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In the starting of quantum mechanics we encounter Plank's relation $E=\hbar \omega$, and from special relativity we have $Ev_p=pc^2$ , where $v_p$ is the velocity of the particle (we are assuming momentum is a particle aspect). Putting the two together, we get $$\hbar \omega v_p = pc^2 \, .$$ Now we assume that $\omega$ is a wave concept and use $\omega / k =v_w $ where $v_w$ is the wave velocity. For now we assume the wave velocity to be different from that of the particle for we don't know if the wave is the particle itself or an excitement in the probability field of the universe. That's a different topic. Here I just want to say that $v_p$ and $v_w$ might be different, or even the same. We don't know so why take risks?

Plugging the expression, we have $$p = \hbar v_p v_w k /c^2 \, .$$

Then, plugging it into the relativity relation $E^2 = p^2c^2 + m^2 c^4$ and finding out $d \omega /dk$, we get $$d\omega / dp = v_p^2v_w/c^2 \, .$$ This is the group velocity which has to be equal to the particle velocity if it has to have some meaning! Therefore, $$v_p = v_p^2v_w/c^2 $$ or $$ v_p v_w = c^2 $$ or $$ v_w = c^2 / v_p$$ which automatically implies $v_w>c$ (assuming $v_p<c$ , which is the very core of special relativity).

What is going on here? The wave velocity has come out more than the speed of light!

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  • $\begingroup$ @Willy Billy Williams Care to explain why? $\endgroup$
    – Yuzuriha Inori
    Commented May 13, 2017 at 14:39
  • $\begingroup$ I erased it, that was correct. I believe the issue is that both $v_w=c$ and $v_p=c$ in vacuum $\endgroup$
    – user126422
    Commented May 13, 2017 at 14:45
  • $\begingroup$ De Broglie wave travel with a group velocity which is less than c $\endgroup$
    – Shashaank
    Commented May 13, 2017 at 14:53
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    $\begingroup$ $E v_p \neq pc^2 $ $\endgroup$
    – Ofek Gillon
    Commented May 13, 2017 at 14:53
  • $\begingroup$ @Shashaank That's the thing... Then why does calculations suggest otherwise? $\endgroup$
    – Yuzuriha Inori
    Commented May 13, 2017 at 14:55

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What you have found is the difference between the phase velocity and the group velocity by using the de Broglie relation that is quantum mechanical and describing a wave nature. The phase velocity can be higher than c because it only appears in the mathematical manipulations.

The phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin. See dispersion for a full discussion of wave velocities.

No information can be carried faster than c, the group velocity.

The de Broglie relation as quantum mechanical has to be treated within those rules, where every observable corresponds to an operator. See the answer by Motl here. Particles can be described by wave packets, and they will have a group and phase velocity, and the phase velocity can be higher than c because it is not observable.

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