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arXiv:2010.05583

An application of a quantum wave impedance approach for solving a nonsymmetric single well problem

Orest Hryhorchak

Published Oct 12, 2020• 8 citations
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Abstract

A short introduction of a relation between a Green's function and a quantum wave impedance function as well as its application to a determination of eigenenergies and eigenfunctions of a quatum-mechanical system is provided. Three different approaches, namely a classical approach based on a direct solving of a Shrödinger equation, a transfer matrix method and a quantum wave impedance technique, for a calculation of eigenenergies and eigenfunctions of a quantum mechanical nonsymmetric single well system are considered. A comparision of these approaches gives the possibility to clarify advantages and drawbacks of each method which is useful especially for teaching and learning purposes.

DOI
10.48550/arxiv.2010.05583

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A short introduction of a relation between a Green's function and a quantum wave impedance function as well as its application to a determination of eigenenergies and eigenfunctions of a quatum-mechanical system is provided. Three different approaches, namely a classical approach based on a direct solving of a Shrödinger equation, a transfer matrix method and a quantum wave impedance technique, for a calculation of eigenenergies and eigenfunctions of a quantum mechanical nonsymmetric single well system are considered. A comparision of these approaches gives the possibility to clarify advantages and drawbacks of each method which is useful especially for teaching and learning purposes.
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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.
This paper is dedicated to an application of a quantum wave impedance approach for a study of infinite and semi-infinite periodic systems. Both a Dirac comb and a δ−δ′δ-δ'δ−δ′ comb as well as a Kronig-Penney model are considered. It was shown how to reformulate the problem of an investigation of mentioned systems in terms of a quantum wave impedance and it was demonstrated how much a quantum wave impedance approach simplifies studying these systems compared to other methods. The illustation of such a simplification was provided by application of classical approach, transfer matrix technique and a quatum wave impedance method for solving Kronig-Penney model.
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