The Modern Revolution in Physics —- Benjamin Crowell

Note that although I encouraged you to think of this derivation in terms of a specific real-world system, the quantum dot, I never made any reference to specific measuring equipment. The argument is simply that we cannot know the particle’s position very accurately unless it has a very well defined position, it cannot have a very well defined position unless its wave-pattern covers only a very small amount of space, and its wave-pattern cannot be thus compressed without giving it a short wavelength and a correspondingly uncertain momentum. The uncertainty principle is therefore a restriction on how much there is to know about a particle, not just on what we can know about it with a certain technique. Example 6: An estimate for electrons in atoms ? A typical energy for an electron in an atom is on the order of (1 volt)?e , which corresponds to a speed of about 1% of the speed of light. If a typical atom has a size on the order of 0.1 nm, how close are the electrons to the limit imposed by the uncertainty principle? ? If we assume the electron moves in all directions with equal probability, the uncertainty in its momentum is roughly twice its typical momentum. This only an order-of-magnitude estimate, so we take ? p to be the same as a typical momentum: ?p?x =ptypical?x =(melectron)(0.01c)(0.1×10-9m) =3×10-34J?s




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