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Integral help needed for Quantum Physics problem

  1. Sep 10, 2008 #1
    1. The problem statement, all variables and given/known data

    integral from - infinity to + infinity of
    N/(k2+a2) * eikx dk

    2. Relevant equations

    this is for a quantum physics problem (chapter 2 problem 1, gasiorowicz) where I am given A(k) = N/(k2+a2) and must calculate psi(x)

    I am using the equation
    psi(x,t) = integral from - infinity to + infinity A(k) ei(kx-wt) dk

    which when t=0 goes to

    psi(x,t) = integral from - infinity to + infinity A(k) eikx dk

    3. The attempt at a solution

    I've tried integrating by parts, substitution and on my TI-89 however I am a little rusty with all these methods

    Any help would be greatly appreciated
     
    Last edited: Sep 10, 2008
  2. jcsd
  3. Sep 10, 2008 #2

    Dick

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    You need to do these sort of integrals in the complex k plane using contour integration. Review the residue theorem and some examples of how to use it and then take another look at the problem.
     
  4. Sep 11, 2008 #3
    Thank you so much! would the residue then be e-ax/2ai ?
     
  5. Sep 11, 2008 #4

    Dick

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    Something like that, yes. If you want more detailed help you should tell us how you got it. What did you get for the integral?
     
  6. Sep 11, 2008 #5
    well, as I am not well-versed in complex analysis I looked up residue theorem and found an example on wikipedia which I modified to fit my situation. The work is as follows

    -∞dk (1/(k+ai)-1/(k-ai)) e^ikx

    Which has a singularity at ai=k

    so Res k=ai = (e^ikx)/2ai

    so I multiply by 2*pi*I to get (pi*e^-ax)/a

    and then put an absolute value on the "ax" which comes from integrating along the bottom of the arc of the line integral

    Does this make sense? is there somewhere I can start to understand exactly how line integrals and residue theorem works?
     
  7. Sep 12, 2008 #6

    Dick

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    You've left out a lot of the details, and in the first line the integrand should be exp(ikx)/((k+ia)(k-ia)) but yes that's it. I don't have any favorite references, but you can probably find a lot more examples on the web or in books on the subject of applied mathematics.
     
    Last edited: Sep 12, 2008
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