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Average Momentum Integral

  1. Dec 13, 2005 #1

    eck

    User Avatar

    I have the physics question along with the solution, but in the solution I don't understand how they evaluated the integral. I can't get my brower to preview TeX input, so I'm going to leave it without formatting, but you can find the problem here and the solution here. The problem I am looking at is number one. I've also got the problem and solution posted here (w/o formatting) but it's kind of hard to read:
    Problem
    --------------------
    A particle's coordinate space wavefunction is square-integrable and real up to an arbitrary multiplicative phase:
    psi(x) = exp(i * alpha) phi (x)
    with alpha real and constant and phi(x) real. Prove that its average momentum is zero.
    Solution
    -------------------
    Setting up the integral is easy, and you can pull out a couple constants. So you have an infinite integral with this inside:
    dx exp(-i alpha) phi(x) exp(i alpha) d/dx[phi(x)]
    Somehow, in the solution, they pull out 1/2 and leave the following in the integral:
    dx d/dx[phi(x)^2]
    When I look at it, I see the exponentials cancelling, but I don't understand where the 1/2 comes from and how the first phi(x) gets pulled into the derivative.
    Can anyone shed any insight on how this integral was simplified?
     
  2. jcsd
  3. Dec 13, 2005 #2

    eck

    User Avatar

    I was looking at the problem some more, and all of a sudden it hit me. It's kind of embarrassing that I didn't see it before. If anyone else looks at it... nothing tricky is involved. It's integration by parts, but it's so obvious I didn't even see it.
     
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