Homework #2 (221B)
Problem 1c. 
Double-click on any graph to see the movie.

We form a wave packet by taking our solution for ψwith a certian momentum q and multiplying it by the packet amplitude e\(TraditionalForm\`\^\(\(-\((q - k)\)\^2\) d\^2\)\)and by (e^(i((q^2)/2  m) 
            t)) for the time evolution (after setting =1). To make the integrate easier we will estimate q~kin the coeficient of the sccatered wave. This is a good approximation if the wave packet is narrow enough in q space.   

It is useful to play around with different values for γ, k, d, m to see how the scattering changes; e.g. γ=10 means the incoming wave is almost totally reflected.  To see the movie, double-click any of the graphs.  

This solution and all of yours participate in a little quantum mechanical fraud.  Wavepacket spreading goes as \(TraditionalForm\`t/md\^2\), and for small enough d you can see it within the time frame of the movie.  But for negative times the packet actually unspreads as time increases.  This does not happen in real life.  Can you see where the problem enters and how to fix it?

scattered = 
      Integrate[((-I m γ)/(k + I m γ)) 
          E^(I q r) (E^(-I  (q^2)  t/(2  m))) 
          (E^(-(((q - k))^2) (d^2))),  {q, -∞, ∞}]

incoming = 
      Integrate[(E^(I q x )) (E^(-I (q^2)  t/(2  m))) 
          (E^(-(((q - k))^2) (d^2))),  {q, -∞, ∞}]

ψ = incoming + scattered;

Do[Plot[
        Abs[ψ]^2 /. {k -> 1,  m -> 1,  
            d -> 5,  γ -> 1, r -> Abs[x], 
            t -> 2  i},  {x, -90, 90},  
        PlotRange -> {0,  .14}], {i, -30,  30}]