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Doklady of the National Academy of Sciences of Belarus

ТУННЕЛИРОВАНИЕ ЧЕРЕЗ ГЛАДКИЙ ПАРАБОЛИЧЕСКИЙ БАРЬЕР КОНЕЧНОЙ ВЫСОТЫ

V. V. Kudryashov, Ágnes Baran

Published Jan 6, 2017• 0 citations
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Abstract

The smooth barrier of finite height and variable shape is constructed by means of joining the central inverted parabolic potential and two side parabolic potentials. The problem of tunneling through this barrier is solved exactly. The dependence of the transmission coefficient on energy is presented. The real and imaginary components of wave functions are shown.

Related papers

ТУННЕЛИРОВАНИЕ ЧЕРЕЗ ГЛАДКИЙ ПАРАБОЛИЧЕСКИЙ ДВОЙНОЙ БАРЬЕР

Ágnes Baran, V. V. Kudryashov

The exact description of tunneling is given for a smooth symmetric double barrier which is constructed with the help of both parabolic and inverted parabolic potentials. The analytical expression for transmission coefficient is found. The resonant tunneling condition is obtained. The dependence of transmission coefficient on incident particle energy is presented for different values of double barrier parameters. It is established that the number of resonances increases with growing the width of barriers and the distance between barriers. The continuous wave functions are expressed in terms of the confluent hypergeometric functions. The real and imaginary components of wave functions are shown at the resonance values of energy. The proposed smooth parabolic potential extends a very limited list of exactly solvable models that describe tunneling through double barriers. The variable shape of the considered double barrier gives the supplementary possibilities to simulate tunneling processes.

Doklady of the National Academy of Sciences of Belarus•Oct 5, 2017•0 citations

SEMICLASSICAL TUNNELING EFFECT

Johan Öhman

This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.

•Jan 1, 2014•0 citations

A Numerical Method for Quantum Tunneling

Ban Shi

A simle numerical method of calculating transmission coefficients across arbitrary potentials is presented by using the combination of a transfer matrix procedure and the solution of the wave functions in arbitrary potentials.This method is applied to a parabolic potential and a double barrier potential in an electric field.The transmission coefficients calculated are in agreement with those obtained using a multistep potential approximation.The wave functions at the resonant states are also given for the two examples.

Acta Scientiarum Naturalium Universitatis Neimongol•Jan 1, 2000•0 citations

Transmission coefficient in double barrier quantum tunnelling effect

Zhongzheng Duan, Wenxi Luo, Xiaohan Xu

At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.

Theoretical and Natural Science•Dec 19, 2023•0 citations

Analytical considerations of barrier transmission

John Heading

Abstract The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.

Mathematical Proceedings of the Cambridge Philosophical Society•Mar 1, 1972•5 citations

Порівняння традиційного та імпедансного методів моделювання квантово-розмірних структур

Є. А. Нелін, M. V. Vodolazka

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.

Вісник Національного технічного університету України Київський політехнічний інститут. Серія: Радіотехніка. Радіоапаратобудування•Jan 1, 2014•0 citations
The smooth barrier of finite height and variable shape is constructed by means of joining the central inverted parabolic potential and two side parabolic potentials. The problem of tunneling through this barrier is solved exactly. The dependence of the transmission coefficient on energy is presented. The real and imaginary components of wave functions are shown.
The exact description of tunneling is given for a smooth symmetric double barrier which is constructed with the help of both parabolic and inverted parabolic potentials. The analytical expression for transmission coefficient is found. The resonant tunneling condition is obtained. The dependence of transmission coefficient on incident particle energy is presented for different values of double barrier parameters. It is established that the number of resonances increases with growing the width of barriers and the distance between barriers. The continuous wave functions are expressed in terms of the confluent hypergeometric functions. The real and imaginary components of wave functions are shown at the resonance values of energy. The proposed smooth parabolic potential extends a very limited list of exactly solvable models that describe tunneling through double barriers. The variable shape of the considered double barrier gives the supplementary possibilities to simulate tunneling processes.
This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.
A simle numerical method of calculating transmission coefficients across arbitrary potentials is presented by using the combination of a transfer matrix procedure and the solution of the wave functions in arbitrary potentials.This method is applied to a parabolic potential and a double barrier potential in an electric field.The transmission coefficients calculated are in agreement with those obtained using a multistep potential approximation.The wave functions at the resonant states are also given for the two examples.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
Abstract The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.
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.