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Wavepackets and group velocity

*Physics 273, Fall 2000, Prof. Ted Jacobson*

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***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.

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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:

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**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.**

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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.

Converted by *Mathematica*
October 29, 2000