Phase velocity and group velocity

The group velocity is the same as the velocity of the particle and can't be higher than the speed of light according to the equations of special relativity.

The phase velocity is the speed of the peaks of the wave and can travel faster than light. The speed of the peaks is more of a mathematical concept; no physical thing is travelling that fast. Since no energy or mass or information is going faster than light, the equations of special relativity are not violated.

It's related to these examples (copied from Wikipedia):
"If a laser is swept across a distant object, the spot of laser light can easily be made to move across the object at a speed greater than c. Similarly, a shadow projected onto a distant object can be made to move across the object faster than c. In neither case does the light travel from the source to the object faster than c, nor does any information travel faster than light."

I recently had a question related to this. The thread is here:
https://www.physicsforums.com/showthread.php?t=693357
Posts #8 and #9 have derivations of phase and group velocities that you might find helpful.

 

Group velocity is a property of waves. Without quantum mechanics a particle is not considered a wave so there is no group velocity or phase velocity, only the velocity of the particle.

You can say that the velocity of the particle can't be higher than the speed of light without quantum mechanics, but then there is no group velocity.

 

See this page : www.mathpages.com/home/kmath210/kmath210.htm