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Problem Group and Phase Velocity

  1. Oct 24, 2006 #1
    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 dont know how to get rid of k here!!!!!!!
     
  2. jcsd
  3. Oct 24, 2006 #2

    OlderDan

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    What is hk/m? What do you have for the group velocity?
     
  4. Oct 25, 2006 #3

    quasar987

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    Is it me or we lost 2 posts here? :-O
     
  5. Oct 25, 2006 #4

    OlderDan

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    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.
     
  6. Oct 25, 2006 #5
    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: Oct 25, 2006
  7. Oct 25, 2006 #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.
     
  8. Nov 28, 2007 #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?
     
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