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Fourier transform of vector potential

  1. Nov 3, 2015 #1
    1. The problem statement, all variables and given/known data
    I have question on doing the following indefinite integral:
    $$\int{d^3x(\nabla^2A^{\mu}(x))e^{iq.x}}$$

    2. Relevant equations
    This is part of derivation for calculating the Rutherford scattering cross section from Quarks and Leptons by Halzen and Martin. This books gives the following result obtained by partial integration of the above integral:
    $$\int{d^3xA^{\mu}(x)(\nabla^2e^{iq.x})}$$

    3. The attempt at a solution
    I tried to use the identity from vector calculus:
    $$\nabla^2(\phi\psi) = \phi\nabla^2\psi + \psi\nabla^2\phi + 2\nabla\phi.\nabla\psi$$
    But not sure how to get rid of the other terms.
    Any help is most welcome.
     
    Last edited: Nov 3, 2015
  2. jcsd
  3. Nov 7, 2015 #2

    Orodruin

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    No, you need to use partial integration as mentioned by the authors and not vector identities. I suggest you start in one dimension and then see how it generalises. Also remember that the fields are assumed to go to zero at infinity such that the boundary terms from the partial integration vanishes.
     
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