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.
The aim of the present lecture note is not to propose an alternative formulation of quantum mechanics, but to emphasize that several apparently distinct descriptions of the double-slit experiment may be understood as complementary consequences of a single geometric projection onto the transmission subspace defined by the apertures. Modeling the double slit as a projection onto the two-dimensional transmission subspace spanned by the localized slit modes provides a common mathematical framework for the wave-based, particle-based and information-theoretic descriptions of interference. The formalism starts from an arbitrary incident localized wave packet and naturally reproduces Duane's momentum-selection law, which emerges directly from the geometric transmission condition rather than being introduced as an independent postulate.