Diffraction phenomenon occurs when the wavelength of the wave is comparable to the size of the obstacle. Tunneling through a barrier occurs when the particle size is comparable to the width of the barrier. The Transmission probability versus potential energy curve is hyperbolic in nature. This main aim of this article is to show that there is no probability of penetrating the particle through the barrier if barrier width is extremely larger and extremely smaller than the particle size. The condition for tunneling is similar to the necessary and sufficient condition for diffraction to occur. The various examples presented here shows that the penetration of the particle through the barrier is only possible when the size of the particle is comparable to the size of the barrier height. This article shows that probability occurrence, Wave-particle duality and uncertainty of a particle exists only when there is comparable size between the particle and its barrier width.
Diffraction phenomenon occurs when the wavelength of the wave is comparable to the size of the obstacle. Tunneling through a barrier occurs when the particle size is comparable to the width of the barrier. The Transmission probability versus potential energy curve is hyperbolic in nature. This main aim of this article is to show that there is no probability of penetrating the particle through the barrier if barrier width is extremely larger and extremely smaller than the particle size. The condition for tunneling is similar to the necessary and sufficient condition for diffraction to occur. The various examples presented here shows that the penetration of the particle through the barrier is only possible when the size of the particle is comparable to the size of the barrier height. This article shows that probability occurrence, Wave-particle duality and uncertainty of a particle exists only when there is comparable size between the particle and its barrier width.
Abstract This paper examines tunnelling through potential well protected by double barriers as well as the motion of a wave packet encountering this type of potential. Various options for the left and right barriers’ penetration factors are taken into consideration. When one of the penetration factors has limiting value of 0.5, the transmission coefficients turns out to be independent of incoming energy of the particle and the width of potential well. When both penetration factors have a limiting value of 0.5, a scenario similar to resonance tunneling is achieved. Furthermore, by doing the wave packet analysis, we found that the width and height of potential barriers have a negligible impact on the probability of finding the particle inside the potential well in the limit of large penetration factors.
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.
In this paper,transmission coefficient that the micro particle penetrate two potential barriers successively is calculated by solving the stationary schrodinger equation.Our numerical results show that the micro particle could penetrate potential barriers no matter energy of particle is taller or smaller than the height of potential barriers.But the transmission coefficient varies in several different circumstances.Particularly,it may arise resonance transmission in conditions.In a word,the transmission coefficient could changes with the height and width of two potential barriers,and have nothing to do with the distance between two potential barriers.
The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.
Quantum tunneling across double potential barriers is studied. With the assumption that the real space is a continuum, it is rigorously proved that large barriers of arbitrary shapes can be penetrated by low-energy particles with a probability of unity, i.e., realization of resonant tunneling (RT), by simply tuning the interbarrier spacing. The results are demonstrated by tunneling of electrons and protons, in which resonant and sequential tunneling are distinguished. The critical dependence of tunneling probabilities on the barrier positions not only demonstrates the crucial role of phase factors but also points to the possibility of ultrahigh accuracy measurements near resonance. By contrast, the existence of a nonzero minimum length puts upper bounds on the barrier size and particle mass, beyond which effective RT ceases. A scheme is suggested for dealing with the practical difficulties arising from the delocalization of particle position due to the uncertainty principle. This work opens a possible avenue for experimental tests of the existence of a minimum length based on atomic systems. Published by the American Physical Society 2024
A tennis ball is not expected to penetrate through a brick wall since a motion under a barrier is impossible in classical mechanics. With quantum effects a motion of a particle through a barrier is allowed due to quantum tunneling. According to usual theories of tunneling, the particle density decays inside a classical barrier resulting in an extremely slow pentration process. However, there are no general laws forbidding fast motion through classical barriers. The problem addressed is investigation of unusual features o quantum tunneling through a classic static barrier which is at least two-dimensional. Here we show that penetration through such barrier can be not slow. When the barrier satisfies the certain conditions, a regime of quantum lens is possible with formation of caustics. De Broglie waves are reflected from the caustics, interfere, and result in a not small flux from under the barrier. This strongly contrasts to the usual scenario with a decaying under-barrier density. We construct a particular example of fast motion through a classical barrier. One can unexectedly conclude that, in principle, nature allows fast penetration through classical barriers which against common sense. The phenomenon may be responsible for a variety of processes in labs and nature. For example, tunneling in solids may occur with a different scenario, in biophysics and chemistry one can specify conditions for unusual reactions, and evanescent optical waves may strongly change their properties. In condensed matter and cosmic physics there are phenomena with misterious reasons of an energy emission, for instance, gamma-ray bursts. One can try to treat them in the context of fast escape from under some barriers.