1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Easy Change of Variables

  1. Feb 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Given the equation
    [tex] U(\mu) = \frac{2}{\sqrt\pi} \exp\left[ -4\mu^2 \right] [/itex]
    find an expression for [itex] \hat U(\hat x) [/itex] given that change of variables
    [tex] x = \frac n2 + \sqrt n \mu, \qquad \hat x = \frac xn [/tex]
    and [itex] \hat U [/itex] is the U under this variable transformation.

    3. The attempt at a solution
    Using the fact that [itex] x= \frac n2 + \sqrt n \mu [/itex] it is easy to re-arrange to find that

    [tex] \mu^2 = \frac1n \left(x-\frac n2\right)^2 = \frac{x^2}n - x + \frac n4 [/itex]

    dividing by n, we get

    [tex] \frac{\mu^2}n = \hat x^2 - \hat x + \frac14 = \left( \hat x - \frac12 \right)^2 [/itex]

    Now I substitute this back into [itex] U(\mu) [/itex] to get

    [tex] \hat U(\hat x) = \frac2{\sqrt\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right] [/tex]

    The problem is that the solution is supposed to be

    [tex] \hat U(\hat x) = 2 \sqrt{\frac n\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right] [/tex]

    I can't seem to deduce where the factor of [itex] \sqrt n [/itex] comes up.
  2. jcsd
  3. Feb 18, 2010 #2
    Nobody? Nothing?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook