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arXiv:quant-ph/0109111 · Journal of Physics A Mathematical and General

Single- and double-slit scattering of wavepackets

G K$auml$lbermann

Published May 18, 2002• 14 citations
arXivHTMLPDF

Abstract

The quantum scattering of wavepackets from a single slit and a double slit with the Schrödinger equation is studied numerically and analytically. The phenomenon of diffraction of wavepackets in space and time in the backward region previously found for barriers and wells is encountered here also. A new phenomenon of forward diffraction that occurs only for packets thinner than the slit, or slits, is calculated numerically as well as in a theoretical approximation to the problem.

DOI
10.1088/0305-4470/35/21/309

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The quantum scattering of wavepackets from a single slit and a double slit with the Schrödinger equation is studied numerically and analytically. The phenomenon of diffraction of wavepackets in space and time in the backward region previously found for barriers and wells is encountered here also. A new phenomenon of forward diffraction that occurs only for packets thinner than the slit, or slits, is calculated numerically as well as in a theoretical approximation to the problem.
The phenomenon of wavepacket diffraction in space and time is described. It consists in a diffraction pattern whose spatial location progresses with time. The pattern is produced by wavepacket quantum scattering off an attractive or repulsive time-independent potential. An analytical formula for the pattern at t →∞ is derived both in one dimension and in three dimensions. The condition for the pattern to exist is developed. The phenomenon is shown numerically and analytically for the Dirac equation in one dimension also. An experiment for the verification of the phenomenon is described and simulated numerically.
The phenomenon of wave packet diffraction in space and time is investigated numerically and analytically for a one-dimensional array of equally spaced finite-depth wells. Theoretical predictions for the lattice at long times and at low scattering energies coincide exactly with the results for a single well. At intermediate and short times compared to the classical passage time, the pattern shows both a broad diffractive pattern and an interference pattern inside each diffractive peak. The diffractive structure persists for this case to infinite time.
The theoretical results for the diffraction of electrons by a disk and a pair of rectangular slits is derived from the Schrödinger equation using results from time-dependent perturbation theory in both the strong and weak scattering limits. The double-slit ‘‘thought’’ experiment is often used in introductions to quantum mechanics as a demonstration that electrons have wave-like properties. It seems desirable to include a derivation of the result both as an interesting example and also as reassurance that the diffraction of ‘‘matter waves’’ leads to the same results as the diffraction of electromagnetic waves by the same obstacle.
Abstract The goal of this paper is to study double-slit diffraction at the nanometric scale. We treat the time evolution of a Gaussian wave packet in quantum mechanics traveling until it reaches a double-slit wall. A detailed investigation and analysis of the diffraction phenomenon and its progress is developed by numerically solving the Schrödinger equation. This is written using a potential intended to model the wall. It is solved by using finite differences in the time domain method, so that the probability amplitude, ψ , is obtained as a function of the spatial coordinates at discretized times. Then, the probability density, ρ , is found in a straightforward way and used to plot graphics or make videos. Furthermore, the time evolution of the wave packet is registered at several fixed points to acquire local information. Fourier transforms are calculated to complement the knowledge obtained. Students interested in didactic approaches, or in applications for nanodevices, could find interest in this approach.
The diffraction pattern by double slit is studied by incoming Gaus-sian wave packet subjected to scattering interaction with the wall and a truncation assumption when crossing the slits. Explicit results are valid for sufficiently smooth scattering amplitude. They concern large and narrow aperture of the incoming wave packet. For wave packet localized in spatial region smaller then the aperture of the slits there is no diffraction and the particle behaves classically. For wave packet peaked in the momentum probability distribution there are maxima, superimposed to the canonical diffraction pattern, in correspondence to the edges of the slits. Previous similar results are generalized and improved.
We performed time-evolved numerical simulation of a two-dimensional electron wave packet through a double slit by solving the time-dependent Schrödinger equation using the finite difference method to simulate the electron interference phenomenon of a single electron. The propagating directions of interference peaks show that the starting point of electron interference is the center between the two slit exits. The number of interference peaks and their angles are found to be independent of potential barrier thickness. These results are compared with the results of our previous calculation of electron diffraction through a single slit.