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Homework Help: Dispersion Relation of a De Broglie Wave

  1. Mar 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Use the dispersion relation to find the group velocity v_group and phase velocity v_phase.

    2. Relevant equations

    omega(k) = [(hbar)k^2]/2m

    3. The attempt at a solution

    v_group = domega(k)/dk = [hbar]k/m = h/m lambda = p/m = v

    This isn't right.

    v_phase = omega / k = 2pi f/(2pi/lambda) = f lamda = v

    again... is this right?, doesn't look it but as far as I can see there is no mistake in the calc as I have doing it several times already, am I goign wrong in my physics?
  2. jcsd
  3. Mar 7, 2009 #2
    Hi there,

    Your first answer is definitely right.

    But then, why did you substitute omega = 2pi f and k = 2pi/lambda in the expression for the phase velocity? When I substitute the expression you were given for omega and divide it by k I get v_phase = 1/2 v.

    I don't think f lambda equals v here, because if omega = (hbar*k^2)/2m I get f = h/(2m*lambda^2) and if you multiply that by lambda you'll get h/(2m*lambda) = p/2m which again gives you 1/2 v.

    Relations like p = h/lambda, omega = 2pi f and k = 2pi/lambda are always true, but in a dispersive medium f lambda = v = v_group is not always true, only when omega = v * k, then v = v_group = v_phase.

    Hope that helps.

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