Potential barrier, wave function, and probability

[judonight]

I have a couple questions about finite potential barriers that I can't seem to figure out on my own...

1) Why does the real part of the wave function collapse inside the barrier (square, rectangular, barrier with V less than the energy of particle)? It seems to me that there should be some probability that the particle, if tunneling, could get trapped inside the barrier, since as the wave collides with the barrier the wave function is non zero.

2) Is the wave function 'expanding' after 'collision' witht the barrier (for the reflected and transmitted waves) because the probability of it's position is changing? Has it lost momentum?

3) How is the real and imaginary components of the probability density dependent on time after the 'collision'? Is it more than just following the wave function? I understand that the amplitude is ever lowering due to the 'expansion' of the wave packets corresponding to the transmitted and reflected waves, but, not sure how each component are dependent on time.


I know these aren't great questions, but in my attempt to understand the quantum world these few things are still bothering me.

 

[Nugatory]

1) The particle will not be trapped in the barrier if its energy is greater than the barrier potential $V$. You are right, the wave function is non-zero there so there is some probability that it will be found there, just as there is everywhere else that the decaying exponential tail of the wave packet reaches - but of course it is much more likely to be found in the high-amplitude areas of the reflected and transmitted wave packets.

2) Clearly if the particle is detected in the reflected wave its momentum will have changed; the discrepancy is made up by a change in the momentum whatever is creating the barrier (which is assumed to be so massive that the effect of the momentum transfer on it can be ignored). This is the same thing we do with momentum conservation in classical physics when bouncing a ball off an immovable wall.
The wave packet is always "expanding". It's a superposition of many different frequencies so it spreads out over time. This is all determined by the time-dependent Schrödinger equation. (It's also computationally a nuisance, so when we just need the probabilities of transmission and reflection we work with a plane wave of a given energy instead of a wave packet).

3) Yes, you just follow the wave function.