quantum mechanics

Probably the single most important and relevant result from quantum scattering theory is our finding that the scattered spherical wave in the firstorder Born approximation is simply the Fourier transform of the potential. That means if we can measure and reconstruct the wave {whether it is using Xray crystallography or electrons in cryoEM], we can recover an image of the potential, or equivalently, of the charge density for our molecule of interest. To prove to ourselves that this is true, we use Poisson's Law {which is a result from classical electromagnetism that follows from Maxwell's eq uations}, vzvr?) = 4rrp(F) (3.1) to show that our primary result from elastic scattering within the FirstOrder Born Approximation, that the scattering amplitude is the Fourier transform of the potential, f(k,k') or: J's? w?) eff-\"f {3.2) can be rewritten in a form that reveals the scattering amplitude is the Fourier transform of the charge density [or electron density when Xrays are used], i.e., fe 36'} oc Id? 190'\") Elly\"'- {3.3) This is much easier than it might at first appear. To do this, we can take advantage of a fundamental result from Fourier Analysis: If the function Wk) has a Fourier Transform that is my, i.e., 1 . it} = Id): VLI) at\" V 2:: and the function Ufa) has a derivative d V(x}fd.x, we can write its Fourier transform as f1lk]: a): I p. M m a; Using integration by parts, and noting that the function 'v'lxl goes to D as 3: goes to ice, it can be shown that fi(k) = (-ik) f(k), and fr(k) = (-ik)" f(k) for the derivative of order n. In particular, this means for the second derivative, we can use f2(k) = (-ik)- f(k) Using the 3-D version of the last result, along with Poisson's Law, enables you to arrive at equation 3.3 from equation 3.2. Do it. Hint: The proportionality sign means you don't have to worry about pre-factors