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Difraction of particle on slit with internal structure

Matěj Hudec

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

In this thesis we study the difraction of a particle on a slit, focusing on the resonance effects. The two-dimensional model containing a delta-function barrier with variable height which represents the slit is solved on a belt with periodic boundary condition. In most cases we use the numerical solution of the problem using our program. The results are compared with elastic approximations. We study the characteristics of the resonances in detail, the dependence of their existence, energy and height on the parameters of the slit, and the relation between resonances and bound states. The last chapter is devoted to the problem of spreading the belt in order to get the results that describe the scattering process on the whole plain. We suggest two ap- proaches of description of this process and demonstrate that the appearance of the resonances is not limited to the given boundary condition. 1

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In this thesis we study the difraction of a particle on a slit, focusing on the resonance effects. The two-dimensional model containing a delta-function barrier with variable height which represents the slit is solved on a belt with periodic boundary condition. In most cases we use the numerical solution of the problem using our program. The results are compared with elastic approximations. We study the characteristics of the resonances in detail, the dependence of their existence, energy and height on the parameters of the slit, and the relation between resonances and bound states. The last chapter is devoted to the problem of spreading the belt in order to get the results that describe the scattering process on the whole plain. We suggest two ap- proaches of description of this process and demonstrate that the appearance of the resonances is not limited to the given boundary condition. 1
Scattering of a quantum particle with internal structure is fundamentally different from that of a point particle and shows quantum effects such as the modification of transmission due to tunnelling and trapping of the particle. As in a preceding paper (Shore et al 2014 New J. Phys. 17 013046 ) we consider a model of a symmetric, rigid rotor travelling through an aperture in a thin but impenetrable screen which is perpendicular to both the direction of motion and the rotation axis. We determine the quantum mechanical properties of this two-dimensional geometrical model using a quasi one-dimensional scattering problem with unconventional boundaries. Our calculations rely on finding the Green's function, which has a direct connection to the scattering matrix. Evaluated on a discrete lattice the Hamiltonian is 'dressed' by a self-energy correction that takes into account the open boundary conditions in an exact way. We find that the passage through the aperture can be suppressed or enhanced as a result of the rotational motion. These effects manifest themselves through resonances in the transmission probability as a function of incident energy and symmetry of the incident wavefunction. We determine the density-of-states to reveal the mode structure of resonant states and to exhibit the lifetimes of temporary trapping within the aperture.
In this article we hold on an analytic solution of the well-known cases of difraction and interference of electrons through one and two slits (simply that, the one-dimensional case is assumed only). In addition, we hold an approximations of the electron distribution which offer the interpretation of the results. Our derivation is based on the Feynman path integral formula and this work could also serve an awesome introduction to multiple slits interference. Then it is comparing between theoretical results and simulation in order to get interference pattern of it.
The immersing method is used to solve the two-channel scattering problem in the case of concrete potential. In particular, we consider the particle scattering by a two-dimensional barrier which is constant in the scattering direction and is arbitrary in the transverse direction. For this case the scattering amplitudes t 1, t 2, r 1, and r 2 are determined. Expressions for the transmission and reflection amplitudes are obtained for the case of δ-potential. The behavior of the scattering amplitudes in the limit k 2 → 0 is studied. It is also shown that the ratio of products of transmission and reflection amplitudes for two channels does not depend on the coordinate of the middle of coordinate.
Classically, rigid objects with elongated shapes can fit through apertures only when properly aligned. Quantum-mechanical particles which have internal structure (e.g. a diatomic molecule) also are affected during attempts to pass through small apertures, but there are interesting differences with classical structured particles. We illustrate here some of these differences for ultra-slow particles. Notably, we predict resonances that correspond to prolonged delays of the rotor within the aperture—a trapping phenomenon not found classically.
Abstract We reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height V 0 with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e. E → V 0 − mc 2, for a given V 0. In such a case, the particle is actually colliding with an impenetrable barrier. In fact, V 0 → E + mc 2, for a given relativistic energy E( < V 0), is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we note that, unlike the nonrelativistic case, the entire eigensolution does not completely vanish, either at the barrier or in the region under the step, but its upper component does satisfy the Dirichlet boundary condition at the barrier. More importantly, by calculating the mean value of the force exerted by the impenetrable wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. We use two different approaches to obtain the latter two results. In one of these approaches, the corresponding force on the particle is a type of boundary quantum force. Throughout the article, various issues related to the Klein energy zone, the transmitted solutions to this problem, and impenetrable barriers related to boundary conditions are also discussed. In particular, if the negative-energy transmitted solution is used, the lower component of the scattering solution satisfies the Dirichlet boundary condition at the barrier, but the mean value of the external force when V 0 → E + mc 2 does not seem to be compatible with the existence of the impenetrable barrier.
Abstract In this paper propagation of matter wave of particles is modeled using the Fourier optics approach. In first step the Schrödinger equation of quantum mechanics is used to find the wave function of the particle. In the second step Fourier optics is employed to model the diffraction of the wave function of the particle through single and double slits. The results of the calculations are presented as graphs of diffraction patterns. The results of the presented method are compared to the existing results in the literature (with different methods) to check the validity of the introduced model. It is shown that the Fourier optics approach is applicable to matter wave of particles in diffraction through slits.