Quantum physics question

[DaLegend]

Hi all,

I started on my physics program this semester. But things begin to puzzle me much, especially quantum mechanics. I hope that you wuold take the time to answer a few of my puzzles.

I used Sakurai's book in my quantum mechanics course. It is said that in the position space, the mometum eigenkets takes on the plane wave form. This result is independent of potentials. So does it mean that if we place a particle in whatever potentials, as long as we are able to measure the momentum accurately, then it should be equally probable to find the particle at any point in space (at later times)? This puzzles me because the result seems to hold even when the particle is in an infinite potential well, where classically the particle cannot penetrate. I know the uncertainty relation requires this, but it just seems strange, because the energy eigenfunction on the other hand cannot penetrate into that region...

The other question is like this. Suppose a particle is subjected to a potential and we measure the position of the particle as it goes along. If we let the time interval tends to zero, we would be able to find a path of such a particle as a function of t. It seems that we would be able to obtain the velocity of the particle from such path. Of course we cannot because in doing so we would ascribe properties to the system additional to that contained in the wavefunction. But this reminds me of some strange brownian motion. Brownian motion is continous yet (almost surely) non-differentiable. Is it possible that as the time interval tends to zero, the particle takes on such a path?

My third question is the quantum description of the double slit experiment. The double slit experiment for particles is always explained in a manner similar to light waves by ascribing the de brogile wave length to the particle. I wonder how might it be described in the quantum language. Is it like the barrier with the two slits are

[Sachs]

Errrrr .... yeah but no but yeah