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Fourier Transform and Wave Function 
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#1
Apr2812, 03:19 PM

P: 196

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{(xx_{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{(xx_{0})^{2}}{2K^{2}}}[/tex] [tex]\int N e^{\frac{(xx_{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{(xx_{0})^{2}}{2K^{2}}}[/tex] [tex]F(\omega)=\frac{1}{\sqrt{2\pi}} \int \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{(xx_{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{(xx_{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{(xx_{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. 


#2
Apr2812, 03:53 PM

P: 196

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] 


#3
Apr2812, 05:07 PM

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Thanks
PF Gold
P: 11,755

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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