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Expectation value of a wave function

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data

    The wave function of a state is Psi(x)= N*a(x)exp(i*p0*x/h)where a(x) is a quadratically integrable real valued function Show that the expectation value of the function is p0.

    2. Relevant equations



    3. The attempt at a solution

    The only thing I'm having a problem with is how to integrate the square of the wavefunction so that I could normalize N and put the operator p=-ih(d/dx) in the integral. So I think I know what to do, just not sure how to do it.
     
  2. jcsd
  3. Sep 22, 2009 #2
    First off the expectation value of what? Position?

    Also what is p0, just a real-valued constant?
     
  4. Sep 22, 2009 #3
    It's the expectation value of the momentum I'm after. Sorry, forgot to put that in. And yes, p0 is a real-valued constant.
     
  5. Sep 22, 2009 #4
    All right so to normalize the wave function the wave function squared (probability) must be one. In this case you need to integrate [itex]\Psi^{*} \Psi[/itex] over all possible values (-inf to inf) and set that equal to 1, then solve for N.
     
  6. Sep 22, 2009 #5
    But that's the thing. I have no idea of how to do the integral.
     
  7. Sep 22, 2009 #6
    Well what have you tried? Start by setting up the integral and working as far as you can.
     
  8. Sep 22, 2009 #7
    well I get stuck when i'm supposed to integrate a(x)^2*exp(i*2*p0*x/h). How do I do that?
     
  9. Sep 22, 2009 #8
    [tex] 1 = \int_{- \infty}^{\infty} \Psi^{*} \Psi dx [/tex]

    Psi* is the complex conjugate of Psi. Do you know how to take a complex conjugate?

    exp(i*2*p0*x/h) makes it seem like you're trying to do:

    [tex] 1 = \int_{- \infty}^{\infty} \Psi \Psi dx [/tex]

    When you need to be doing:

    [tex] 1 = \int_{- \infty}^{\infty} \Psi^{*} \Psi dx [/tex]
     
  10. Sep 22, 2009 #9
    For instance the complex conjugate of [tex]f(x) = A e^{i x}[/tex] is [tex]f(x)^{*} = A e^{-i x}[/tex] where A is a real constant.
     
  11. Sep 22, 2009 #10
    Thanks, I think I got it right.
     
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