I have read for example here that the single slit experiment can be seen as a visualisation(?) of Heisenberg's Uncertainty principle. Basically, the photons passing through the slit are given a fixed value for the standard deviation of position. Therefore, the standard deviation for momentum is "fixed" and, given a statistically large amount of photons, the standard deviation for their momentum is made "visible" as the width of the central fringe, where the central fringe is the expected value of momentum for the photons.
If this is correct, why isn't the intensity pattern on the screen distributed according to the normal distribution, but shows individual fringes?