Fourier Transform and Wave Function

[atomicpedals]

Homework Statement

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.

 

[atomicpedals]

If I crank through it, I think the result is

$$F(\omega)=\frac{1}{\pi^{1/4}\sqrt{1/K^{}2}\sqrt K}e^{\frac{-K^{2}k{2}}{2}+i x_{0} \omega}$$

 

[vela]

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.