1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: How to fourier invert a plane wave

  1. Mar 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Ok so got a solution to the Klein-Gordon equation and need to solve for a(k)

    [tex] \varphi(x) = \int \tilde{dk} \left[ a(\bold{k}) e^{ikx} + a^{\dagger}(\bold{k} ) e^{-ikx} \right] [/tex]

    [tex] \tilde{dk} = \frac{d^{3}k}{(2 \pi)^{3} 2 \omega} [/tex]

    The way it's done in Srednicki p.26 has me confused when taking the fourier transform of [itex] \varphi [/itex]

    [tex] \int d^3x e^{-ikx} \varphi(x) = \frac{1}{2\omega} a(\bold{k}) + \frac{1}{2\omega} e^{2i\omega t} a^{\dagger}(\bold{-k} ) [/tex]

    2. Relevant equations

    [tex] kx = \bold{k} \cdot \bold{x} - \omega t [/tex]


    3. The attempt at a solution


    [tex] \int d^3x e^{-ikx} \varphi(x) =\int d^3x e^{-ikx} \int \tilde{dk} \left[ a(\bold{k}) e^{ikx} + a^{\dagger}(\bold{k} ) e^{-ikx} \right] [/tex]


    [tex] = \int d^3x \int \tilde{dk} a(\bold{k}) + \int d^3x \int \tilde{dk} e^{-2kx} a^{\dagger}(\bold{k} ) [/tex]


    so the problem is how do these integrals with respect to dx and dk disappear?
     
  2. jcsd
  3. Mar 29, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    dummy!

    Hi waht! :smile:

    It's a different k …

    you have ∫∫ eikx eik'x … for a dummy k' :wink:
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook