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Zeitschrift für Naturforschung A

Transverse Momentum Distribution of Atoms in an Interferometer

Mirijana Božić, D. Arsenović, Lepša Vušković

Published Feb 1, 2001• 7 citations
Publisher

Abstract

Abstract From the stationary solution of Schrödinger's equation in an interferometer we derive the wave functions of the longitudinal and the transverse motion. The former function is a plane wave. The wave function of the transverse motion is a one dimensional wave packet representing a super­ position of components with various values of the particles transverse momentum. The particles transverse momentum distribution in an interferometer is time independent and is determined by the aperture wave function. Consequently, it is independent of the distance from the slits. As such, it is a very important characteristic of the quantum state. Experimental determination of the mo­ mentum distribution would support the particle and wave interpretation of quantum interference in a new manner.

DOI
10.1515/zna-2001-0128

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Abstract From the stationary solution of Schrödinger's equation in an interferometer we derive the wave functions of the longitudinal and the transverse motion. The former function is a plane wave. The wave function of the transverse motion is a one dimensional wave packet representing a super­ position of components with various values of the particles transverse momentum. The particles transverse momentum distribution in an interferometer is time independent and is determined by the aperture wave function. Consequently, it is independent of the distance from the slits. As such, it is a very important characteristic of the quantum state. Experimental determination of the mo­ mentum distribution would support the particle and wave interpretation of quantum interference in a new manner.
Quantum theory of interference phenomena does not take the diameter of the particle into account, since particles were much smaller than the width of the slits in early observations. In recent experiments with large molecules, the diameter of the particle has approached the width of the slits. Therefore, analytical description of these cases should include a finite particle size. The generic quantum interference setup is an asymmetric double slit interferometer. We evaluate the wave function of the particle transverse motion using two forms of the solution of Schrodinger's equation in an asymmetric interferometer: the Fresnel-Kirchhoff form and the form derived from the transverse wave function in the momentum representation. The transverse momentum distribution is independent of the distance from the slits, while the space distribution strongly depends on this distance. Based on the transverse momentum distribution we determined the space distribution of particles behind the slits. We will present two cases: a) when the diameter of the particle may be neglected with respect to the width of both slits, and b) when the diameter of the particle is larger than the width of the smaller slit.
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In the experiments considered here, we measure the y-component of momentum for a particle passing through a system of slits. The source-slit system is the preparation apparatus that determines the state vector. Recognizing that a system of slits is a position-measuring device allows us to ascertain that the state vector is a position state. Then, writing the state vector in momentum space provides a straightforward calculation for the probability amplitude and its corresponding probability function. Interference effects, if any, are inherent in the probability function We determine the statistical distribution of scattered particles for four different slit systems. The results are in agreement with the well-known interference patterns obtained in classical wave optics.
The Schrodinger equation based on the de Broglie wave is the most fundamental equation of the quantum mechanics. There can be no doubt about it's prediction validity. However, the probabilistic interpretation on the quantum mechanics has insoluble semantic interpretations like reduction of wave packet on observations of physical values. Especially, it is not clear that the wave function which is described by complex function, is whether formality or reality to express the state of particle motion. On this paper, we interpret the wave nature of particle as not the inherency of particle itself, but the motional property of particle in fluctuated space-time due to the kinetic energy and momentum the belief that the kinetic energy and momentum fluctuates the microscopic space-time, and the particle move through the fluctuated space-time adversely. Then, through the particle motion in the Euclidean space-time, the particle will be recognized as if it has the wave nature. We estimate the governing equation of fluctuations of microscopic space-time based on the macroscopic law of motion. On this paper, the equivalence between the governing equation and the Schrodinger equation is indicated.
Abstract This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.
Time-independent and time-dependent forms of Schrodinger equations are presented in one and three dimensions along with their interpretations. Also, the time-independent version is shown to follow from the time-dependent version. For better conceptualization of wave motion, the general equation of wave motion is discussed and solved, explaining the physical significance of variables involved. Born interpretation of the wave function as a measure of the probability of finding the particle in a region of space at a given instant is described. The requirements to be fulfilled by the wave function for representing a physical world system are outlined.