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I Problem during finding <x> in p space

  1. Oct 18, 2016 #1
    Thank you advance for help!

    So the problem is find <x> in P space.
    The solution's logic is use formula: upload_2016-10-18_9-35-45.png


    The submit with upload_2016-10-18_9-35-18.png


    Then one will get the integral upload_2016-10-18_9-36-33.png

    by noticing that upload_2016-10-18_9-36-48.png , one can replace xeipx/ħ in the integral

    so that upload_2016-10-18_9-38-0.png

    and then after solving the integral one can find the expectation value for x.

    The part I don't understand is why upload_2016-10-18_9-39-1.png , firstly I am not sure where is the minus sign comes from. Next I don't think I can just change the order of eipx/ħ and Φ because the term d/dp.

    Could anyone help me figure the integral out?

    Thank you everybody.
    Shell
     

    Attached Files:

  2. jcsd
  3. Oct 18, 2016 #2
    I think that integration by parts comes into this one.
     
  4. Oct 18, 2016 #3

    DrClaude

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    Staff: Mentor

  5. Oct 18, 2016 #4
    @Jilang @DrClaude Thank you for help! I mislead by the integral sign to think it gained by doing some rearrangement inside the integral. Everything become clear if I do integral by part and use the fact that momentum operator is hermitian. Thanks a lot!
     
  6. Oct 18, 2016 #5

    dextercioby

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    The tacit assumption is that any wave function decreases at infinity slower than any polynomial, thus can be safely assumed to be a Schwartz test function. The Schwartz space is invariant under the Fourier transform and provides a domain of essential self-adjointness for the momentum operator over the entire real line.
    Because of this explanation, you can use partial integration in the improper definite integral and discard the so-called "surface term" (a misnomer, because we're in 1D).
     
    Last edited: Oct 19, 2016
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