If you have any questions, please ask. I love yapping.
yap
Let's say we have some periodic function f. The goal is to represent it as a bunch of sine waves. This is called a Fourier series (FS).
The FS coefficients are given by
⌠2⋅π
1 ⎮ -i⋅k⋅x
f̂(k) = ─── ⎮ f(x)⋅e dx
2⋅π ⌡0
for whole numbers k. We can then approximate the original function with
n
___
╲ -i⋅k⋅x
╱ f̂(k)⋅e
‾‾‾
k = -n
and make the approximation better by making n larger.
For non-periodic functions, we can still get an approximation! The Fourier Transform (FT) uses a continuous range of frequencies instead of whole numbers, and the sum is replaced by an integral (the inverse FT).
FT's are used everywhere, for example in signal processing and spectroscopy. If a recording contains an annoying high-pitched noise, we can apply the FT and filter the unwanted frequencies. Spectroscopy separates light into its different wavelengths. By comparing the resulting spectral lines with known measurements, we can determine which elements and molecules are present in stars and planets, for example.
idk what more to add ._.
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The one that makes my noodle is why is it momentum and position and not velocity and position
Why is mass in it???? Is momentum the more fundamental quantity??
To be specific:
The fourier transform of position is wavenumber.
Momentum = plank's constant * wavenumber
Phase velocity = angular frequency / wavenumber
Group velocity = derivative of angular frequency wrt wavenumber
All of these are important concepts for physics and waves. Heisenberg's uncertainty principle also actually applies to the phase and group velocities of waves. The relationship is just less straight forward because instead of being multiplied by a constant, you have division and derivatives.
The frequency of a wave relates to the energy. Energy is also not a constant, so uh ... yeah.
But basically, the schrodinger equation helps you find the allowed frequency and wavenumber of particles in a system.
Then you can interpret them in whatever way you want. Those 2 things are the actually fundamental quantities. Well ... that and the phase of waves.