What properties determine whether one variable can be fourier-transformed into anothe

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Simfish
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Frequency can be fourier-transformed into time and vice versa. They're inversely related.

In QM, momentum can be fourier-transformed into position and vice versa. But they're not necessarily inversely related. The uncertainty in time and the uncertainty in position merely have a lower bound.

So how do you determine how one variable can be fourier-transformed into another variable?
 

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Frequency can be fourier-transformed into time and vice versa. They're inversely related.
Not exactly. Fourier analysis works on groups of information items (such as position at a particular time for many different points of time) So it can translate a group of positions at different times into a group of frequencies and vice versa.
 
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Given any continuous function of a variable x, we can apply a Fourier transform that will decompose the function into a sum of sin(nx) and cos(nx) (in this case, n can take on any real value at all, and we can give the sines and cosines any complex weighting)

The n in this case is a "frequency" with respect to x.

In quantum systems, we can take the position wave function (which gives us the probability amplitude at each position) and Fourier transform it with respect to position. The "frequency" of the sines and cosines with respect to position is called the "wave length", and according to deBroglie, this is hbar * momentum.

So when we do a Fourier transform, we get some quantity like a frequency or wavelength. That this is related to any other physical quantity like momentum is put in by physics.

On the other hand, we can formally define a momentum via the Fourier transform of the wave function of some "position type measurement", and this generalized momentum can have many of the same properties of the usual momentum.
 

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