What’s the underlying principle on why you can’t know the position and momentum of a particle at the same time? Is there an explanation? I don’t have any knowledge of calculus or the math behind anything so I’m asking here, thanks!
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This requires a bit of knowledge about Fourier transforms in mathematics.
Say you have a signal that varies with time, e.g., a pulse of some sort like a short voltage pulse in electronics. The pulse can be decomposed as a sum of sinusoidal waves with varying frequencies, amplitudes and phases. You can compute the frequency components from the signal using a mathematical technique called the Fourier transform. The shorter and sharper the pulse in time (on the signal vs time graph), the more spread in frequency (i.e., signal vs frequency graph shows more spread). This is a mathematical consequence when using the Fourier transform to go from time to frequency. **There is a minimum value in the product of the ‘spread’ of the pulse in time and its ‘spread’ in frequency.**
In quantum mechanics, a particle has a probability wave (e.g., in position). Imagine a signal in position e.g. a pulse that gives the probability of finding a particle at a specific position. This pulse can be decomposed of a sum of sinusoids with different momentum. If you do the Fourier transform of a signal vs position, you get a signal vs momentum. As a consequence of the Fourier transform, there is a minimum in the product of the ‘spread’ of the pulse in position and the ‘spread’ in momentum. This ‘spread’ is basically an uncertainty in the position or momentum of a particle.
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