Velocity and Phase Velocity of Matter Wave

[hschia]

Main Question or Discussion Point

In quantum mechanics, what is the difference between velocity and phase velocity of matter wave? How can it also be that phase velocity of matter wave always exceeds the speed of light?

 

[mfb]

Wikipedia has to nice illustrations how phase and group velocity can be different, see the article about phase velocity.
In addition, it answers your second question.

 

[jtbell]

Yes, the animations on that Wikipedia page are very nice! :!!)

 

[hschia]

What then, is the difference between the velocity,v and phase velocity, vp in the lowest section? What is the physical implication behind these?

 

[mfb]

Group velocity is similar to the classical velocity - the speed of particles and signals (this is an approximation, but I don't want to go into details). You emit something at point a, measure it after time t at point b, and calculate the group velocity as (b-a)/t.
Phase velocity and wavelength are important in quantum mechanics and interference effects.

 

[hschia]

No. I understand the difference between PHASE and GROUP velocities, but find it difficult to visualise the difference between PHASE velocity and ''PURELY'' velocity. The following is an equation copied from the wikipedia link:

Using relativistic relations for energy and momentum, we have
v_p=c²/v.

My essential question is, what is the difference between v_p and v here? Isn't v_p=ω/κ, the same idea as the ''PURE'' velocity of wave?

 

[mfb]

Sorry, what is that "purely velocity" you are talking about? The wikipedia page uses group velocity there.

The variable v can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave.

 

[PhilDSP]

My essential question is, what is the difference between v_p and v here? Isn't v_p=ω/κ, the same idea as the ''PURE'' velocity of wave?

v = v_g = $\frac{\partial \omega}{\partial k}$, that is, group velocity equals the velocity of the particle associated with the wave or wave packet.

(Signal velocity can be different from group velocity and doesn't necessarily have the invariant relationship group velocity has with phase velocity)


v_p = ω/κ = c = v_g for a non-dispersive medium only. Otherwise v_p increases as v_g decreases and v_p v_g = c².

Phase velocity, v_p = ω/κ, is the velocity in which a surface in the medium (across which the phase is constant) travels.
That is, the surface is mathematical - not associated with any physical surface.