Physics

The Scoop on the Uncertainty Principle



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The uncertainty principle, like many aspects of physics, is often misunderstood. Mathematically, it is a consequence of the 'non-commutivity' of the operators (mathematical constructs of things we can observe like momentum and position) in quantum mechanics. What this means is the order you do the math in matters. Just like division and subtraction do not commute, certain pairs of operators do not commute. This amounts to a non zero minimum in the product of the uncertainty with which we can simultaneously know the values the quantities take. The uncertainty principle is not in any way related to the precision with which we can build our detectors and instruments. It is one of the rules that nature plays by, and there does not appear to be any way around it, to my knowledge.

As a handle to hold, one often talks of the position and momentum (or velocity) of a particle. If we want to know where a particle is, we have to see it. We can see it by shining light on it, and having that light bounce off and strike our detector. For our purposes the detector can be the eye. However, the light we use has a certain wavelength. We can only know the particle's location to within about one wavelength of the light. If we want to know where the particle is with more certainty we must use a shorter wavelength light. So we can shrink the uncertainty in location considerably by using a short wavelength.

However, light carries energy and momentum (but not mass). When some light bounces off our particle there is a transfer of momentum according to Newton's third law. (Yes, this law is still valid in quantum mechanics because it is really a statement about the conservation of momentum.) The way this momentum transfer works depends on the initial momenta of both the particle and the light. The higher the momentum of the incoming light, the greater the uncertainty in how it pushes off the particle during the 'bouncing' into the detector. Nature has conspired such that the momentum of light is inversely proportional to wavelength. Shorter wavelengths (or, equivalently, higher frequencies) have larger momenta. Our gains in the knowledge of the particle's position are offset by the loss of knowledge regarding the particle's momentum.

The most interesting aspect of the uncertainty principle is that all such pairs of quantities (position & momentum, excited state energy & lifetime, rotational position & angular momentum) have an uncertainty product on the order of Planck's constant divided by 2 pi (h-bar, for those in the know). The size of this constant determines when it is necessary to use quantum mechanics (h-bar is significant) or if classical mechanics will suffice (h-bar is effectively zero).

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