Wave function in quantum mechanics I was wondering about something while studying quantum mechanics. If the wave function collapses when measuring a particle and assumes a single position, how do we know that it was a wave in the first place? 
P.S.: sorry if it is absurdly simple, I was just confused and couldn't come up with any explanation. 
 A: It was not a wave.
The wavefunction is not a wave. It fulfills the Schrödinger equation in the position representation, and although that looks similar to what one usually writes as "wave equation", and produces similar interference phenomena, it is not a wave in any physical sense. The wave function is not a physical object, it is merely a way of writing the coefficients for a quantum state in the position basis. It is not measureable, and there is, in general, no physical quantity oscillating that would be associated with it.
Any object in quantum mechanics is described by an abstract state in a Hilbert space, and the abstract Schrödinger equation tells you that there is a basis of "stationary states" that evolve in time just by being multiplied by a phase $\mathrm{e}^{\mathrm{i}Et}$, i.e. essentially doing nothing. If you now add several of these states with different energies/Hamiltonian eigenvalues $E$, the overall time evolution of the state is not a simple multiplication anymore, and the state is indeed changing. Essentially, this is all "interference" means - you have phases $\mathrm{e}^{\mathrm{i}Et}$ with different $E$ that can be added, and then some non-trivial kind of evolution appears. (Since usual solutions to wave equation also contain $\mathrm{e}^{\mathrm{i}\omega t}$ in this  way, this explains the name)
But that doesn't mean quantum states are waves. It also doesn't mean they are particles. They are quantum objects, states in a Hilbert space. Not waves. Not particles. When we look at them in some ways, e.g. at their time evolutions, and their properties of superposition and interference, they look like our inutitive notion of waves. When we look at them in detectors, they often look like our intuitive notion of particles.1 They are neither.

1It should be noted that trying to describe that actual occurence of such measurements is still a topic of some debate. Nevertheless, "collapse" is not a necessary interpretation of the math - decoherence approaches to measurements/emergence of classical physics do not need that concept.
A: This is a supplement to my original answer. For rigor, we have to make a distinction between the reality that travels in our apparatuses, and the mathematical description that we give it. However, the mathematical description proved to be so successful, that sometimes we place a sign of equality between them. 
About what happens with a quantum object when it interacts with a macroscopic apparatus, we don't know. At present, we don't have a better tool to handle this problem than the collapse (reduction postulate of von Neumann). And we simply use it because we have to go on, to work.
Now, the wave-form for the wave-function works well in some cases, and works badly in other cases. But in most of the cases in which interference in involves, it works well.
For instance if we put on the way of the particle a beam-splitter, we believe that we get a splitting of the wave, into a reflected wave and a transmitted wave. I.e. although we speak of one particle, we believe that we get two waves. Then, if we redirect, with mirrors, the two waves to cross the path of one another, we get an interference pattern (see experiments with the Mach-Zender interferometer in Wikipedia) if in the crossing region we place a photographic plate. 
However, the interference tableau doesn't appear for one single particle. We have to prepare many particles carefully, in an identical way, i.e. same type of particle, same velocity, etc.
So, interference pattern is produced by waves, while a single particle is detected on the photographic plate in a single place, as any particle.
Though, we incline to admit that before the detection on the plate, we had for each particle and particle, the two waves as I said above, and at the detection all the energy of the particle is delivered to one single molecule on the plate.
(The process of impressing the photographic plate is some more complicated but I restricted myself to a simple line. What is most important is that at the detection on the plate, the particle doesn't impart its energy to all the region covered by the interference pattern. No, the energy is delivered to a single point (e.g. a certain molecule is decomposed) ).
