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The precise mathematical form of this relation between the uncertainties of position and momentum of a particle is known as Heisenberg’s uncertainty relation, or uncertainty principle. It means that, in the subatomic world, we can never know both the position and momentum of a particle with great accuracy. The better we know the position, the hazier will its momentum be and vice versa. We can decide to undertake a precise measurement of either of the two quantities; but then we will have to remain completely ignorant about the other one. It is important to realize, as was pointed out in the previous chapter, that this limitation is not caused by the imperfection of our measuring techniques, but is a limitation of principle. If we decide to measure the particle’s position precisely, the particle simply does not have a well-defined momentum, and vice versa.

The relation between the uncertainties of a particle’s position and momentum is not the only form of the uncertainty principle. Similar relations hold between other quantities, for example between the time an atomic event takes and the energy it involves. This can be seen quite easily by picturing our wave

packet not as a pattern in space, but as a vibrational pattern in time. As the particle passes a particular point of observation, the vibrations of the wave pattern at that point will start with small amplitudes which will increase and then decrease again until finally the vibration will stop altogether. The time it takes to go through this pattern represents the time during which the particle passes our point of observation. We can say that the passage occurs within this time span, but we cannot localize it any further. The duration of the vibration pattern represents therefore the uncertainty in the temporal location of the event. 

Now, as the spatial pattern of the wave packet does not have a well-defined wavelength, the corresponding vibrational pattern in time does not have a well-defined frequency. The spread in frequency depends on the duration of the vibrational pattern, and since quantum theory associates the frequency of the wave with the energy of the particle, the spread in the pattern’s frequency corresponds to an uncertainty in the particle’s energy. The uncertainty in the location of an event in time thus becomes related to an uncertainty in energy in the same way as the uncertainty of a particle’s location in space is related to an uncertainty in momentum. This means that we can never know both the time at which an event occurs and the energy involved in it with great accuracy. Events occurring inside a short time span involve a large uncertainty in energy; events involving a precise amount of energy can be localized only within a long period of time.

Excerpted from Pages 158-159 of ‘The Tao of Physics’ by Fritjof Capra