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Calculating variance of momentum infinite square well

  1. May 18, 2015 #1
    1. The problem statement, all variables and given/known data
    Work out the variance of momentum in the infinite square well that sits between x=0 and x=a


    2. Relevant equations
    Var(p) = <p2> - <p>2

    $$ p = -i\hbar \frac{{\partial}}{\partial x} $$
    3. The attempt at a solution
    I've calculated (and understand physically) why <p> = 0

    Now I'm calculating $$<p^2> = \int_{0}^{a} sin(\frac{nπx}a)(-\hbar^2)\frac{{\partial}^2}{\partial x^2}sin(\frac{nπx}a) dx$$

    $$<p^2> = ({\frac{n\pi\hbar}{a}})^2 \int_{0}^{a} sin(\frac{nπx}a)sin(\frac{nπx}a) dx$$

    $$ <p^2> = ({\frac{n\pi\hbar}{a}})^2 * \frac{a}2 $$

    I'm out here by a factor of a/2 because of the integral and I'm not sure why, does anybody have any suggestions?
     
  2. jcsd
  3. May 18, 2015 #2

    Orodruin

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    You forgot to normalise your eigenstates.
     
  4. May 18, 2015 #3
    Oh yes, that's exactly right, thanks very much. Was staring at this for ages, much appreciated :)
     
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