When studying time series analysis, the answer is usually ‘the FT variables are defined such that they are duals of one another’ but this isn’t a very satisfying answer and doesn’t explain why the same pairs come up again and again.
Are the uncertainty principle pairs the same because the uncertainty principle relies on Fourier transforms in some way or vice versa?
Or is there some underlying mechanism or symmetry which causes then to pop up all over the place, and if so, how does it cause them to appear?
In: Mathematics
They’re all essentially equivalent. I’ll cover one way to see the relationship between them all, though it’s definitely not the most general.
Let’s say you have some state defined in the basis of some coordinate (for example x position) and you want to find the operator that translates the state by s. In the that coordinate basis, the translation operator will end up being T(s) = exp(-s∂/∂x) = exp((s/ih)p) where p is some hermitian operator called the *generator of translation in x*. In the case of x position, that’s x momentum. In the case of angle, it’s angular momentum, etc…
Now **Noether’s Theorem:** Assume the hamiltonian of your system does not change with translation in your chosen coordinate. Then HT(s) = T(s)H and so the commutator [H, T(s)] = 0. You can prove that this must imply [H, p] = 0 as well, and as the change in the expectation value of an observable over time is proportional to this commutator, it implies that <p> is constant.
For the **commutator** relationship, it’s relatively simple to prove that the commutator of a coordinate with the generator of its translation is [x, p] = [x, -ih∂/∂x] = ih.
The **uncertainty principle** is actually defined in terms of commutators. It’s not too hard to prove that for any two operators A and B, stdev(A)stdev(B) ≥ |E([A, B])|/2. Note that since [x, p] = ih, this implies the usual uncertainty principle you’re probably used to, but it’s more general.
Finally, for **fourier trasnforms**, using the whole p=-ih∂/∂x thing we found earlier, you can try to find eigenfunctions of this operator. If you do, you end up seeing that plane waves at different frequencies are each eigenfunctions of p. Pretty much the entire point of Fourier transforms is seeing how you can decompose arbitrary functions into integrals of plane waves, so it’s quite straightforward to see why you swap back and forth between the x and p bases using Fourier transforms.
Btw, in the future, r/AskPhysics might be better for this kind of question!
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