Can You Help With Phase and Group Velocity for Relativistic Electron Waves?

In summary: Thanks!In summary, the dispersion relation for free relativistic electron waves is w(omega)=(c^2k^2+(m(mass of electron)c^2*2pi/h)^2)^0.5Obtain expression for the phase velocity Vp and group velocity Vg of these waves and show that their product is a constant ,independent of k.
  • #1
somebody-nobody
12
0
I am really stucked with my homework problem.Can anybody help me.

The dispersion relation for free relativistic electron waves is

w(omega)=(c^2k^2+(m(mass of electron)c^2*2pi/h)^2)^0.5

Obtain expression for the phase velocity Vp and group velocity Vg of these waves and show that their product is a constant ,independent of k.

Solution:

I know that Vp=w/k=c(1+4m^2c^2pi^2/h^2k^2)^0.5

but I don't know how to get rid of k here!
 
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  • #2
somebody-nobody said:
I am really stucked with my homework problem.Can anybody help me.

The dispersion relation for free relativistic electron waves is

w(omega)=(c^2k^2+(m(mass of electron)c^2*2pi/h)^2)^0.5

Obtain expression for the phase velocity Vp and group velocity Vg of these waves and show that their product is a constant ,independent of k.

Solution:

I know that Vp=w/k=c(1+4m^2c^2pi^2/h^2k^2)^0.5

but I don't know how to get rid of k here!
What is hk/m? What do you have for the group velocity?
 
  • #3
Is it me or we lost 2 posts here? :-O
 
  • #4
quasar987 said:
Is it me or we lost 2 posts here? :-O
It's not you. They are gone. Here is my part of it

hk/m = p/m = v It would just be a shorter way to write all those terms. You don’t need it to do the problem. I’m sorry I mentioned it.

Do not try to eliminate k from the individual velocities. The problem is asking you to show that their product is independent of k, not that each of them are independent of k.
 
  • #5
somebody-nobody said:
I am really stucked with my homework problem.Can anybody help me.

The dispersion relation for free relativistic electron waves is

w(omega)=(c^2k^2+(m(mass of electron)c^2*2pi/h)^2)^0.5

Obtain expression for the phase velocity Vp and group velocity Vg of these waves and show that their product is a constant ,independent of k.

Solution:

I know that Vp=w/k=c(1+4m^2c^2pi^2/h^2k^2)^0.5

but I don't know how to get rid of k here!
As OlderDan said, the problem clearly ask to show that the product Vp*Vg is a constant, independent of k, not to show that Vp or Vg are independent of k!

Their product is c^2:

Vg = dw/dk = c^2*k/SQRT[c^2*k^2 + (m*c^2*2*pi/h)^2] =

c/SQRT[1 + (2*pi*m/h*k)^2] --> Vg*Vp = c^2.

But, as OlderDan said (again!) there is no need to make these computations, since Vg = p/m = h*k/m, so: Vp*Vg = (w/k)*h*k/m = h*w/m = E/m = c^2.
 
Last edited:
  • #6
i got it

Sorry,

I was reading problm 1000 times ,and I didnt realize that they ask for products.

Thank you all for help.
 
  • #7
I got the same question, except it's worded slightly differently. It wants us to show that a relativistic electron of velocity v=hk/m (h is hbar) with dispertion relation

w^2/c^2 = k^2 + m^2c^2/h^2 (slightly different from the one from the previous question)

satisfies

group velocity x particle velocity = c^2.

Like discussed above, I can find that group velocity x *phase* velocity = c^2, but if I take the particle velocity as the velocity of the electron v (above), then I can't get the same thing. Do you think this may have been a typo?
 

Related to Can You Help With Phase and Group Velocity for Relativistic Electron Waves?

What is the difference between group velocity and phase velocity?

Group velocity refers to the speed at which a wave packet (a group of waves) travels through a medium. It is determined by the average speed of the individual waves within the packet. Phase velocity, on the other hand, refers to the speed at which an individual wave within the packet travels. It is determined by the wavelength and frequency of the wave.

How do group and phase velocities relate to each other?

In most cases, the group velocity is slower than the phase velocity. This is because the individual waves within a wave packet can interact with each other, causing the overall speed of the packet to decrease. However, there are some cases where the group and phase velocities are the same, such as in certain types of non-dispersive media.

What is the significance of group and phase velocities in physics?

Group and phase velocities are important concepts in the study of wave phenomena. They help us understand how waves travel through different mediums and how they interact with each other. These velocities also play a crucial role in fields such as optics, acoustics, and electromagnetics.

How are group and phase velocities measured?

Group and phase velocities can be measured using various techniques, depending on the type of wave being studied. In general, group velocity can be measured by observing the movement of a wave packet, while phase velocity can be measured by analyzing the interference patterns of individual waves.

Can group and phase velocities be manipulated?

Yes, it is possible to manipulate group and phase velocities through various methods. For example, in optics, the use of different materials with varying refractive indices can alter the group and phase velocities of light. Additionally, in some cases, external forces such as magnetic fields or electric fields can also affect the velocities of waves.

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