Wavepacket Evolution: Questions on Time Evolution and Plane Waves

[nrivera1]

I had a few questions regarding the time evolution of wavepackets of the form

∫ dk*A(k)*cos(kx-wt), where w = w(k)

If the group velocity is zero, i.e; dw/dk evaluated at k' = 0, where k' the central wavenumber of the narrow packet, then do we essentially see complete constructive interference? I would imagine if all the k values are near k' that this would be the case.

And furthermore based on this logic, the smaller the group velocity is, the more constructive interference you get for a narrow band of wavelengths because in the limit as w doesn't change as k varies the phase difference between the component waves gets close to zero assuming kx can be approximated as k'x. Is this also right?

And lastly, something on the existence of plane waves. So I understand that monochromatic plane waves don't exist because they have to be infinite in extent so when we talk about observing the individual waves of a wavepacket, say in a water wave, are we really talking about observing various narrow wavepackets that travel at roughly the same speed? Because it seems based on the Fourier representation of a wave above that the amplitude of each component wave is actually infinitesimal and that you would not be able to observe this.

Thanks!

 

[Valerie Park]

Yes, if the group velocity is zero then you will essentially see complete constructive interference. This is because when the group velocity is zero and all the k values are near k', the phase difference between the component waves is close to zero, leading to constructive interference. Yes, the smaller the group velocity is, the more constructive interference you get for a narrow band of wavelengths. This is because in the limit as w doesn't change as k varies, the phase difference between the component waves gets close to zero assuming kx can be approximated as k'x.When we talk about observing the individual waves of a wavepacket, we are talking about observing various narrow wavepackets that travel at roughly the same speed. This is because monochromatic plane waves don't exist as they have to be infinite in extent, so what we observe are wavepackets with a range of wavelengths and amplitudes. The Fourier representation of a wave does involve infinitesimal amplitudes, however these amplitudes can be amplified through constructive interference.