# Fourier Transform and Wave Function

by atomicpedals
Tags: fourier, function, transform, wave
 P: 196 1. The problem statement, all variables and given/known data a) Find the normalization constant N for the Gaussian wave packet $\psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}$. b) Find the Fourier Transform and verify it is normalized. 2. The attempt at a solution a) I think I've got $$\psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}$$ $$\int |N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}|^{2}dx = 1$$ $$N^{2}=\frac{1}{\sqrt{\pi}K}$$ $$N = \frac{1}{\pi^{1/4}\sqrt K}$$ b) This is where the trouble starts... $$\psi (x) = \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}$$ $$F(\omega)=\frac{1}{\sqrt{2\pi}} \int \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} e^{i \omega x}dx$$ I think I can pull the normalization constant out of the integrand and get $$F(\omega)=\frac{1}{\sqrt{2\pi}}\frac{1}{\pi^{1/4}\sqrt K} \int e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} e^{i \omega x}dx$$ And the exponents should combine (here I'm not so sure) $$F(\omega)=\frac{1}{\sqrt{2\pi}}\frac{1}{\pi^{1/4}\sqrt K} \int e^{\frac{-(x-x_{0})^{2}}{2K^{2}}+i \omega x}dx$$ Assuming I've not gone horribly wrong earlier, evaluation of this integral stumps me. Any help and suggestions are much appreciated.
 P: 196 If I crank through it, I think the result is $$F(\omega)=\frac{1}{\pi^{1/4}\sqrt{1/K^{}2}\sqrt K}e^{\frac{-K^{2}k{2}}{2}+i x_{0} \omega}$$