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

[Higgsono]

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?

 

[Andy Resnick]

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?

 

[Higgsono]

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?

 

[Nugatory]

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).

 

[Andy Resnick]

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.

 

[Higgsono]

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?

 

[Andy Resnick]

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.