Understanding the Double Slit Experiment without Path Integrals

I did once look at path integrals, but they are really just a complicated way of doing wave mechanics and I never found them useful. The double slit is much easier using basic methods, and I am not sure why your books would approach it any other way. There is no potential (or constant potential if you prefer), so the solution of the Schrodinger equation is a simple wave. The wave functions from the two slits sum, so one gets exactly the same pattern as for the classical Young's slit experiment.

In the figure, where wave crests (light grey) meet crests and troughs (dark grey) meet troughs, the amplitude of the wave increases (white crests, black troughs) creating a bright area on the screen. Where crests meet troughs, the waves cancel out (mid-grey), leading to a dark region.

The only difference between the classical interpretation and quantum mechanics is that in quantum mechanics the wave function is simply a way of calculating the probability for where a particle will be observed on the screen, as shown when particles pass through the slits one at a time.

Result of the Young’s slits experiment using individual electrons, as carried out by Dr. Tonomura in 1989, showing the build-up of an interference pattern of single electrons. Numbers of electrons are 200 (b), 6000 (c), 40000 (d), 140000 (e).