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Helmholtz equation and Multislice approach

  1. Sep 29, 2014 #1
    Hi there!
    I have a problem about the proof of an equation in microscopy. I think this is the right section because it is about solving the Helmholtz equation.

    I'm looking to Wikipedia page
    where they try to solve the Schrödinger equation for an electron passing through a medium.
    In the theory section they write
    [tex]\phi(\vec r) = 1-i\frac{\pi}{E\lambda}\int\int_{z'=-\infty}^{z'=z}V(\vec X', z')\phi(\vec X', z')\frac{1}{i\lambda(z-z')}\exp\left(ik\frac{|\vec X-\vec X'|^2}{2(z-z')}\right)d\vec X' dz'[/tex]

    At the end, removing the convolution with the Fresnel propagator (i.e. discarding the Fresnel scattering) they say that in a multislice approach (which I think that mathematically is a sort of successive approximations method) where in every slice I consider V to be constant (the slices are on the z axis), the solution is
    [tex]\phi(\vec X, z_{n+1})=\phi(\vec X, z_n)\exp\left(-i\sigma\int _{z_n}^{z_{n+1}}V(\vec X, z')\right)dz'[/tex]

    I don't understand how this solution can be derived from the first equation. I think is something like the Dyson series, but I cannot decouple the potential V from the function phi.
    Please help!
    Last edited: Sep 29, 2014
  2. jcsd
  3. Oct 4, 2014 #2
    Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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