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Fourier Transform and Wave Function

  1. Apr 28, 2012 #1
    1. The problem statement, all variables and given/known data

    a) Find the normalization constant N for the Gaussian wave packet [itex]\psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/itex]. b) Find the Fourier Transform and verify it is normalized.

    2. The attempt at a solution

    a) I think I've got
    [tex]\psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/tex]
    [tex]\int |N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}|^{2}dx = 1[/tex]
    [tex]N^{2}=\frac{1}{\sqrt{\pi}K}[/tex]
    [tex]N = \frac{1}{\pi^{1/4}\sqrt K}[/tex]
    b) This is where the trouble starts...
    [tex]\psi (x) = \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/tex]
    [tex]F(\omega)=\frac{1}{\sqrt{2\pi}} \int \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} e^{i \omega x}dx[/tex]
    I think I can pull the normalization constant out of the integrand and get
    [tex]F(\omega)=\frac{1}{\sqrt{2\pi}}\frac{1}{\pi^{1/4}\sqrt K} \int e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} e^{i \omega x}dx[/tex]
    And the exponents should combine (here I'm not so sure)
    [tex]F(\omega)=\frac{1}{\sqrt{2\pi}}\frac{1}{\pi^{1/4}\sqrt K} \int e^{\frac{-(x-x_{0})^{2}}{2K^{2}}+i \omega x}dx[/tex]
    Assuming I've not gone horribly wrong earlier, evaluation of this integral stumps me. Any help and suggestions are much appreciated.
     
    Last edited: Apr 28, 2012
  2. jcsd
  3. Apr 28, 2012 #2
    If I crank through it, I think the result is

    [tex]F(\omega)=\frac{1}{\pi^{1/4}\sqrt{1/K^{}2}\sqrt K}e^{\frac{-K^{2}k{2}}{2}+i x_{0} \omega}[/tex]
     
    Last edited: Apr 28, 2012
  4. Apr 28, 2012 #3

    vela

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    There should be an ##\omega^2## somewhere in the exponent. Your answer seems to have other typos as well. It would help to see what you actually did, but I think you have it under control.
     
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