Heisenberg Uncertainty Principle

An odd aspect of Quantum Mechanics is contained in the Heisenberg Uncertainty Principle (HUP). The HUP can be stated in different ways; let me first talk in terms of momentum and position (it can also be thought of in terms of energy and time).

If there is a particle, such as an electron, moving through space, I can characterize its motion by telling you where it is (its position) and its velocity (more precisely, its momentum).

• Classically, i.e., in our macroscopic world, I can measure these two quantities to infinite precision (more or less). There is really no question where something is and what its momentum is. Consequently, I can tell with infinite precision where that particle will at the next instant in time.

• In the Quantum Mechanical world, the idea that we can know things exactly breaks down. More precisely, suppose a particle has momemtum p and a position x. In a Quantum Mechanical world, I would not be able simultaneously to measure both p and x precisely. There is an uncertainty associated with this pair of measurements, e.g., there is some dp and dx, which I can never get rid of even in a perfect experiment!!!. The size of the uncertainties are not independent, they are roughly related by
  
  The preceding is a statement of The Heisenberg Uncertainty Principle. So, for example, if I measure x exactly (that is, dx is 0!), the uncertainty in p, dp, must be infinite in order to keep the product dp dx constant.

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  This uncertainty leads to many strange things. For example, in a Quantum Mechanical world, I cannot then predict where a particle will be with 100 % certainty. I can only speak in terms of probabilities. For example, I can say that an atom will be at some location with a 99 % probability, but there will be a 1 % probability it will be somewhere else (in fact, there will be a small but finite probabilty that it will be found across the Universe). This is strange.

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  We do not know if this indeterminism is actually the way the Universe works because the theory of Quantum Mechanics is probably incomplete. That is, we do not know if the Universe actually behaves in this probabilistic manner (there are many possible paths a particle can follow and the observed path is chosen probabilistically) or if the Universe is deterministic in the sense that I can predict the path a particle will follow with 100 % certainty.

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  A consequence of the Qunatum Mechanical nature of the world, is that particles can appear in places where they have no right to be (from an ordinary, common sense [classical] point of view)!

  This notion has interesting consequences for nuclear fusion in stars.