We have talked about particles and waves, and the fact that in quantum mechanics everything is a wave. But what is the important property that waves have but particles do not?
Before we come to this important difference, let us note that there are many similarities between particles and waves:
The relevant property of waves is that waves can cancel. Suppose two water waves are coming to the same place. Let one have a crest at this place, and the other a trough. Then the waves will cancel at this place; i.e., that the water will neither go up nor down.
The waves in quantum mechanics have a similar cancellation property. This is in sharp contrast to what would happen with particles: if two particles reach a given place then we will have two particles there; they cannot cancel out to leave zero particles.
Let us now see the cancellation of waves more explicitly.Since electrons are waves, the cancellation of Fig.1c will happen for electron waves as well. But what will this cancellation mean?
The wave going to the right describes an electron beam headed to the right. The wave going up describes a beam of electrons headed up. Suppose we have arranged these beams to have the wave structure of Fig 1c; i.e., the crests of one beam falls on the troughs of the other in the center of the intersection region.
Now suppose that we keep an electron detector at the location marked by the circle. Since the two waves cancel out here, this detector will detect no electrons.
By contrast, if electrons were particles (i.e. not waves), then there would be no such cancellation, and the detector would pick up particles form both beams.
This difference is the essence of the wave nature of the electron. To prove that the electron is a wave, we set up an experiment analogous to Fig.1c, and check that the detector in the center indeed picks up no particles.
 
 
 
 
 
 
 
 
 
 
 
 
Fig.1a: A wave moving right
 
Fig.1b: A wave moving up
 
Fig.1c: Both waves present together. The waves cancel at the center. If these are electron waves, an electron detector kept at this center will detect no electrons.