A common misconception is that in a double slit experiment, the electrons or the photons go through both slits at the same time and interfere with themselves.
Nothing of the sort ever happens and that idea doesn’t come from quantum mechanics. It is just meaningless to ask the question (from which slit did the particle pass?) in the context of the particular experiment and even more meaningless to say that it went through both. The particle always passes from the one or the other slit if you setup an experiment that asks the question and in the case that you shoot the particles one at a time you always get one hit on the screen and not an interference of portions of the particle.
I think that the misconception has its roots in the wave analogy of the wave function description of quantum mechanics, where you must have a wave passing from both slits in order to have interference on the other side. Of course that picture doesn’t translate to having a particle passing from both slits, but it only states that there exists interference between the wave functions of the two independent events that translates to a distribution of probability for finding a particle on a particular point on the screen. That is a statement of Born’s interpretation that has been recently tested with a triple slit experiment.
One could ask at this point, “Ok, I get it for the electrons, but light behaves like a wave in everyday experience. What happens in that case?”, and the answer is that light behaves like a wave only when you have a large flux of photons and you can go to the continuous field approximation. It is then that the wave-like properties become apparent both for photons and electrons.
Update: A very interesting video on Quantum Mechanics
Update2: QM in your face
Update3: The Feynman Lectures on Physics vol3: Scattering from a crystal (neutrons).
... Let’s review the physics of this experiment. If we could, in principle, distinguish the alternative final states (even though you do not bother to do so), the total, final probability is obtained by calculating the probability for each state (not the amplitude) and then adding them together. If you cannot distinguish the final states even in principle, then the probability amplitudes must be summed before taking the absolute square to find the actual probability. The thing you should notice particularly is that if you were to try to represent the neutron by a wave alone, you would get the same kind of distribution for the scattering of a down spinning neutron as for an up spinning neutron. You would have to say that the “wave” would come from all the different atoms and interfere just as for the up spinning one with the same amplitude. But we know that this is not the way it works. So as we stated earlier, we must be careful not to attribute too much reality to the waves in space. They are useful for certain problems but not for all.