This applet shows what happens with a single electron - modelled by a Gaussian wave function – that is going through a double-slit.
Running the simulation shows a wealth of features: an interference pattern in front of the slits that is due to reflections, and two "bubbles" coming up immediately behind the slit. After a while a third bubble appears in the middle, which together with the other two develops into an interference pattern on the screen.
To better understand where this central part comes from, restart the simulation, this time looking at the real part of the wave function instead of the absolute value. Compare the behaviour with the diffraction in the classical case.
Now increase the particle velocity and notice, how the interference pattern on the screen changes. Can you explain the behaviour qualitatively?