- #1
thomas19981
Homework Statement
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}##
Homework 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}##
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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