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arXiv:2010.07632

Application of a quantum wave impedance method for study of infinite and semi-infinite periodic media

Orest Hryhorchak

Published Oct 15, 2020• 5 citations
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Abstract

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 $δ-δ'$ 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.

DOI
10.48550/arxiv.2010.07632

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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.
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.
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.
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.
On the base of a 1D Shrödinger equation the non-linear first-order differential equation (Ricatti type) for a quantum wave impedance function was derived. The advantages of this approach were discussed and demonstrated for a case of a single rectangular barrier. Both the scattering and the bound states problem were reformulated in terms of a quantum wave impedance and its application for solving both these problems was considered. The expressions for a reflection and a transmission coefficient were found on the base of a quantum wave impedance approach.
Some discontinuous potentials commonly used in quantum mechanics are briefly considered in order to highlight some unusual conclusions one may draw from the solutions of the associated Schrödinger equation. Classical analogs of these potentials are given; in particular, it is shown that a boundary condition necessary for the quantum system may have physical meaning of its own in the classical counterpart. A classical example of the Kronig–Penney model displaying the appearance of frequency bands, as opposed to the energy bands found in the quantum domain, is also discussed.
The relation between finite square well and laser is paramount in quantum mechanics. This paper focuses on establishing the relationship between finite square well and laser through the most fundamental theory of quantum mechanics. This paper first introduced the importance of finite square well and the Kronig-Penney model in the development of the theory of laser and Quantum mechanics briefly. Then, this paper gives a detailed derivation of energy level finite square well through the Schrodinger equation. In the application section, we first introduced Bloch’s Law, a vital theorem for the following subsection. It describes the form of wave function under periodic potential. Then, the energy level of the finite square well is put into the application of the Kronig-Penney model, which is an infinite repetition of the finite square well. This part focus on the derivation of the energy level of the Kronig-Penney model. It applies both the energy level of the finite square well and Bloch’s Law. In the end, we note that the energy level of the Kronig-Penny model exhibits a band gap. It is an important phenomenon for lasers.