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## Homework Statement

Consider a propagating wavepacket with initial length ## L_{0}##. Use the bandwidth theorem to show that the minimum range of angular frequencies present in the wavepacket is approximately:

$$ \Delta{\omega}\approx \frac{v_{g}}{L_{0}} $$

## Homework Equations

Bandwidth theorem:

$$ \Delta f \Delta t > 1$$

## The Attempt at a Solution

Use the following approximation for the group velocity ## v_{g}##

$$ v_{g}=\frac{\Delta \omega}{\Delta k} $$

Using bandwidth theorem:

$$ \Delta \omega \Delta t > 2 \pi $$

$$ \Delta t =\frac{\Delta v_{g}}{\Delta l} $$

$$ \therefore \Delta \omega > \frac{2\pi \Delta v_{g}}{\Delta L} $$

If $\Delta v_{g}$ is small: $$\Delta v_{g} \approx v_{g}, \Delta L \approx L_{0} $$

Therefore:

$$ \Delta \omega > \frac{2 \pi v_{g} }{L_{0}} $$

So

$$ \Delta \omega \approx \frac{v_{g}}{L_{0}} $$

But this seems a bit of a fudge? Can anyone explain how I might get a more kosher version? Thanks!

*EDIT: No this is completely wrong - sorry - the limit of ##\Delta L ## as ##\Delta v_{g} \rightarrow 0 ## is zero! How do I fix this? Thanks :)