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arXiv:1001.0661

N-slit interference: Path integrals, Bohmian trajectories

Valeriy I. Sbitnev

Published Jan 5, 2010• 2 citations
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Abstract

Path integrals give a possibility to compute in details routes of particles from particle sources through slit gratings and further to detectors. The path integral for a particle passing through the Gaussian slit results in the Gaussian wavepacket. The wavepackets prepared on N slits and superposed together give rise to interference pattern in the near-field zone. It transforms to diffraction in the far-field zone represented by divergent principal rays, at that all rays are partitioned from each other by (N-2) subsidiary rays. The Bohmian trajectories in the near-field zone of N-slit gratings show wavy behavior. And they become straight in the far-field zone. The trajectories show zigzag behavior on the interference Talbot carpet (ratio of particle wavelength to a distance between slits are much smaller than 1 and N>>1). Interference from the the N-slit gratings is simulated by scattering monochromatic neutrons (wavelength=0.5 nm). Also we have considered simulation of interference fringes arising at scattering on an N-slit grating of fullerene molecules (according to the real experiment stated in e-print 1001.0468).

DOI
10.48550/arxiv.1001.0661

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Path integrals give a possibility to compute in details routes of particles from particle sources through slit gratings and further to detectors. The path integral for a particle passing through the Gaussian slit results in the Gaussian wavepacket. The wavepackets prepared on N slits and superposed together give rise to interference pattern in the near-field zone. It transforms to diffraction in the far-field zone represented by divergent principal rays, at that all rays are partitioned from each other by (N-2) subsidiary rays. The Bohmian trajectories in the near-field zone of N-slit gratings show wavy behavior. And they become straight in the far-field zone. The trajectories show zigzag behavior on the interference Talbot carpet (ratio of particle wavelength to a distance between slits are much smaller than 1 and N>>1). Interference from the the N-slit gratings is simulated by scattering monochromatic neutrons (wavelength=0.5 nm). Also we have considered simulation of interference fringes arising at scattering on an N-slit grating of fullerene molecules (according to the real experiment stated in e-print 1001.0468).
Scattering cold particles on an NNN-slit grating is shown to reproduce an interference pattern, that manifests itself in the near-field region as the fractal Talbot carpet. In the far-field region the pattern is transformed to an ordinary diffraction, where principal beams are partitioned from each other by (N−2N-2N−2) weak ones. A probability density plot of the wave function, to be represented by a gaussian wavepacket, is calculated both in the near-field region and in the far-field one. Bohmian (geodesic) trajectories, to be calculated by a guidance equation, are superimposed on the probability density plot well enough. It means, that a particle, moving from a source to a detector, passes across the grating along a single bohmian trajectory through-passing one and only one slit. Keywords: Gaussian wavepacket, neutron scattering, guidance equation, bohmian trajectory, near-field interference, far-field diffraction, Talbot carpet, fractal
Paths of particles, emitted from distributed sources and passing out through slits of two gratings, G0G_{0}G0​ and G1G_{1}G1​, up to detectors, have been computed in details by the path integral method. The slits are represented by Gaussian functions that simulate fuzzy edges of the slits. Waves of matter be computed by this method show perfect interference patterns both between the gratings and behind the second grating. Coherent and noncoherent the distributed particle sources reproducing the interference patterns are discussed in details. Paraxial approximation stems from the wave function when removing the distributed sources onto infinity. The more hard-edged slits of the grating G1G_{1}G1​ are examined by simulating those slits by a superposition of more hard the Gaussian functions. As for the particles here we consider fullerene molecules. De Broglie wavelength of the molecules is adopted equal to 5 pm.
Particle paths, emitted from distributed sources and passing out through slits of two gratings, G0 and G1, up to detectors, have been computed in detail by the path integral method. The particles under consideration are fullerene molecules with a De Broglie wavelength equal to 5 pm. The slits are Gaussian functions that simulate fuzzy edges of the slits. Waves of the matter computed by this method show perfect interference patterns both between the gratings and behind the second grating. Coherent and non-coherent distributed particle sources reproducing the interference patterns are discussed in detail. Paraxial approximation results from removing the distributed sources onto innity. This approximation gives a wave function reproducing an exact copy of the Talbot carpet. PACS numbers: 03.75.-b, 03.75.Dg, 42.25.Hz
Single particle interference experiments have documented the existence of trajectory of particles ’ during the accumula-tion of an interference pattern. The expression for probability of particle’s arrivals to the screen was derived previously [22] by approximating the trajectories with straight lines, which start at various points of the slits, and using wave function of the transverse motion in the momentum representation. We use this expression to determine particle distribution at various distances along the lines parallel to a multiple slit grating. The agreement with usual probability density given by the mod-ulus square of the wave function in coordinate representation is very good in the far field. In the near field the agreement is poor. This result supports the view that the trajectories of the particles do exist and that they are straight lines in the far fields, but further study on the form of the trajectories of particles ’ in the near field is necessary. 164 Concepts of Physics, Vol. II (2005) On the (non)existence of quantum particles’...
Diffraction and interference of matter waves are key phenomena in quantum mechanics. Here we present some results on particle diffraction in a wide variety of situations, ranging from simple slit experiments to more complicated cases such as atom scattering by corrugated metal surfaces and metal surfaces with simple and isolated adsorbates. The principal novelty of our study is the use of the so-called Bohmian formalism of quantum trajectories. These trajectories are able to satisfactorily reproduce the main features of the experimental results and, more importantly, they provide a causal intuitive interpretation of the underlying dynamics. In particular, we will focus our attention on: (a) a revision of the concepts of near and far field in undulatory optics; (b) the transition to the classical limit, where it is found that although the quantum and classical diffraction patterns tend to be quite similar, some quantum features are maintained even when the quantum potential goes to zero; and (c) a qualitative description of the scattering of atoms by metal surfaces in the presence of a single adsorbate.
For various experiments to investigate quantum effects, matter-wave diffraction is an apparent example. Particle diffractions have been measured even at the large molecular level, such as the C 60 molecules. In order to describe the matter-wave diffraction of C 60 beam, we assume the initial wave functions behind a grating in form of the Fourier series with a Gaussian distribution function. By applying the Feynman path integral, the exact wave functions of the molecule diffraction in the near-field regime can be derived analytically. The obtained probability density distributions are corresponding to interference fringes as found in the Fresnel diffraction. The probability of finding C 60 molecules, behind the grating at the Talbot length, consistency with the mentioned experimental data of C 60 diffractions.