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The Heisenberg uncertainty principle is a limit on how precise the position and momentum of an object is defined, specifically a limit on the product of their precise values such that the more precise the momentum is known to be, the less determined the location is. The large mass of everyday items makes this lower limit unnoticeably small for them, but the limit is significant at atomic scale and smaller. A concise statement of the principle is the inequality:
σxσp ≥ ℏ/2
Since the object's momentum is directly related to its mass, another way of stating the principle is that the definition of the object's speed and position are limited by an amount inversely proportional to the object's mass. The principle is derived from quantum mechanics (QM), essentially a theorem that can be used in QM calculations, and is one of QM's aspects that strike people as very strange. The question of whether the object really has no precise position and momentum versus the notion that we are merely unable to determine both (hit the object with a photon to attempt to locate it and you've pushed it a bit) is under continual discussion and depends upon interpretation of QM, but whatever the answer, it must be consistent with the established facts. The principle says that a particle placed near a narrow barrier from one side, may actually be on the other side of the barrier, and in fact, this is sometimes the case. It doesn't matter how strong the barrier is, only that the barrier is so narrow that the calculated lack-of-knowledge of exactly where the particle is located is extensive enough to traverse the barrier. Furthermore, if you place the particle near the barrier, you might find it on the other side, and if you wish it to be on the other side, you could repeatedly place it near the barrier, improving the chances that it turns out to be on the other side. This phenomenon is termed quantum tunneling, and as nonsensical as it seems, it is not merely suggested by theory and laboratory experiments, but is a key point in current scientific and technological knowledge, among other things, widely used in current technology (tunnel diodes), in which an electric current is allowed through such a barrier via such quantum tunneling.
The phrase uncertainty principle (i.e., without "Heisenberg") also refers to other pairs of quantities which, in analogous manner, are not simultaneously defined within QM. The phrase Heisenberg uncertainty principle is sometimes used with this general QM phenomenon in mind and sometimes specifically referring to the above-mentioned phenomenon, which is the one that Heisenberg showed in his paper.
Outside of science, the meaning of the phrase Heisenberg uncertainty principle is often stated as "experiments always affect what is being experimented upon". While this statement may be true (right down at the atomic level) and arguably is a corollary to the inequality Heisenberg actually stated (and meanings are ultimately based upon usage and this meaning is widely used), it is also clear that Heisenberg's inequality precisely explains consequences that this popular meaning does not, e.g., quantum tunneling.