# In Fourier transforms, can momentum space be analogized to frequency, and position space be analogized to wavelength?

We know that in quantum mechanics, momentum space is the fourier transform of position space (and vice versa)

And also, in time-series analysis, that frequency (of cycles) is the fourier transform of the distribution of all cycle lengths.

What about electromagnetic radiation? Is the distribution of frequencies the Fourier transform of the distribution of wavelengths?

Is it physically feasible to think of a distribution of positions (each position value with a certain count), and then to take a fourier transform of that, and end up with a distribution of momentum values? Even in JPG compression, you have frequency and position (each position value has a certain count that corresponds to the color value on a scale of $0$ to $255$)

• Your question is not clear. The Fourier transform is a very well defined mathematical operation that decomposes a function into sines and cosines. It is up to you (or the field of science) to assign meaningful names/interpretations to the original domain and the transformed domain. So you end up with time/frequency or spacing/wavenumber, etc. If I have misunderstood your question, please clarify. – Lorem Ipsum Sep 7 '11 at 5:04
• Ah yes - it is up to the field of science to assign meaningful names/interpretations to the original domain and the transformed domain. What I was trying to ask was this: are there common features within the original domain, as compared to within the transformed domain? – InquilineKea Sep 7 '11 at 5:28
• In other words, is "momentum" a "frequency" of position? – InquilineKea Sep 7 '11 at 5:29

## 1 Answer

The notions of position and momentum are not fundamental to the uncertainty principle, but the fact that position and momentum are analogous to instantaneous time and instantaneous frequency is. There is no necessity to translate the spatial domain of an image and its fourier representation in terms of position and momentum. The notion of frequency in this case expresses how fast an image changes or where one finds sharp discontinuities/edge like structures in an image.

This question might be motivated by a misunderstanding of the origins of the Heisenberg Uncertainty principle. The basis of any uncertainty princple is time-frequency uncertainty, i.e. given any two functions where one of them can be expressed in terms of the fourier transform of the other, both cannot be localized in their respective domains. See 1 for other types of discrete-time uncertainty principles that don't have the classical time-frequency interpretation.