Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Dec 6, 2017 #1
    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.
     
    Last edited by a moderator: Dec 6, 2017
  2. jcsd
  3. Dec 6, 2017 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    How are we going to find out where you have gone wrong in your work if you do not post your work?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted