uncertainty principle, physical principle, enunciated by Werner Heisenberg in 1927, that places an absolute, theoretical limit on the combined accuracy of certain pairs of simultaneous, related measurements. The accuracy of a measurement is given by the uncertainty in the result; if the measurement is exact, the uncertainty is zero. According to the uncertainty principle, the mathematical product of the combined uncertainties of simultaneous measurements of position and momentum in a given direction cannot be less than Planck's constant h divided by 4π. The principle also limits the accuracies of simultaneous measurements of energy and of the time required to make the energy measurement. The value of Planck's constant is extremely small, so that the effect of the limitations imposed by the uncertainty principle are not noticeable on the large scale of ordinary measurements; however, on the scale of atoms and elementary particles the effect of the uncertainty principle is very important. Because of the uncertainties existing at this level, a picture of the submicroscopic world emerges as one of statistical probabilities rather than measurable certainties. On the large scale it is still possible to speak of causality in a framework described in terms of space and time; on the atomic scale this is not possible. Such a description would require exact measurements of such quantities as position, speed, energy, and time, and these quantities cannot be measured exactly because of the uncertainty principle. It does not limit the accuracy of single measurements, of nonsimultaneous measurements, or of simultaneous measurements of pairs of quantities other than those specifically restricted by the principle. Even so, its restrictions are sufficient to prevent scientists from being able to make absolute predictions about future states of the system being studied. The uncertainty principle has been elevated by some thinkers to the status of a philosophical principle, called the principle of indeterminacy, which has been taken by some to limit causality in general. See quantum theory.

principle of indeterminacy: see uncertainty principle.

exclusion principle, physical principle enunciated by Wolfgang Pauli in 1925 stating that no two electrons in an atom can occupy the same energy state simultaneously. The energy states, or levels, in an atom are described in the quantum theory by various values of four different quantum numbers; the exclusion principle holds that no two electrons can have the same four quantum numbers in an atom. One of these quantum numbers describes one of the two possible directions for the electron's intrinsic spin. As a result of the exclusion principle, two electrons that are in the same energy level as described by the other three quantum numbers are differentiated from each other because they have opposite spins. This principle applies not only to atoms but to other systems containing particles as well, and it applies not only to electrons but also to a large class of particles collectively known as fermions (see elementary particles).

correspondence principle, physical principle, enunciated by Niels Bohr in 1923, according to which the predictions of the quantum theory must correspond to the predictions of the classical theories of physics when the quantum theory is used to describe the behavior of systems that can be successfully described by classical theories. Technically this principle means that the results of a quantum theory analysis of a problem that involves the use of very large quantum numbers must agree with the results of a classical physics analysis. Such correspondence is known as the classical limit of the quantum theory. Ordinarily the quantum theory is used to describe the behavior of bodies that are so small that they cannot be seen under an optical microscope, while the theories of classical physics are used to analyze the behavior of large-scale bodies. The correspondence principle provided an important theoretical basis for the development of a detailed correlation between the newer quantum theory and the classical physics that preceded it.

complementarity principle, physical principle enunciated by Niels Bohr in 1928 stating that certain physical concepts are complementary. If two concepts are complementary, an experiment that clearly illustrates one concept will obscure the other complementary one. For example, an experiment that illustrates the particle properties of light will not show any of the wave properties of light. This principle also implies that only certain kinds of information can be gained in a particular experiment. Some other information that is equally important cannot be measured simultaneously and is lost.

Mach's principle [for E. Mach], assertion that the inertial effects of mass are not innate in a body, but arise from its relation to the totality of all other masses, i.e., to the universe as a whole. Thus, the inertial forces experienced by a body in accelerated motion have the same physical origin as the gravitational forces it experiences near mass concentrations, namely the mass-energy field described by the general theory of relativity. Inertial forces have a much longer range than gravitational forces, so the role of very distant matter becomes preponderant. According to Mach's principle, a body experiences no inertial forces when it is at rest or in uniform motion with respect to the center of mass of the entire universe. When its motion is nonuniform (accelerated) with respect to the total mass of the universe, it experiences forces such as centrifugal force (see centripetal force and centrifugal force) and the Coriolis effect. Hence, the "local" behavior of matter is influenced by the "global" properties of the universe, i.e., those properties that describe the universe as a whole, which are studied in cosmology.

