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arXiv:quant-ph/0305189

An asymmetric double-slit interferometer for small and large quantum particles

Bozic, Mirjana, Arsenovic, Dusan, Vuskovic, Lepsa

Published May 30, 2003• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0305189

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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.
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 before the rise of molecular interferometry. In the experiments with large molecules, the diameter of the particle has approached the width of the slits. Therefore, an analytical description of these cases should include a finite particle size. An asymmetric double-slit grating seems to be very suitable for the study of the influence of a particle's size on the interference pattern. We identify three characteristic cases for the ratio of slit widths ${\ensuremath{\delta}}_{1}$ and ${\ensuremath{\delta}}_{2}$ and the particle diameter DDD: $D⪡{\ensuremath{\delta}}_{1}$ and $D⪡{\ensuremath{\delta}}_{2}$, ${\ensuremath{\delta}}_{1}>D>{\ensuremath{\delta}}_{2}$, and $D>{\ensuremath{\delta}}_{1}>{\ensuremath{\delta}}_{2}$. Taking into account the influence of both slits on the particle wave function, regardless of through which slit the particle passed, we treat the particle-wall interaction in a simple fashion, such that if the particle size is greater than the slit opening there is no transmission. The transverse momentum distribution is independent of the distance from the slits and the particle size, while the space distribution strongly depends on this distance and the particle size. We found that the interference is absent only when the particle's diameter is larger than both slit widths, $D>{\ensuremath{\delta}}_{1}>{\ensuremath{\delta}}_{2}$.
In the experiments with large molecules, the diameter of particles has approached the width of the slits. Therefore, a theoretical description of these cases should include a finite particle size. We are developing a theory of quantum interference which takes into account the trajectories and the size of particles. The theory uses a distribution of transverse momenta of particles to describe emergence of an interference pattern during accumulation of single particle arrivals to the screen. In this way we are accomplishing the task which Barut assigned to theoretical physicists by writing: “It now becomes a task of theoretical physics to build models and theories of single events, and a task of experimental physics to record more of single events”.
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.
The standard quantum theory has not taken into account the size of quantum particles, the latter being implicitly treated as material points. The recent interference experiments of Zeilinger [3] with large molecules like fullerenes and the thought experiments of Bozic et al [7] with asymmetrical Young slits make it possible today to take into account the particle size. We present here a complete study of this phenomenon where our simulations show differences between the particles density after the slits and the modulus square of the wave function. Then we propose a crucial experiment that allows us to reconsider the wave-particle duality and to test the existence of the Broglie-Bohm trajectories for indistinguishable particles.
The particle- interference-experiments, with the photons and electrons have clearly shown that they are always detected as localized „particles‟; and the density of their detections exhibit a pattern similar to the interference of waves. This experimental-observation is now explained here, making the current, „probabilistic interpretation‟, a past history. The „particle ‟ is first mathematically characterized as an „impulse function ‟ in space; and then Fourier-transformed into the wave-number-domain, as a wide band of constituent-waves as shown in the fig. 1&2. The „particle ‟ is explained as an „event ‟ of constructive-superimposition of a wide band of „pre-quantum-waves‟. This „event ‟ occurs in space at regular distances; and at regular instances in time. We know that even the purest monochromatic laser source has a „line-width ‟ of a few Giga-Hertz, or at least a few kilo-Hertz. So in the particle-interference-experiments, they are these wide bands of waves which pass through both the slits, even in the case of single photon states, and interfere physically. This physical interference of the wide bands of pre-quantum-waves determines the position and time of emergence of the „particle‟. Thus, form the theoretical view-point, there are only the waves, and waves alone; and the „particle ‟ is a specific situation of constructive-superimposition of all the spectral-components of pre-quantum-waves; so there is no wave-particle-duality.
Unlike the usual interference experiment with two slits separated in space, a time double-slit experiment uses the equivalent of a single slit covered by a shutter that is opened and closed and then quickly opened and closed again. Such experiments have recently been performed, providing further confirmation of wave-particle duality. In either type of experiment, the wave function beyond the slit(s) initially has two peaks that then spread and interfere. This article derives the results of these experiments in a schematic, but analytic solution of the time-dependent Schrödinger equation with appropriate initial conditions. In the space double-slit case, in addition to the transverse interference pattern, the time evolution at a fixed position exhibits an oscillating transient behavior similar to that obtained in the time double-slit experiment. Finally, the schematic model also reproduces the observed progressive visibility of the interference pattern obtained as a movable mask is displaced in front of the double-slit.