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3D-Fourier Transform of a delta-function?

  1. Apr 29, 2007 #1
    1. The problem statement, all variables and given/known data
    hi
    im trying to the integral int(delta(r-b)*exp(ikr)d^3r). but im not really getting anywhere.
    I´m trying to integrate over all space in spherical coordinates.
    The r part is easy i just do:

    delta(r-b)*exp(ikr)r^2*sin(a)*b*dr*da*db -> b^2*exp(ikb*cos(someangle??)*sin(a)*da*db

    (sorry that i´m not familiar with tex :( )

    I kinda need some help how to do the angular part.
    My idea was that the solution should not invole the angles in any sense that´s sure and i think that it´s some kind of trigonometric function but i got no clue how to get somewhere i have some feeling that it´s somethink link sin(bk) * normalization factor but how to get there ?

    Any help would be appreciated
    thanks :)
     
  2. jcsd
  3. Apr 29, 2007 #2

    Dick

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    The whole purpose in life of a delta function is to satisfy the condition int(delta(x-a)*f(x))=f(a). So your integral had better come out to be exp(ikb).
     
  4. Apr 30, 2007 #3
    but im integrating in polar coordinates not in one dimension?
    my intetgral (leaving out angular parts!) is int(delta(x-b) * exp(ikx))*r^2 dr isn`t it ?
     
  5. Apr 30, 2007 #4

    Dick

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    If it's a real 3d delta function the coordinates can't matter, can they?
     
  6. Apr 30, 2007 #5
    hmm yes but it`s a one dimensional delta function in front of a 3d object so you gotta take that into account and take dV=r^2*sin(a)*da*db dont you ?
     
  7. Apr 30, 2007 #6

    Dick

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    Ok, so it's a delta over the coordinate r, not the vector r integrated d^3r. Then what's exp(ikr)? Is r the coordinate r or is it a dot product? If the latter then you have more than one meaning for r in your expression.
     
  8. Apr 30, 2007 #7
    oh yea sorry for that it think i wrote that somehere in parentesis :)
    But youre right i got 2 meanings for one variable sorry :(
    you got any idea how to solve that anyways can´t find any solution anywhere even though it seems to be a pretty common problem in stat. mechanics :(
     
  9. May 1, 2007 #8
    :D can be deleted i solved it was really easy just hat to set my coordinate system so that k hat only a z-component :)
     
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