What does the inverse Fourier transform represent in quantum scattering studies?

[sol66]

So its been awhile since I've taken PDE, and forgot a lot about Fourier transforms. Anyways I'm trying to understand what the inverse of the Fourier transform actually represents. I understand perfectly how the infinite sum of periodic functions can be used to create any periodic function when summing in respects to your wave number or frequency k, however I don't understand what the inverse of that transform actually represents. It seems that through the inverse you integrate over x to find a function of frequency. What exactly does this function actually represent? Is it a function that can be represented by a sum of x functions ( this statement doesn't even make sense to me )?

This question is important to me because in my studies of quantum scattering ( or really just any other scattering problems regarding waves ) we deal with k-space which is the inverse Fourier transform and I don't completely understand it because of my lack of comprehension of the inverse fourier.

Thanks you all for the replies

 

[marcusl]

The inverse transform of the spectrum is the original function or sequence. In signal processing, the IFT gives the time-domain waveform from the spectrum.

k-space is also called reciprocal space, and it is the space of the forward (not inverse) transform. Given a crystal lattice in physical space, the lattice in reciprocal space governs scattering, etc. The wavenumber k is a spatial frequency and has units of (1/length). Thus a function (1, 2 or 3-D) in k-space is the spatial spectrum of a physical function (also 1, 2 or 3-D) in real space.