(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show, by completing the square in the exponent, that the Fourier transform of a Gaussian wavepacket ##a(t)## of width ##\tau## and centre (angular) frequency ##\omega_0##:

##a(t)=a_0e^{-i\omega_0t}e^{-(t/\tau)^2}##

is a Gaussian of width ##2/\tau##, centred on ##\omega_0##, given by:

##a(\omega)=\frac{a_0\tau}{2\sqrt \pi}e^{-(\frac{\omega-\omega_0}{2/\tau})^2}##

2. Relevant equations

I just used the Fourier transformation:

##\frac{1}{ \sqrt{2\pi}}\int \psi(t)e^{-i\omega t} \, dt## the limits of integration is all ##\Bbb{R}##

3. The attempt at a solution

Well I subbed in ##a(t)## for ##\psi(t)## and then carried the integral through and I got:

##\frac{a_0\tau}{\sqrt2}e^{-(\frac{\omega+\omega_0}{2/\tau})^2}##. As you can see the exponent should be ##\omega- \omega_0## and there is a missing factor of ##\frac{1}{\sqrt{2\pi}}##. I have looked through my work endlessly and can't find any mistakes so is that the right formula that I am using above or is there an alternative? If that is the right formula above then does that mean the solution is wrong?

So I completed the square of a(t) which gave me:

##a_0e^{-(t/\tau+(1/2)\tau i\omega_0)^2}e^{-\tau^2 \omega_0^2 /4}##

I then plugged this into the integral which gave me:

##\frac{1}{ \sqrt{2\pi}}\int a_0e^{-(t/\tau+(1/2)\tau i\omega_0)^2}e^{-\tau^2 \omega_0^2 /4}e^{-i\omega t} \, dt##

I completed the square of this which gave me:

##\frac{1}{ \sqrt{2\pi}}e^{-\tau^2 \omega_0^2 /4}\int a_0e^{-(t/\tau+(\omega+\omega_0)i\tau/2)^2}e^{-\omega^2 \tau^2 /4}e^{-\omega_0 \omega \tau^2 /2} \, dt##.

Then I simplified this down to:

##\frac{a_0\tau}{\sqrt2}e^{-\frac{\tau^2}{4}(\omega+\omega_0)^2}##.

I then simplified this down further to give me my incorrect answer at the top of the page.

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# Homework Help: Show the Fourier transformation of a Gaussian is a Gaussian.

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