- #1

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[itex]\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int dk A(k)e^{i(k x - \omega(k) t)}[/itex],

can be expressed in two ways.

First, we have that it's the time derivative of the mean position (i.e., its mean group velocity):

[itex]\frac{d \langle x\rangle}{dt}=\int dk |A(k)|^{2} \frac{d\omega(k)}{d k}\approx \frac{d\omega}{dk}[/itex] at center frequency.

Second, we have that it is the averaged velocity of all the plane wave components of the wavepacket (i.e., the mean phase velocity):

[itex]\langle \frac{\omega(k)}{k}\rangle=\int dk |A(k)|^{2} \frac{\omega(k)}{k}.[/itex]

My questions are these:

When are these two formulations of "the mean velocity" equivalent?

Which (if either) best corresponds to the speed of information transfer, or energy flow?

Thanks for reading.