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I'm trying to get my head round modelling particles in free space in quantum mechanics. I appreciate that we can "build" wavepackets by superposing many plane waves with different k-numbers (i.e. with different frequencies & momentums & energies I think). The greater the number of phase waves making up the packet, the greater the localisation too giving a narrower packet.

I wonder if anyone could clarify some issues regarding dispersion of this packet please. I understand that the phase velocities will vary for each component wave and this spread causes dispersion, whilst the envelope of the packet travels at the group velocity,

v_g = d\omega / dk

My questions arise in terms of relating this to the original particle of mass m and travelling at velocity v, so momentum, p = mv.

1) Is the average k-number, k_av of the component waves in the packet related to the particle's momentum by p= h-bar * k_av? So therefore, is it right to say that the particle's momentum doesn't affect the rate of dispersion.

2) We can relate particle energy to momentum by E = (p^2) / 2m and the planar wave energy is given by E = (h-bar * k)^2 / 2m. How do we relate wavepacket energy to the particle energy and will this affect dispersion?

Any help would be much appreciated.