Understanding Phase & Group Velocities in Different Contexts

[The Head]

I am trying to understand phase and group velocities in a few different contexts, but require some assistance. From pictures I have seen how these speeds can be different, and I have come to understand that the phase velocity can be greater than the speed of light because it does not actually transport material or energy, that function is reserved for the group velocity of the wave.

1) Now, I am trying to make this a little more concrete in my mind. In the case of an electron, the particle's propagation is governed by the group velocity. So what effect does the phase velocity have on the particle? My book gives an example where an electron's phase velocity (>c) is twice that of its group velocity. What exactly does this velocity represent? Is it possible that this value can be something other than twice the group velocity, and if so, what would actually change?

2) Also, I may be incorrect in saying what follows, so forgive me. Regarding The Uncertainty Principle, a particle can only be found in locations where the group's amplitude is non-zero. So does phase have any relationship to the location of a particle. In some sense, I would think not, because it does not govern the propagation of the particle, but then I don't see what relationship it has to matter waves.

3) Finally, one last clarification of this concept with regards to dispersion. In dielectrics, when the refractive index of light travels through a material at a particular frequency, at times there is a great deal of absorption. This tends to occur at similar frequencies as anomalous dispersion, where the refractive index plummets below one. Again, no laws of physics are violated because only the phase velocity exceeds 'c.' So in this case, what exactly does this phase velocity relate to? Is this purely related to an EM wave, or does this phase velocity describe something related to the material in which radiation is crossing?

Thanks in advance!

 

[Alkim]

Hi,
When speaking about phase and group velocity it is better to think about waves and not about single particles. The latter can be found anywhere in space with certain probability.
Have you seen the animation in http://en.wikipedia.org/wiki/Phase_velocity ? I think it is worth much more than 1000 words :-)

 

[The Head]

Thanks. I have seen something similar before, but that is a nice animation.

Still confused about the things I mentioned above. Even if we are talking about the electron as a matter wave, I don't have any context with how the phase velocity is significant though. To me right now, it is purely a mathematical construct with no interpretation. Same goes for an n<1.

 

[Alkim]

No, in no way these parameters are abstract mathematical constructs. In fact they are extensively used in signal processing and especially important in the theory of ultrashort laser pulses. About your questions:

1) Phase velocity has no effect on the particle. Rather it is the result of the wave nature of the particle. The phase velocity is the speed of propagation of the wavefront, while the group velocity is the wave packet speed. In general they are different, and can have opposite signs. There is no fixed coefficient relating them, although there is a relationship between them (see http://en.wikipedia.org/wiki/Group_velocity). vph=2vgr is just an example of a possible relation.

2) Uncertainty principle has nothing to do with phase velocity. These are wave parameters and it does not make much sense to relate them to particle position, although a correlation can be found between them applying statistics on sufficiently weak signals. Again group velocity does not govern the behavior of a particle, it is rather a consequence of the same.

3) Group velocity and phase velocity have each their own dispersion and both depend on the material. Negative refraction is related to negative phase velocity and is extensively studied. It is employed in some metamaterials, which are often nanostructured metal-dielectric composites.

 

[pmacias]

Dear curious scientist,

I am glad to hear that you are exploring the concept of phase and group velocities in different contexts. These velocities play an important role in understanding the behavior of waves, including electromagnetic waves and matter waves. I will do my best to provide some clarification on your questions.

1) When it comes to particles, such as electrons, their propagation is indeed governed by the group velocity of the wave associated with them. The phase velocity, on the other hand, represents the speed at which the phase of the wave is moving. In the case of an electron, the phase velocity can be greater than the speed of light, but this does not violate any laws of physics because the electron itself is not traveling at this speed. The phase velocity simply describes the movement of the wave associated with the electron. It is possible for the phase velocity to be different from twice the group velocity, depending on the specific properties of the wave and the medium it is traveling through. In this case, the phase velocity would not have a direct effect on the particle, but it is still an important characteristic of the wave.

2) The phase of a wave does not have a direct relationship with the location of a particle. As you mentioned, the Uncertainty Principle states that a particle can only be found in locations where the group's amplitude is non-zero. The phase of the wave does not determine this amplitude, so it does not have a direct impact on the particle's location. However, the phase of the wave does contribute to the overall behavior and properties of the wave, which can indirectly affect the particle's location.

3) In the context of dispersion, the phase velocity can provide information about the behavior of waves as they travel through a material. In the case of dielectrics, when the refractive index decreases below one, this can lead to anomalous dispersion and increased absorption of the wave. The phase velocity in this case would represent the speed at which the phase of the wave is moving, and it can give insight into the properties of the material that are causing this behavior. So the phase velocity is not just related to the electromagnetic wave itself, but it can also describe the characteristics of the material that the wave is passing through.

I hope this helps to clarify the concept of phase and group velocities for you. Keep exploring and asking questions, as this is an important area of study in understanding the behavior of waves in various contexts. Good luck in your research!