Fourier Transforms: Proving Proportionality

[smallgirl]

1. We consider the on shell wave packet:
$$\varphi(t,x)=\int\frac{dk}{2\pi}exp(-\frac{(k-k_{0})^{2}}{\Delta k^{2}}+ik(t-x))dk$$

I need to show it is proportional to:
$$exp(ik_{0}(t-x)-\frac{\triangle k^{2}}{4}(t-x)^{2})dk$$
through a Fourier transform of the gaussian


3. I used a Fourier transform of the form e^(ikx) but this doesn't seem to give me the right answer as I end up with something proportional to $$exp(-\frac{(k-k_{0})^{2}}{\triangle k^{2}}+ikt)dk$$ before integrating

 

[vela]

Show us what you think the integral for $\varphi(t,k)$ is.

 

[smallgirl]

Solved it! :-)...

However I now need to solve this:

$$\int\frac{dk}{2\pi}exp(-\frac{(k-k_{o})^{2}}{\triangle k^{2}}+ik(pt-x)$$


where $$p=1-\frac{h_{00}}{2}-h_{01}-\frac{h_{11}}{2}$$

by using Fourier transforms

 

[smallgirl]

Solved this one too now :-)

Not sure how to graph it though...