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Expected value of x^2

  1. Oct 3, 2015 #1
    1. The problem statement, all variables and given/known data
    My teacher made up this question, but I think there's something wrong.

    Consider the wave packet in momentum representation defined by Φ(p)=N if -P/2<p<P/2 and Φ(p)=0 at any other point. Determine Ψ(x) and uncertainties Δp and Δx.

    2. Relevant equations
    Fourier trick and stuff...

    3. The attempt at a solution
    I found Ψ(x)=(2ħN/x√2πħ)sin(Px/2ħ), where N=±√1/P
    <x>=0, <p>=0, <p^2>=(P^2)/12

    But when I try to calculate <x^2>, I get a strange integral, which goes to infinity. Am I doing anything wrong?
     
  2. jcsd
  3. Oct 5, 2015 #2

    BvU

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    Hello rgalvao, welcome to PF :smile: !

    I see you didn't get a reply yet, so perhaps I can put in my five cents:

    Your wave function ##sin x\over x## is the Fourier transform of a rectangular function and you can indeed see that ##\int \Psi^* \,x \, \Psi dx ## yields zero, but ##\int \Psi^* \, x^2 \, \Psi dx ## diverges.

    I don't see anything wrong with what you do. The wave function simply doesn't fall off fast enough with |x| to give a finite expectation value for x2.
    Let us know if you or teacher finds otherwise (i.e. correct me if I am wrong .... :wink: ) !
     
  4. Oct 5, 2015 #3

    mfb

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    I'm not even sure about <x>. Sure, you can argue with symmetry, but the integral is not well-defined.
     
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