Velocity and Phase Velocity of Matter Wave

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Discussion Overview

The discussion centers on the differences between velocity and phase velocity of matter waves in quantum mechanics. Participants explore theoretical implications, mathematical relationships, and conceptual understandings related to these velocities.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the difference between velocity (v) and phase velocity (vp) of matter waves, particularly why phase velocity can exceed the speed of light.
  • Another participant references Wikipedia illustrations to clarify the differences between phase and group velocity, suggesting that the article addresses the second question posed.
  • A participant expresses difficulty in visualizing the difference between phase velocity and what they refer to as "purely velocity," seeking clarification on the relationship between vp and v.
  • It is noted that group velocity is akin to classical velocity, representing the speed of particles and signals, with a formula provided for calculating group velocity.
  • Participants discuss the mathematical expressions for phase velocity and group velocity, with one participant asserting that vp is not the same as the "pure" velocity of the wave.
  • There is a mention of the relationship between phase velocity and group velocity in non-dispersive media, highlighting that vp increases as vg decreases under certain conditions.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of phase velocity and velocity, with no consensus reached on the relationship between these concepts. Some participants agree on the mathematical definitions but differ in their interpretations and visualizations.

Contextual Notes

Discussions include unresolved questions about the definitions of "purely velocity" and how it relates to phase and group velocities. There is also a lack of clarity on the implications of these velocities in various contexts.

hschia
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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?
 
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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.
 
Yes, the animations on that Wikipedia page are very nice! :!)
 
What then, is the difference between the velocity,v and phase velocity, vp in the lowest section? What is the physical implication behind these?
 
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.
 
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
vp=c2/v.

My essential question is, what is the difference between vp and v here? Isn't vp=ω/κ, the same idea as the ''PURE'' velocity of wave?
 
Sorry, what is that "purely velocity" you are talking about? The wikipedia page uses group velocity there.
Wikipedia said:
The variable v can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave.
 
hschia said:
My essential question is, what is the difference between vp and v here? Isn't vp=ω/κ, the same idea as the ''PURE'' velocity of wave?
v = vg = \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)vp = ω/κ = c = vg for a non-dispersive medium only. Otherwise vp increases as vg decreases and vp vg = c2.

Phase velocity, vp = 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.
 
Last edited:

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