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Mathematical misconception in scattering: switching from cartesian to spherical

  1. Dec 22, 2012 #1
    If we were to consider a nucleon-nucleon interaction:
    We know that the incident wave (plane wave) is ψ= Ae[itex]^{ikz}[/itex], propagating in z direction

    But for some mathematical facilities, we tend to use spherical coordinates, the wave becomes = [itex]\frac{A}{2ik}[/itex][e[itex]^{ikr}[/itex]/r - e[itex]^{-ikr}[/itex]/r]

    How come?
    Where did the '2ik' in the denominator come from?
     
  2. jcsd
  3. Dec 23, 2012 #2
    where you have seen that.
    writing e(ikr) - e(-ikr)=2i sin(kr),we have second one as
    sin(kr)/kr which is different from e(ikz)=e(ikr cosθ)
     
  4. Dec 24, 2012 #3
    What? Sorry, but I didn't get where you're pointing to.
    What was written is as follows:

    The incident plane wave traveling in z direction:
    ψ=Ae[itex]^{ikz}[/itex]

    They then mentioned that it was mathematically easier to work with spherical waves e[itex]^{ikr}[/itex]/r and e[itex]^{-ikr}[/itex]/r.

    Lastly, for l=0,
    ψ=[itex]\frac{A}{2ik}[/itex]( e[itex]^{ikr}[/itex]/r - e[itex]^{-ikr}[/itex]/r)

    That was what's written in some nuclear course.
     
  5. Dec 24, 2012 #4
    e(ikz)lil(2l+1)jl(kr)pl(cosθ)
    lil(2l+1)(i/2k)[e-i(kr-l∏/2)/r-ei(kr-l∏/2)/r] pl(cosθ)
    jl(kr) is spherical bessel function and Pl you know.this form of spherical bessel can be gotten from spherical hankel functions of first and second kind(their addition).for l=o this reduces to the required form.
     
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