First of all, its helpful to appreciate what the induvial parts of the Fourier transform expression do. One important thing to realise is that for a single Fourier coefficient (F(Q) evaluated at one value (frequency) of Q) we must evaluate the entire real space function that we wish to sample (in this case the electron density p(r), at all possible values of r) .
Fourier Transforms
This is a video I created during lock-down to help describe visually what a Fourier transform does. It follows the idea that part of the mathematical expression for a Fourier transform can be described as a "winding machine", and it makes use of common cylindrical house hold items, which have different diameters and hence winding frequencies.
Fourier Transform Expression
Winding machines
The “exp(i Q ∙ r)” part of the expression act like a “winding machine”, winding the function (p(r)) back on itself about a cylinder with a frequency for the selected Q. This will make more sense when you watch the video below.
The other part of the function is an integral that evaluates the sum of the individual part of the expression once they have been wound back round on themselves. In this context it is easier to think about a real object (like the string in the video below). If when wind an object back on itself, characteristic features all line up together, this will produce a non-zero integral indicating that the sampled Fourier coefficient contributes to the overall Fourier transform. If however feature in the object oppose each other when they are wound at a given frequency, the integral will evaluate to zero. Again, hopefully this will all make more sense when you see me winding string around some cylindrical household object below!
Fourier Transforms with string!
Watch the video below to discover to gain a visual understand of how Fourier Transforms work. All you need is a piece of string and some cylindrical objects to do your own Fourier transform!
Spectrum of Fourier amplitudes
So what does our Fourier transform look like? In this instance we have done a very low resolution Fourier transform only considering a few possible values of Q. Obviously in reality you would do this for continuous values of Q between some sensible range that would denote the highest and lowest frequencies you might expect to find in your sampled function. However, as in this case I know my function (lines on the piece of string) only contain 3 inherent frequencies, I may still draw a complete spectrum of the Fourier amplitudes as I have done above.
