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Homework Help: Troublesome Integral

  1. Aug 17, 2005 #1
    A quantum mechanics problem calls for the reader to find the momentum space wave function of [tex] \Psi(x,0) = A/(x^2 + a^2) [/tex]. But I do not know how to resolve the fourier transform:

    [tex] \Phi(p, 0) = \frac{1}{\sqrt{2 \pi \hbar}}\int_{-\infty}^\infty e^{-i p x/\hbar} \frac{A}{x^2+a^2}dx.[/tex]​

    The problem implies an exact solution can be found since it subsequantly asks you to check normalization and compute the expected values of p and p2 using the transformed fn. Mathematica evaluates the transform in terms of a special fn MeijerG.
     
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  3. Aug 17, 2005 #2

    Tide

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    The integral has singularities at x = ia and x = -ia. To evaluate the integral you need evaluate the residues resulting from closing the path either in the upper half plane (Im z > 0) or the lower half plane (Im x < 0) depending on the sign of x (i.e. with a semicircle whose radius approaches infinity).
     
    Last edited: Aug 17, 2005
  4. Aug 17, 2005 #3
    Singularities? I don't understand. The integrand is well-defined for all real values x and the integral almost certainly exists over any range. Sorry, I have not had a course in complex analysis.
     
  5. Aug 17, 2005 #4

    Tide

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    genxhis,

    Sorry about that - I edited my original post to reflect the correct locations of the singularities.

    I assumed you were familiar with complex analysis but, since you're not, what I said won't make sense to you. I'll have to think a bit about how to do it without invoking complex analysis.
     
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