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Expectation value of position of wavepacket

  1. Mar 23, 2010 #1
    Hello, this is just a general question, how is <x^2> evaluated, if

    <x> = triple integral of psi*(r,t).x.psi(r,t).dr (this is the expectation value of position of wavepacket)

    Is it possible to square a triple integral? Is <x^2> the same as <x>^2 ?

    I'm only wondering how the squared works in this situation, I would understand how to use <x> if the square wasn't there.

    Thank you!
  2. jcsd
  3. Mar 23, 2010 #2
    You cannot square the integral. The way it is written is (in 1 dimension):

    [tex]\left<x^2\right> = \int \psi(x)^{\dagger}x^2\psi(x) dx[/tex]

    It will be different in most cases from <x>^2. For example,

    [tex]\int x^2 dx = \tfrac13 x^3 \neq \left(\int x dx\right)^2 = \tfrac14 x^4[/tex]

    So you are unable to take the square outside of the integral.
  4. Mar 23, 2010 #3
    Got it, thank you very much!
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