1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Using given Fourier transform to find the equation for the wave packet.

  1. Sep 30, 2012 #1
    1. The problem statement, all variables and given/known data
    Any wavepacket can be obtained by the superposition of an infinite number of plane waves using the so-called Fourier integral or Fourier transform
    [itex]f(x,t) = \frac{1}{\sqrt{2\pi}} _{-\infty}\int^\infty A(k)e^{i(kx-\omega t)} dk[/itex]

    Find at t=0 the representation of the wavepacket f(x) associated with the flat distribution given by:

    A(k) =
    0 for k<-K and k>K
    [itex]\frac{1}{\sqrt{2K}}[/itex] for -K < k < K

    2. Relevant equations
    The textbook I found that isn't leaving me entirely confused has replaced k with p (momentum), but that doesn't seem to be overly relevant to my lack of understanding. The one I've found that seem to be in the ball park is:
    [itex]\left|A(p,t)^2\right| = \left|A(p)e^{\frac{ip^{2}t}{2mh-bar}}\right|^2 = \left| A(p,0) \right|^2[/itex]


    3. The attempt at a solution
    I have figured out that A(k) is the Fourier transform, but after that I run into a brick wall. I can't even seem to get far enough to be able to make useful searches. I'm getting the impression that I need to do some more manipulation so that I can use the above equation, but right now it'd just be blind hammering without understanding why.

    I feel like there may have a linking concept I'm not getting, even just a nudge in the right direction would be extremely helpful!
     
  2. jcsd
  3. Sep 30, 2012 #2
    You've been given [itex]A(k)[/itex], and a formula that describes [itex]f(x,t)[/itex] in terms of it. So finding [itex]f(x,0)[/itex] should just be a matter of plugging things in.

    Since [itex]t=0[/itex], you can drop the [itex]\omega t[/itex] term from the integral. So right off the bat, we have:
    [tex]f(x, 0) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} A(k)e^{ikx} dk[/tex]

    Now, see if you can determine how the integral is affected by substituting in the definition of [itex]A(k)[/itex].
     
  4. Sep 30, 2012 #3
    Wow, I was so focused on trying to figure out what the transform did that I completely overlooked the basics. Did not cross my mind once about the omega disappearing.

    Thanks a bunch! You're right, it's just simple substitution now. Lesson learned!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Using given Fourier transform to find the equation for the wave packet.
Loading...