In double slit experiment, It's the wave function who represent particles, travelling?

I believe the electrons possess a position wavefunction which is initially concentrated at the edge of the gun (think of a gaussian) with a momentum concentrated at "moving forward". As time passes, the peak of the position wavefunction advances towards the screen, while becoming more and more spread out.

Note that this wavefunction occupies all of the room, it includes the values in the slits and on the detector screen at all times. As the peak of the wavefunction approaches the slits, its shape begins to change more noticeably, showing the effects of diffraction. Once it passes them, the section of it on the detector will become more and more like the expected interference pattern.

So at each point in time, there is a position wavefunction that occupies all the room. The evolution of this wavefunction is given by the time dependent Shrodinger equation. Thus, it would be a travelling wave-pulse.

First we need to clarify:

1. the electron when traveling from the gun to the screen, does not have a well defined position

2. the electron is in a superposition of states

3. it is the electron's wavefunction that describes its probability distribution in space

4. the electron is in between the gun and the screen

5. as the electron interacts with the screen, its position becomes well known, that is why we see a point on the screen

6. the screen is in a superposition of states too, but we see it as well defined, because complex systems are harder to be in superpositions (they have rest mass and do not move fast)

7. the wavefunction as per QM describes the probability of finding the electron at a certain position

8. when the electron interacts with the screen, it's wavefunction collapses, and the electron's position will be known on the screen

9. the wavefunction describes the probability distribution of this, and many electrons need to be fired and measured on the screen to see the same pattern as what the wavefunction describes