Given a $L^2$ function $f$ with $\int_\mathbb{R}xf(x)dx=0$, define its variance to be $\sigma_f^2=\int_{\mathbb R}x^2f(x)dx$. The uncertainty principle states that $\sigma_f\sigma_\hat f\geq 1/4\pi$, where hat denotes the Fourier Transform.
The most famous application of this lies in quantum mechanics, but I have heard of the following other "applications" that don't sound entirely right.
- "Time and frequency (actually energy) are Fourier Transform conjugates. So if we hear a long sound (for example from a flute), the time is long, so the time domain is very spread out. As a result, the frequency domain is likely to be very sharp and concentrated. Therefore, we can easily determine the pitch (and write it on a five-line stave) when we hear it. Vice versa, when the sound is short (e.g. a click on the mouse), we cannot easily tell its frequency, unless you are a genius, because the frequency domain is spread out." This theory is awkward for me because our brain cannot really perform Fourier transform on the infinite interval $(-\infty,\infty)$. We really judge the frequency based on the signals in a finite interval of time. What is the uncertainty principle for functions defined on finite intervals?
- " Radars measure positions by measuring the time taken for signals to return. Radars measure velocities of objects by noting the change in frequencies of the signals, reflected from objects. So if the radar wants to measure position accurately, it cannot measure velocity accurately, because frequency and time are Fourier conjugates." This is not very convincing. If the radar needs to send along signal for accurate velocity measurements, it can still measure the position accurately, by taking the time of the very beginning of the return signal. "Spreading out" doesn't imply that we have large uncertainty in time. 3Blue1Brown explain this by saying that noises makes signal random and "the very beginning" unclear, but this is not very convincing.
Source: The above two points are ideas from 3Blue1Brown videos.
I have a third from somewhere else:
- " In music, consonant sounds last long, because it contains fewer frequencies, and thus more spread on the time domain. Dissonant sounds, for the same reason, don't tend to last long, and usually keep changing."
Are those three statements about uncertainty principle correct?
If any of them are valid statements, just outline how I can prove or explain them mathematically.
