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Phase velocity contradiction

  1. Mar 6, 2013 #1
    So for the phase velocity of a massive particle we have

    Vph = Vg/2 for non-relativistic case

    Vph = c^2/Vg for the relativistic case

    Vg is the group velocity or particle velocity
    But there seems to be a contradiction in that for the non-relativistic case the phase velocity is predicted to be less than the group velocity whereas in the relativistic case the phase velocity must be greater than the group velocity (greater than the speed of light even).
    Furthermore a taylor series expansion of c^2/Vg will not converge to Vg/2 for small velocities. Whats the deal with this - why are these predictions so vastly different? Im guessing it has something to do with not having absolute speed in relativity but can't connect the dots.
  2. jcsd
  3. Mar 7, 2013 #2


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    Different choices for the zero point of energy. The phase velocity in both cases is vph = ω/k = E/p. Clearly this will change if a constant is added to E. In relativity, E = √(m2c4 + p2c2) ≈ mc2 + p2/2m, while in nonrelativistic QM we drop the rest energy mc2.

    Basically the same as this other thread.
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