Wavepackets and group velocity
Physics 273, Fall 2000, Prof. Ted Jacobson

We form wavepackets with Gaussian weighting functions and evolve them according to the Schrodinger equation, which is imposed via its dispersion relation ω = / 2  (in units with ℏ = m = 1).

The group velocity is dω/dk = k. Our packets are peaked around k=10, with a spread in k small compared to 10, so the packet velocity is reasonably well-defined and equal to 10. We can see the packet move at this speed by evaluating
the integral over k for different values of t.

Square window in k:  In class we evaluated this case analytically.  The plot looks like this. The sharp cutoff in k in this example produces the wiggles at large x in the wavepacket:

With a smoother weighting function, the wavepacket goes to zero more definitively,  although it is wider in x  in the following example since the width in k is smaller. First it is plotted for t = 0, and next for both t = 0 and t = 1. You can see that the wavepacket has propagated a distance of 10 units in one unit ot time, as expected since the group velocity is 10.

Now we make a wavepacket that is broader in k, but has sharp cut-offs in k. This produces a wavepacket that is narrower in x than the previous one, but which also has long distance wiggles due to the sharp cut-offs. The first plot is at t = 0, the  second at both t = 0 and t = 1, and the third at both t = 0 and t = 5. At t = 5 the spreading of the wavepacket can be clearly seen.