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You can have a burp or a note, but not both.

The Heisenberg uncertainty principle is an interesting non-classical phenomenon, which states that certain pairs of physical properties cannot be known with arbitrary accuracy.  In it’s most common form, it is expressed as an uncertainty between position and momentum, where the position and momentum of a particle cannot both be known simultaneously with arbitrary accuracy.

The idea sounds very foreign to us, as at the macro level, we are familiar with being able to know both the position and momentum of say, a car at the same time.  What is even more interesting, is that the uncertainty between position and momentum is not a limitation on measurements or observation, but actually a description of physical nature itself.  It is a result of quantum mechanics and the wave-like nature of matter.

There is also an uncertainty principle between energy and time.  This uncertainty is commonly known in Fourier analysis as a frequency-time uncertainty.  One can specify a signal with arbitrary time or arbitrary frequency, but not both.  An example of this uncertainty would be the contrast between a burp and a note. For a burp, this signal has a very narrow standard deviation in time-domain (just a single bump), but the signal is comprised of a many different frequencies and has a large uncertainty (standard deviation) in frequency-domain.  The shorter the pulse, the larger the range of frequencies that are used to make up this pulse.  On the other hand, for a flute playing a note, the frequency of that signal has a narrow standard deviation (a single frequency or note), but in the time-domain, it looks like a sine wave and has a large uncertainty (standard deviation).

Similarly, the wave function associated with a particle such as an electron may have a definite position at some time but the momentum will have some uncertainty, or it may have a definite momentum at a given time, but it’s position will be uncertain.

Mathematically speaking, the position-momentum uncertainty principle is expressed as \Delta x \Delta p_x >= \frac{\hbar}{2} where \Delta x is the standard deviation of the position, and \Delta p_x is the standard deviation of the momentum.  While the position and momentum of a particle can be known with arbitrary accuracy, they cannot both be known at the same time.  \hbar = 10^{-34} kg m^2/sec, which is very small.  It is much too small for objects of macroscale size, but is significant for very small objects such as electrons.

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