Signal Processing with Heisenberg
Transmission Control -
Anyone who has done much audio signal processing likely has a better than average intuitive grasp of the Heisenberg‘s Uncertainty Principle assuming you know what to look at.
The Fourier transform is used widely in signal processing to convert a signal between time and frequency domain. You can change from a time series of amplitudes (waveform on an oscilloscope) to a frequency series of amplitudes (histogram on a graphic equalizer) and vice versa. Other pairs of complex conjugate properties of particles (like position vs momentum) are also related by Fourier transform. Uncertainty is a fundamental property of all waves, not just the ones in quantum mechanics.
Imagine for a second that you have something that produces a nice ideal single-frequency tone with no noise. You’ve also got a perfect microphone hooked up to an analog oscilloscope of infinite precision.. Both are in an ideal noiseless and echo free environment. Assume none of the materials are heat or radiation sensitive and none of the electronics introduce any noise. Basically cool the whole room down to absolute 0, take out all the air and fill the room with some sort of non-quantum medium that sound can travel though. Also assume I gave you a way to do a Fourier transform on an analog signal without taking discreet measurements at some sample rate.
Now then, in order to measure the frequency, in the best case you have to be able to measure just over a half wavelength to be sure you’ll see both a peak and the point it crosses 0 which is a quarter wavelength. That means to measure the frequency, you now have a time uncertainty of +/- a quarter wavelength since the position of what you’ve just measured is “spread out” over the period between the peak and crossing. So what is the amplitude of this frequency? An average of the amplitude over that quarter wavelength. That’s going to be a very bad approximation to the amplitude at any given point in time. You can’t know both the time and frequency domain amplitudes to the same precision simultaneously because they can’t be simultaneous if one represents an infinitesimally small time and the other is a quarter wavelength of time.
NOTE: There’s a bit of hand-waving approximation in that last paragraph because of all the idealization and the hypothetical continuous Fourier transform. The actual fuzziness would likely be governed by the Nyquist-Shannon sample theorem and limits of the Fourier transform itself. Hopefully it gets the idea across.
The ideal quarter wavelength limit is a necessary quantization of the time for the frequency. Of course us digital signal processing folks know all about quantization error. We introduce a lot more of it in the analog to digital conversion, resampling and both sorts of compression.
What does it do? Right, it adds noise. So now you’ve got both uncertainty over the time and additional apparent randomness even though we isolated all the usual sources of noise in a highly impossible way. In the real world, you’ve also had to quantize the initial time series of amplitudes because the Fourier transform requires a time series of discreet values and can’t operate on a continuous waveform. So both series have noise you can’t get rid of and the transform itself gives you two related but different amplitude values that can’t really be compared any more than apples and oranges can.
Physicists have it even harder because once they’ve measured one, while they know they can get an approximation from the Fourier transform, because of the Observer Effect they can’t actually compare that to a real-world measured value. Not unless you’ve got entangled particles, anyway, and then things really get weird. In any case, it’ll randomly almost never match your calculated value even if you could account for all sources of noise. Also, things on the quantum level are, of course, quantized so no matter how good your measurements or even if you found a better way to do the transform, you’d still be stuck with uncertainty and noise from quantization error.
Don’t ask me about the Measurement Problem. Wave function collapse is all gobbledygook to me. I tend to agree with the theory that there’s a single very complex wave function that describes the entire universe which supposedly obviates the need for collapse. I don’t have the skills to even go there so I’ll just take their word for it.