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Effective Potential

  1. Apr 5, 2009 #1
    1. The problem statement, all variables and given/known data

    I am trying to do problem 5, I seem to be having a hard time with the algebra.
    http://img24.imageshack.us/img24/4224/landau.th.png [Broken]

    2. Relevant equations



    3. The attempt at a solution

    To find maximum value of the effective potential we just do: dU/dr = 0.
    I get (which I have verified to be correct)
    [tex]r_m^{n-2} = \frac{n*\alpha }{m \rho^2 v^2}[/tex]

    I rearrange that into
    [tex]r_m = (\frac{n*\alpha }{m \rho^2 v^2})^{1/(n-2)}[/tex]

    Plugging that into the Ueff equation for r, I get (and simplifying a bit, let x = n/(n-2) ).

    [tex]\frac{m^x v^{2x} \rho^{2x}}{2 n^{2x/n} \alpha^{2x/n}} - \frac{\alpha m^x v^{2x} \rho^{2x}}{n^x \alpha^x}[/tex]

    At this point I kind of get stuck, I can do the following, but it never seems to turn out 'right'.

    [tex]\frac{m^x v^{2x} \rho^{2x}}{\alpha^x n^x} ( \frac{1}{2 \alpha^{2/n} n^{2/n}} - a)[/tex]

    Anyone know what I am doing wrong? it must be something simple because I am getting very close to the correct answer, it's just those terms on the right don't seem to become 'nice'.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Apr 5, 2009 #2
    Whoops, I seem to have figured it out, apparently it's n^(-1) not n^2/n
     
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