For a general wave pulse, neither E nor p (i.e., neither ω nor k) are well defined, but th...

For a general wave pulse, neither E nor p (i.e., neither ω nor k) are well defined, but they have approximate values E0 and p0. Although it comprises many plane waves, the general pulse has an overall phase velocity corresponding to these values.

If the pulse describes a “large” massive particle, the uncertainties are reasonably small, and the particle may be said to have energy E^and momentumpQ. Using the relativistically correct expressions for energy and momentum, show that the overall phase velocity is greater than c and given by

Note that the phase velocity is greatest, for particles whose speed is least.