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One important consequence of the wave-particle duality of nature was
discovered by Heisenberg, and is called the **uncertainty principle**.
To formulate it, let us imagine that we want to measure the position and
the momentum of a particular particle. To do so we must ``see'' the particle,
and so we shine some light of wavelength on it.
We recall in the last chapter that there is a limit to the resolving power
of the light used to see the particle given by the wavelength of light used.
This gives an *uncertainty* in the particle's position:

This results from considering the light as a wave. However, viewed as
a photon, the light when striking the particle could give up some or
all of its momentum to the particle. Since we don't know how much it gave
up, as we don't measure the photon's properties, there is an uncertainty
in the momentum of the particle; using Eq.(27.8), we find

Combining Eqs.(27.11,27.12), we find

Note that this is independent of the wavelength used, and says there is
a limit *in principle* as to how accurately one can simultaneously
measure the position and momentum of a particle - if one tries to measure
the position more accurately by using light of a shorter wavelength, then the
uncertainty in momentum grows, whereas if one uses light of a longer wavelength
in order to reduce the uncertainty in momentum, then the uncertainty in
position grows. One cannot reduce both down to zero simultaneously - this
is a direct consequence of the wave-particle duality of nature.
The arguments used in deriving Eq.(27.13) are somewhat rough. A more refined
treatment, developed by Heisenberg, results in the following relation:

As with de Broglie waves, for everyday macroscopic objects such as
bowling balls the uncertainty principle plays a negligible role in limiting
the accuracy of measurements, as we shall see in some examples. However,
for microscopic objects such as electrons in atoms the uncertainty principle
does become a very important consideration.

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** Previous:** de Broglie Waves

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10/9/1997