Undergrad Fourier transform -- what physical variables am I allowed to transform between?

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The discussion centers on the use of Fourier transforms in physics, particularly in transforming between momentum-space and position-space. It highlights that transformations are typically made between time-varying signals and temporal frequencies, or spatially-varying signals and wavenumber. The key consideration is whether the transformed variables correspond to physically meaningful functions, as not all transformations yield useful insights. Additionally, it emphasizes that the product of the conjugate variables in the transformation must be dimensionless for the functions to be valid. Understanding these relationships is crucial for applying Fourier transforms effectively in various physical contexts.
Higgsono
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A common use of the Fourier transform in physics is to transform between momentum-space and position-space. But what physical variables am I allowed to transform between? For instance can I use the Fourier transform to go from momentum space to frequency space or whatever?
 
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Usually, one transforms between a time-varying signal and a temporal frequency, or a spatially-varying signal and 'wavenumber' (equivalently, spatial frequency or angle).

Does that help?
 
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Andy Resnick said:
Usually, one transforms between a time-varying signal and a temporal frequency, or a spatially-varying signal and 'wavenumber' (equivalently, spatial frequency or angle).

Does that help?

But why? What pair of variables are allowed and why?
 
Higgsono said:
But why? What pair of variables are allowed and why?
A Fourier transform is a way of writing a given function as a sum of sinusoids, so I can Fourier transform just about any function that meets some minimal standards for well-behavedness. The interesting question is whether that's useful: do the sinusoids correspond to any physically interesting function? For example, Fourier transforming a sound signal tells me what frequencies have been superimposed to produce that signal... but Fourier transforming the elevation above sea level along a path is unlikely to tell me anything interesting (unless the topography happens to include some unusually evenly spaced and symmetrical hills, which would show up as a spike in the transform).
 
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Higgsono said:
But why? What pair of variables are allowed and why?

I'm not sure what you are getting at. For one thing, the product of whatever conjugate variables you choose (call them 'K' and 'L') must be dimensionless.
 
Andy Resnick said:
I'm not sure what you are getting at. For one thing, the product of whatever conjugate variables you choose (call them 'K' and 'L') must be dimensionless.

What about the dimensions? RHS should have the same dimension as the LHS?
 
Higgsono said:
What about the dimensions? RHS should have the same dimension as the LHS?

In order for the functions sin(KL), cos(KL), exp(iKL), etc. to be evaluated, 'KL' must be a pure number.
 

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