Problem Group and Phase Velocity

[somebody-nobody]

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!

 

[OlderDan]

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?

 

[quasar987]

Is it me or we lost 2 posts here? :-O

 

[OlderDan]

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.

 

[lightarrow]

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.

 

[somebody-nobody]

i got it

Sorry,

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

Thank you all for help.

 

[Manicwhale]

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?