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Understanding single-slit experiments on the basis of Galton board experiments

In a single-slit experiment conducted for microparticles, the well-aligned rough structure of the slit wall can be viewed as a Galton board. Thus, when microparticles pass through the single slit, both the particle probability density (PPD) and particle direction of motion have a normal
 distribution. Therefore, when the distance between the slit and the receiving film becomes large, particles with different directions of motion will separate into different particle groups. By the nature of a normal distribution, the PPD for any particle group should also be normal-distributed.
 Obviously, between any two neighboring particle groups, there should be a valley in the PPD and thus the particle groups are observed as discrete fringes. All phenomena observed in the single-slit experiment can be explained reasonably well from the above viewpoint. In particular, analysis
 shows that the PPD can be described by the square of the modulus of the average least action of particles at a given location.