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International Journal of Nonlinear Sciences and Numerical Simulation

The Feynman Path Integral and Ε-Infinity from the Two-slit Gedanken Experiment

Mohamed Saladin El Naschie

Published Jan 1, 2005• 45 citations
Publisher
DOI
10.1515/ijnsns.2005.6.4.335

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In this paper we describe a complete numerical treatment of the two-slit experiment for quantum mechanics. We first introduce the two-slit problem and the Fraunhofer approximation. We describe a number of experimental approaches, including the experiment of F. Shimizu, K. Shimizu, and H. Takuma in which they subject ultracold neon atoms to the two-slit experiment. We then present the Feynman Path Integral Method and its application to the two-slit problem. Using this method, we create a toolkit to simulate this experiment, before expanding this program. We explore the visualizations created by our program, and we compare the behaviors of particles simulated in different experiments and compare the behaviors to the Fraunhofer approximation. Finally, we note how this program can then be used to simulate a variety of particles and initial conditions and their subsequent behaviors with different gratings in both the near and far regions.
An extension of Feynman's rule, relating probability amplitudes and probabilities in two-photon interferometry, has been proposed recently [G. Jaeger et al., Phys. Rev. A 48, 1023 (1993)]. It would be interesting to analyze the validity of this rule in other physical situations not considered by these authors. In particular, we show in this paper that the extended rule cannot be applied to the gedanken two-slit experiment. This result suggests the existence of two quantum regimes, differentiated by the validity of the generalized rule. The distinguishability of one particle and the visibilities of one-particle and two-particle interferences are calculated. \textcopyright{} 1996 The American Physical Society.