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Вісник Національного технічного університету України Київський політехнічний інститут. Серія: Радіотехніка. Радіоапаратобудування

Impedance models of double well structures

М. А. Гіндікіна, M. V. Vodolazka, Ю. Ф. Адаменко, Евгений Андреевич Нелин

Published Mar 30, 2015• 3 citations
Publisher

Abstract

Introduction. In this paper the impedance models double well structures (DWS) which is based on δ-inhomogeneities and rectangular potential are developed. The models of double well potential which is based on δ-functions and rectangular depending allow to obtain analytical solutions and to investigate important features of DWS. Analogy of double well structure and coupled oscillatory circuits. The analogy of DWS and coupled oscillatory circuits are considered. It is presented comparison the Schrödinger equation for the wave function and the equation for the current circuit without loss. It is shown that DWS as the system with two states performs the function of logical operations similar bits in classical computer. Models based on the impedance δ-inhomogeneities. Two models of DWS based on the impedance δ-inhomogeneities: δ-barrier in the potential well and two δ-wells are developed. The analytical expressions for input impedance and eigenvalues are received and investigated. It is shown that characteristics of DWS with finite size and δ-inhomogeneities agree well. Eigenvalues of asymmetric double well structure. The most general case of asymmetric DWS with a rectangular potential is considered. The eigenvalues of such a structure on the basis of a generalized model of barrier structures are found. Dependences of eigenvalues of symmetric and asymmetric DWS are presented. Conclusions. Impedance models allow to obtain the analytical solutions with substantial generalization problems in comparison with known traditionally solved problems. By analysis of input impedance characteristics conditions for the eigenvalues DWS placed between environments with different wave impedance are received.

DOI
10.20535/radap.2015.60.131-140

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Introduction. In this paper the impedance models double well structures (DWS) which is based on δ-inhomogeneities and rectangular potential are developed. The models of double well potential which is based on δ-functions and rectangular depending allow to obtain analytical solutions and to investigate important features of DWS. Analogy of double well structure and coupled oscillatory circuits. The analogy of DWS and coupled oscillatory circuits are considered. It is presented comparison the Schrodinger equation for the wave function and the equation for the current circuit without loss. It is shown that DWS as the system with two states performs the function of logical operations similar bits in classical computer. Models based on the impedance δ-inhomogeneities. Two models of DWS based on the impedance δ-inhomogeneities: δ-barrier in the potential well and two δ-wells are developed. The analytical expressions for input impedance and eigenvalues are received and investigated. It is shown that characteristics of DWS with finite size and δ-inhomogeneities agree well. Eigenvalues of asymmetric double well structure. The most general case of asymmetric DWS with a rectangular potential is considered. The eigenvalues of such a structure on the basis of a generalized model of barrier structures are found. Dependences of eigenvalues of symmetric and asymmetric DWS are presented. Conclusions. Impedance models allow to obtain the analytical solutions with substantial generalization problems in comparison with known traditionally solved problems. By analysis of input impedance characteristics conditions for the eigenvalues DWS placed between environments with different wave impedance are received.

Вісник Національного технічного університету України Київський політехнічний інститут. Серія: Радіотехніка. Радіоапаратобудування•Jan 1, 2015•0 citations

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Introduction. In this paper the impedance models double well structures (DWS) which is based on δ-inhomogeneities and rectangular potential are developed. The models of double well potential which is based on δ-functions and rectangular depending allow to obtain analytical solutions and to investigate important features of DWS. Analogy of double well structure and coupled oscillatory circuits. The analogy of DWS and coupled oscillatory circuits are considered. It is presented comparison the Schrödinger equation for the wave function and the equation for the current circuit without loss. It is shown that DWS as the system with two states performs the function of logical operations similar bits in classical computer. Models based on the impedance δ-inhomogeneities. Two models of DWS based on the impedance δ-inhomogeneities: δ-barrier in the potential well and two δ-wells are developed. The analytical expressions for input impedance and eigenvalues are received and investigated. It is shown that characteristics of DWS with finite size and δ-inhomogeneities agree well. Eigenvalues of asymmetric double well structure. The most general case of asymmetric DWS with a rectangular potential is considered. The eigenvalues of such a structure on the basis of a generalized model of barrier structures are found. Dependences of eigenvalues of symmetric and asymmetric DWS are presented. Conclusions. Impedance models allow to obtain the analytical solutions with substantial generalization problems in comparison with known traditionally solved problems. By analysis of input impedance characteristics conditions for the eigenvalues DWS placed between environments with different wave impedance are received.
Introduction. In this paper the impedance models double well structures (DWS) which is based on δ-inhomogeneities and rectangular potential are developed. The models of double well potential which is based on δ-functions and rectangular depending allow to obtain analytical solutions and to investigate important features of DWS. Analogy of double well structure and coupled oscillatory circuits. The analogy of DWS and coupled oscillatory circuits are considered. It is presented comparison the Schrodinger equation for the wave function and the equation for the current circuit without loss. It is shown that DWS as the system with two states performs the function of logical operations similar bits in classical computer. Models based on the impedance δ-inhomogeneities. Two models of DWS based on the impedance δ-inhomogeneities: δ-barrier in the potential well and two δ-wells are developed. The analytical expressions for input impedance and eigenvalues are received and investigated. It is shown that characteristics of DWS with finite size and δ-inhomogeneities agree well. Eigenvalues of asymmetric double well structure. The most general case of asymmetric DWS with a rectangular potential is considered. The eigenvalues of such a structure on the basis of a generalized model of barrier structures are found. Dependences of eigenvalues of symmetric and asymmetric DWS are presented. Conclusions. Impedance models allow to obtain the analytical solutions with substantial generalization problems in comparison with known traditionally solved problems. By analysis of input impedance characteristics conditions for the eigenvalues DWS placed between environments with different wave impedance are received.
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.
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.
Introduction. Comparative analysis of traditional and impedance approaches in modeling asymmetric potential quantum-mechanical barrier is fulfilled. Features of waves tunneling. The effect of waves tunneling is considered. The features of quantum-mechanical wave tunneling through asymmetric potential barrier are illustrated. Traditional approach. Quantum-mechanical wave transmission through asymmetric potential barrier on the basis of quantum-mechanical approach is explored. The expression for reflec-tion coefficient is obtained. Impedance approach. The expression of the reflection coefficient using the impedance meth-od, which is based on the concept of quantum-mechanical impedance and the theory of transmission lines is obtained. The identity of expressions received by two approaches is demonstrated. Tunnling regime. The expressions for transmission coefficient for tunneling regime in exact and approximate forms are obtained. Applicability of the approximate formula is presented. Matched overbarrier transmission. The conditions of reflectionless resonant and nonresonant overbarrier transmission are obtained. Dependences of transmission coefficient for asymmet-ric barrier that illustrate unmatched and matched overbarrier transmission are presented. Conclusions. Impedance method significantly simplifies modeling of quantum-mechanical structures in comparison with the traditional method of solving quantum-mechanical prob-lems.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.