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.

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 what its velocity is (more precisely, its momentum).

Classically, that is, in our macroscopic world, I could, in principle, measure the position and momentum of the object to infinite precision (more or less). There is really no question about a particle's position and momentum.
In the Quantum Mechanical world, the idea that we can locate objects exactly breaks down. Let me state this idea more precisely. Suppose a particle has momemtum p and position x. In a Quantum Mechanical world, I would not be able to measure p and x precisely. There would be an uncertainty associated with each measurement that I could never get rid of, even in a perfect experiment!!! The size of the uncertainties are not independent; they are related as

dp x dx > h / (2 x pi) = Planck's constant / (2 x pi)

The preceding is a statement of the Heisenberg Uncertainty Principle. A consequence of the Uncertainty Principle is that if an object's position x is defined precisely then the momentum of the object will be only weakly constrained, and vice versa. One cannot simultaneously find both the position and momentum of an object to arbitrary accuracy.

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

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 a 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 could predict the path a particle will follow with 100 % certainty.

A consequence of the Quantum 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 has interesting consequences for nuclear fusion in stars.