
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.
The approximate numerical method for a calculation of a quantum wave impedance in a case of a potential energy with a complicated spatial structure is considered. It was proved that the approximation of a real potential by a piesewise constant function is also reasonable in a case of using a quantum impedance approach.The dependence of an accuracy of numerical calculations on a number of cascads by which a real potential is represented was found. The method of including into a consideration of zero-range singular potentials was developed.