Think a double-slit experiment.
From a source of single slit to the two slits, a cylinderical wave function is radiated. From the two slits, two cylinderical wave functions are emitted secondarily. They interfere with each other. But what a dramatic effect the interference causes. Attached please open a file.
The figure 1 is the case in which there is no interference. There is only almost uniform distribution of cloud. On the contrary, the figure 2 is the case in which there is intereference. There can be seen several vivid beams of probability density. In figure 1, the following equation was calculated.
In figure 2, the following equation was caculated.
where Y1R is the real part of the wave function emitted from the slit 1, Y1I the imaginary part, Y2R the real part of the wave function emitted from the slit 2 and so on. So the dramatic effect seems to have originated in the componentwise addition in both of the real and imaginary parts before squaring.
P.S. Figure 3 is the hydrodynamic surface wave case where the slit 2 was closed. It is just the ripples spreading on the surface of a still pond caused by one dropped stone. Figure 4 is the case in which two stones were dropped on a pond simultaneously.
The interference pattern in a double slit-experiment is different between the quantum mechanical case and hydrodynamic case.