Wave properties of matter
That the waves have particle properties has been experimentally demonstrated by the Photo-electric effect. In order to understand a particle, let us recap the relevant part concerning particle from the last discussion. Intuitively, particle implies an object having unique position in space at a particular time and also, some physical properties viz. mass etc. can be attributed to it. It is found that as we go microscopic, all these attributes disappear. However, the premise of a unique position at a particular time remains (to the best of my understanding). It becomes rather impossible, as we go microscopic, to imagine the particle in the terms of physical attributes like mass.
It was De Broglie, who said that a moving body behaves as though it has a wave nature. He said that momentum p for a photon of wavelength λ is h/λ. It was found that the general formula is λ = h/p = h/ϒmv where m is the rest mass of the object and v is the speed. It is important to remind that rest mass of an object is invariant.
Thus, it is clear that De Broglie says that every moving object has a wavelength and that is equal to h/ϒmv. However, it is important to understand as to what this wave is made of. I mean, the way water wave is that of height of water surface, sound wave is that of pressure and light wave is that of electric field and magnetic field, this matter wave is made of which oscillation? The answer to this question is rather mathematical. It says that the matter wave is of ψ. This ψ is called wave function. The value of ψ for a moving object at a unique point in space at a unique time is basically the probability of finding that object there at that point of time. Thus, we see that matter waves are a purely mathematical construct and have got very little to do with physical reality. That is, it doesn’t object to the fundamental premise that a particle has to be at one place at a particular point of time. Here comes the Heisenberg’s uncertainty principle which says that it is impossible to know both the exact position in space and exact momentum at the same time of a moving object.
Let me also mention here that momentum is a more fundamental quality of an object than mass. For e.g., photons don’t have a rest mass and yet they display momentum. Let us also remind ourselves that we are dealing with subatomic levels, which are actually the reality behind the myriad forms of matter perceived in the universe. In my limited understanding, therefore, particles do behave as waves and show all the characteristics such as interference, but they do so in a mathematical form rather than in a physical manner.