Because of this, the HUP has played a SIGNIFICANT role in any scientific theory or technology that follows from quantum mechanics.
For example, classical electrodynamics very effectively models light as a wave that can oscillate in different ways. A wave that is oscillating up and down is said to be "vertically polarized." Take two light sources that emit vertically polarized light in sync (that is, the "up" and "down" movements of the light are matched). Now rotate one of these sources 45 degrees clockwise and the other one 45 degrees counter-clockwise and position them so that the two resulting "in phase" DIAGONAL waves form an "X" as they travel away from you. It turns out these two waves will interact with each other to produce vertically polarized light yet again. In other words, the "left to right" and "right to left" motion of the two waves will cancel each other out and the remaining motion will be purely "up and down." Because of this, ALL VERTICALLY POLARIZED LIGHT spreads 50% of its power along clockwise polarizations and 50% of its power along counter-clockwise polarizations. Every vertically polarized wave can be thought of as a superposition of two diagonally polarized waves. If you fix polarization in one direction, you necessarily spread it over the two 45 degree rotations clockwise and counter-clockwise.
Now move into quantum electrodynamics. Your light source emits photons that have a 100% certainty of being vertically polarized. The HUP states that because you have COMPLETE certainty of VERTICAL polarization, you can have NO KNOWLEDGE of DIAGONAL polarization. This does not mean that a measurement of one polarization will destroy the other polarization. This means that every photon emitted will have vertical polarization, but 50% of them will have clockwise polarization and 50% of them will have counter-clockwise polarization. Here the PROBABILITY has been spread out over the polarizations. If you took enough measurements and every single one of them was clockwise polarized, then that would mean that your emitter is broken. 100% clockwise polarization would show 50% vertically polarized and 50% horizontally polarized. Knowledge of a diagonal polarization NECESSARILY prevents knowledge of a rectilinear polarization.
In other words, the WAVE-LIKE INTERFERENCE is a **RESULT** of the HUP.
This does not mean that things cannot have (for example) a specific position and a specific momentum at the same time. You can measure specific position and specific momentum without "disturbing" either one. The position and momentum you find will be the position and momentum of the particular particle in question. However, if that particle was generated by some physical process that is meant to restrict positions to a particular small space, then if you take many more measurements of many more particles emitted by the same process, then each of those particles will have a predictable position but a completely unpredictable velocity (or, more generally, momentum).
So the HUP is more than just a statement about measurement. It is a statement about existence. It is a statement about all of quantum mechanics. If ANY physical process restricts a variable to a small set of possibilities, it does so by "adding up" a relatively large number of possibilities of some other "conjugate variable," and the "interference" of those different possibilities is what allows for the few possibilities of the variable in question. In order to have vertically polarized light for certain, DIAGONALLY POLARIZED **POSSIBILITIES** MUST INTERFERE in such a way to cancel out all horizontal possibilities. Without BOTH diagonal components, the horizontal components would necessarily show up.
In fact, it is the HUP itself that prevents quantum mechanics from cleanly integrating with general relativity. General relativity is used to describe the mechanics of motion of very massive objects and makes very special use of the relative velocity of different objects through space. Quantum mechanics is used to describe the behavior of objects at very small distances from each other. Some of the most interesting aspects of cosmology are in the study of black holes and the conditions at the beginning of the universe; both of these things involve very massive objects occupying extremely tiny spaces, and thus both of these situations suggest the use of both general relativity and quantum mechanics. However, because of the HUP, the restriction of the component particles to extremely small regions of space means that the velocities of the ensemble of particles must necessarily take on a wide range of very different velocities. Due to the dependence of velocity in relativity, this makes it impossible to combine the two cleanly. String theory gets around this problem by replacing infinitesimal point particles with finite-sized strings and thus keeps the uncertainty in velocity to a manageable size.
So the HUP is pretty important. Also note that the HUP is actually a specialized version of the more general "uncertainty principle" that applies to all waves. Because quantum mechanics involves waveform mathematics, then it inherits this principle. What's ironic is that the formalization of this very general principle really came from Heisenberg with respect to quantum mechanics and then was later generalized to all waves. (take a snapshot of a pendulum swinging back and forth. With a single snapshot, it is impossible to know the frequency (period) of the pendulum. This is a result of the more general uncertainty principle)
Answered by: Ted Pavlic, B.S., Electrical Engineering Grad Student, Ohio State U.
The uncertainty principle formally limits the precision to which two complementary observables can be measured and establishes that observables are not independent of the observer. It also establishes that phenomena can take on a range of values rather than a single, exact value. This necessitates a probabilistic interpretation of the behavior of matter on the molecular level, and a statistical mechanics to explain aggregate behavior of molecular and atomic systems as they give rise to macroscopic properties and behaviors.
So, the Uncertainty Principle finds its main importance in the everyday world by the change in perspective it brings about in fundamental physical principles, which must necessarily have consequences in the interpretation of physical phenomena on all length scales. It suggests that one of the most influential and fundamental physical laws is one of chance. This is not so important with respect to empiricism in the everyday world, but is an integral idea in connecting behavior within the population of an atomic or molecular system and the observable properties of that system.
Answered by: Jay Foley, Chemistry Undergrad Student, Georgia Tech
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