In this paper, the results of the analysis of the 1-dimensional potential barrier case are presented when the energy of the incident beam is equal to the potential of the system (𝐸 = 𝑉 0 ).The analysis was carried out using a computational simulation based on Python programming by reviewing the values of the reflection and transmission coefficients and their influence on the amplitude of the wave function.As an example case for simulation, the potential value of the system 𝑉 0 = 60 × 10 -39 J is taken.First, an approach is made when the system energy E approaches the potential 𝑉 0 from below (𝐸 ≲ 𝑉 0 ), then an approach is made when the system energy E approaches the potential 𝑉 0 from above (𝐸 ≳ 𝑉 0 ).From these two approaches, the result is that for both the 𝐸 ≲ 𝑉 0 and 𝐸 ≳ 𝑉 0 cases, the graph shape is almost the same with slightly different amplitudes.The amplitude for the system with 𝐸 = 61 × 10 -39 J is larger than for the system 𝐸 = 59 × 10 -39 J.An important event is that when the energy of the incident beam is equal to the potential of the system (𝐸 = 𝑉 0 ), the reflection and transmission coefficient values are undefined so that no amplitude appears in the visualization of the programming results.Thus, it is impossible to obtain information about the particle beam in the potential region 𝑉 0 .
In this paper, the results of the analysis of the 1-dimensional potential barrier case are presented when the energy of the incident beam is equal to the potential of the system (𝐸 = 𝑉 0 ).The analysis was carried out using a computational simulation based on Python programming by reviewing the values of the reflection and transmission coefficients and their influence on the amplitude of the wave function.As an example case for simulation, the potential value of the system 𝑉 0 = 60 × 10 -39 J is taken.First, an approach is made when the system energy E approaches the potential 𝑉 0 from below (𝐸 ≲ 𝑉 0 ), then an approach is made when the system energy E approaches the potential 𝑉 0 from above (𝐸 ≳ 𝑉 0 ).From these two approaches, the result is that for both the 𝐸 ≲ 𝑉 0 and 𝐸 ≳ 𝑉 0 cases, the graph shape is almost the same with slightly different amplitudes.The amplitude for the system with 𝐸 = 61 × 10 -39 J is larger than for the system 𝐸 = 59 × 10 -39 J.An important event is that when the energy of the incident beam is equal to the potential of the system (𝐸 = 𝑉 0 ), the reflection and transmission coefficient values are undefined so that no amplitude appears in the visualization of the programming results.Thus, it is impossible to obtain information about the particle beam in the potential region 𝑉 0 .
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x) exceeds the value of the total energy E, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=a at about the time at which the incident packet reaches the point x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.
Abstract We consider a quantum particle with energy E incident upon a one-dimensional potential. We show that the probabilities of transmission and reflection are the same for incidence upon a general potential from either side (from ‘the left’ or ‘the right’). This equality holds true for any potential which goes to constant values as x → ± ∞ and is finite for all x . We present a remarkably simple proof that the probabilities are equal. The simplicity of our proof is the most important pedagogical result of our paper, and will be easily understood by undergraduate students in second to fourth year. We discuss several cases, including the step potential and the potential barrier.
The interaction between incident and target particles is inferred from the numbers and angular distribution of scattered particles and from their energy distribution. It is important to recognize that a scattering region, as represented for example by a rectangular barrier, is a region where, in classical terms, forces are applied to the incident particles. A full analysis of scattering events must, in principle, embody the description of a localized free particle that travels toward a center of force and is scattered by it. Quantum-mechanically, that implies a theoretical analysis in terms of packet states, with an explicit account of their change of position with time. One of the most striking effects described by quantum mechanics is the transmission of incident particles across a potential barrier whose height is greater than the particle energy. Classically, these particles lack the energy to get over the barrier, yet for quantum-mechanical systems some fraction of incident particles are transmitted.
Strongly singular potentials in one dimension are analyzed, namely the n th derivative interaction x n .For the repulsive case, the reflection and transmission coefficients are derived.It is shown that for n = even, these coefficients satisfy the unitarity of the scattering matrix for all values of the incident energy and the barrier becomes perfectly reflective for a specific value of the energy.For n=odd 1 , however, the coefficients do not satisfy the unitarity of the scattering matrix except for one specific value of the incident energy at which the barrier becomes perfectly reflective.For the n=1 case, the results showed that the coefficients satisfy the unitarity of the scattering matrix.For the attractive case, both the bound states and the scattering states are examined.It is shown that, for n=even 0 , there exist two bound states and only one bound state for n=0.For the scattering states, it is demonstrated that one recovers the same coefficients that were obtained for the repulsive case.
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.
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.