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American Journal of Physics

Particle Diffraction by a Single Slit as Error Propagation

J. E. Purcell

Published Dec 1, 1974• 3 citations
Publisher

Abstract

An introductory approach to wave packet spreading is discussed which uses error propagation methods generally known to undergraduates. The specific example of particle diffraction by a single slit is used to illustrate the method. The corresponding quantum theory results are reviewed and sufficient conditions for the vanishing of a term linear in time are discussed.

DOI
10.1119/1.1987949

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An introductory approach to wave packet spreading is discussed which uses error propagation methods generally known to undergraduates. The specific example of particle diffraction by a single slit is used to illustrate the method. The corresponding quantum theory results are reviewed and sufficient conditions for the vanishing of a term linear in time are discussed.
We study the quantum slit diffraction problem in three dimensions. In the treatment of diffraction of particles by a slit, it is usually assumed that the motion perpendicular to the slit is classical. Here we take into account the effect of the quantum nature of the motion perpendicular to the slit using the Green function approach [18]. We treat the diffraction of a Gaussian wave packet for general boundary conditions on the shutter. The difference between the standard and our three-dimensional slit diffraction models is analogous to the diffraction in time phenomenon introduced in [16]. We derive corrections to the standard formula for the diffraction pattern, and we point out situations in which this might be observable. In particular, we discuss the diffraction in space and time in the presence of gravity.
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.
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.
We revisit multi-slit diffraction with a scaling of space and time to exploit an equivalence of wave packets describing quantum free and harmonic oscillator (HO) motion. We introduce a co-moving frame of space and time coordinates to define an effective, time-independent, HO potential that confines and directs initially displaced wave packets along the classical phase space of the oscillator. NNN-slit diffraction in the lab frame from the slits to a distant detector and the familiar spreading of the propagating wave front is then described as the propagation of HO wave packets over just a quarter cycle of the oscillator and confined to the HO potential well. This connection with the co-moving frame suggests a simplified time of flight experiment wherein the arrival time distribution of diffracted particles hitting a \emph{single point detector} in the lab frame images in the co-moving frame as the \emph{full} diffraction pattern from grating to detector.
The evolution of the free-particle wave function in one dimension is the same as scalar Fresnel diffraction from a one-dimensional structure. Quantum mechanics courses often explore the propagation of Gaussian wave packets, but the diffractionlike mathematics is sufficiently tractable to investigate the propagation of other wave packets, both numerically and analytically. More importantly, the diffraction analogy facilitates the development of an intuitive understanding of the role that the initial phase plays in free-particle wave packet evolution. This article considers some of the effects of the initial phase function on the subsequent evolution of free-particle wave packets in the position representation. These considerations reinforce the idea that the classical mechanics limit embodied in the correspondence principle and formalized in the Ehrenfest theorem is necessarily an incomplete representation of quantum behavior.