Description of a wave packet incident on a square potential barrier

[jmm5872]

A wave packet of mean energy E₀ is incident on a potential square barrier. The figure below shows stills from a film showing 9 successive stages in interaction.

a) True or false? The incident packet at time t = 0 describes many particles, each in a
different momentum state. If false, tell what the incident pulse does describe.

b) In the times from 400 − 700, what is the reason for the fine structure of maxima and
minima, outside the barrier?

c) If the well was attractive instead, do you expect adjacent maxima to be more closely spaced inside or outside the well? Why?


I'm a little confused here because I thought that a wave packet could describe a particle or particles, by a superposition of plane waves each with different wavenumbers (k).

The momentum is p = \hbark
Therefore, a range of wavenumbers describes a range of momenta. But if you measure it, the wavefunction breaks down and you will measure one specific value of momentum.

So, I am thinking the answer is false, because this could describe one particle. Also because it describes the probability of measuring the particle(s) position and momentum, meaning there are not different momentum states.


For part (b), I'm not sure what is going on here. I know that the wave is partially transmitted and partially reflected, and the reflected portion will interfere with the incident wave that hasn't reached the barrier yet. This explains the peak in amplitude just outside the barrier because the waves are adding together. But I'm not sure about the fine structure. Does this have something to do with the superposition of plane waves and their different wavenumbers?

Sorry, I couldn't get just the image attached on here since it is in a pdf. I'm not sure if I can attach a whole pdf on here, but I will if someone needs to see the picture.

 

[member 553083]

For part (c), I think that adjacent maxima would be more closely spaced outside the well. This is because the wave packet is spread out more outside the well, and so it takes less time for the wave to travel between maxima.