Fourier transformation [closed]

Question

I have recently studied Fourier and Laplace transformation in maths. I wanted to understand the utility in physics with some examples that requires this change in dimension and the reason why.

Answer 1

This question is pretty broad, but I would summarise thus: there are three, related ways that these integral transforms are important:

1. Both serve as tools for "splitting up" a problem (forwards transform) into simpler problems, analysing the latter, and then using the principle of linear superposition to build (inverse or backwards transform) the whole analytical description up from the simpler solutions. In the used of the Fourier transform, you are analysing a system's response to harmonic excitation (sinusoidal waves) and then building up its response to a general pulse by summing up the responses with the inverse Fourier transform. The Laplace transform is like the Fourier transform with a half-infinite domain ($[0,\,\infty)$ instead of $(-\infty,\,\infty)$ and with a generalised, complex frequency. It is useful for "causal" systems where the excitation has a definite beginning ($t=0$);

2. The transforms transform differential equations in desirable ways: converting the differential operator $\psi\mapsto\mathrm{d}_t\psi$ into a multiplication operator $\psi\mapsto i\,\omega\,\psi$;

3. The Fourier transform is central to quantum mechanics: in particular to the canonical commutation relationship and the Heisenberg uncertainty principle. The FT is the unitary (norm and inner product preserving, i.e. probability-preserving) transformation between position co-ordinates and momentum co-ordinates. The mathematical uncertainty product relationships, the related Paley-Weiner theorem and the special case observation that a function and its FT cannot both have compact support (domain wherein they are nonzero) are all manifestations of the kinds of "mathematical mechanics" that beget the uncertainty principle. Put simply: if you confine a wavefunction (i.e. quantum state) to a small range of positions, its Fourier transform is the same quantum state written in momentum co-ordinates, so the spread over momentums increases as you confine the positions more and more.

I say more about the application of Fourier transforms to electromagnetic theory in this answer here, more about the relationship between the CCR and the FT in this answer here and more about the mathematics of the uncertainty product in this answer here and here

Answer 2

Just to give 3 simple examples:

Someone is playing piano. Every key he hits, will produce not only the desired tone but also a full range of resonants and higher harmonics. Those will show up in fourier space.

In image analysis, sometimes you have periodic patterns overlaying your image (e.g. Moiré fringes) that disturb image quality. In Fourier space, those patterns might show in a very confined frequency domain, where they can be filtered to enhance image quality.

When working in biomedical physics, you come in touch a lot with projection integrals if it comes to attenuation measurement. Solving those inverse problems is a lot easier in Fourier Space. (See for example X-ray CT and the Fourier Slice Theorem).