Le Châtelier's principle, chemical principle that states that if a system in equilibrium is disturbed by changes in determining factors, such as temperature, pressure, and concentration of components, the system will tend to shift its equilibrium position so as to counteract the effect of the disturbance (see chemical equilibrium). For example, at a given temperature a covered beaker partly filled with water constitutes a system in which the liquid water is in equilibrium with the water vapor that forms above the surface of the liquid. While some molecules of liquid are absorbing heat and evaporating to become vapor, an equal number of vapor molecules are giving up heat and condensing to become liquid. If stress is put on the system by raising the temperature, then according to Le Châtelier's principle the rate of evaporation will exceed the rate of condensation until a new equilibrium is established. At the new equilibrium point a greater proportion of molecules will exist in the vapor phase. Le Châtelier's principle is evident in chemical systems, as in the common-ion effect and in buffer solutions (see also separate article on pH). Le Châtelier's principle can be used to encourage formation of a desired product in chemical reactions. In the Haber process for the industrial synthesis of ammonia, nitrogen gas and hydrogen gas react to form ammonia gas in the reaction N₂+3H₂→2NH₃ ; the process is exothermic, i.e., one that gives off heat. Since four molecules—three of hydrogen and one of nitrogen—react to form two molecules of ammonia, the reactants have a higher gas pressure than the products. When the reaction is run under high external pressure, up to 1000 atmospheres, and relatively low temperature, about 500°C; (932°F;), the system favors formation of the substance that will result in a lower total number of molecules, i.e., the ammonia. Running the reaction at relatively low temperature causes it to go far to completion, although if the temperature is too much below 500°C; the rate of reaction is too slow.

D'Alembert's principle, in mechanics, principle permitting the reduction of a problem in dynamics to one in statics. This is accomplished by introducing a fictitious force equal in magnitude to the product of the mass of the body and its acceleration, and directed opposite to the acceleration. The result is a condition of kinetic equilibrium. Jean le Rond d'Alembert, a French mathematician, introduced the principle in 1742 and established it the next year in his Traité de dynamique. The principle shows that Newton's third law of motion applies to bodies free to move as well as to stationary bodies.

Bernoulli's principle, physical principle formulated by Daniel Bernoulli that states that as the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases. The phenomenon described by Bernoulli's principle has many practical applications; it is employed in the carburetor and the atomizer, in which air is the moving fluid, and in the aspirator, in which water is the moving fluid. In the first two devices air moving through a tube passes through a constriction, which causes an increase in speed and a corresponding reduction in pressure. As a result, liquid is forced up into the air stream (through a narrow tube that leads from the body of the liquid to the constriction) by the greater atmospheric pressure on the surface of the liquid. In the aspirator air is drawn into a stream of water as the water flows through a constriction. Bernoulli's principle can be explained in terms of the law of conservation of energy (see conservation laws, in physics). As a fluid moves from a wider pipe into a narrower pipe or a constriction, a corresponding volume must move a greater distance forward in the narrower pipe and thus have a greater speed. At the same time, the work done by corresponding volumes in the wider and narrower pipes will be expressed by the product of the pressure and the volume. Since the speed is greater in the narrower pipe, the kinetic energy of that volume is greater. Then, by the law of conservation of energy, this increase in kinetic energy must be balanced by a decrease in the pressure-volume product, or, since the volumes are equal, by a decrease in pressure.

Archimedes' principle, principle that states that a body immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid. The principle applies to both floating and submerged bodies and to all fluids, i.e., liquids and gases. It explains not only the buoyancy of ships and other vessels in water but also the rise of a balloon in the air and the apparent loss of weight of objects underwater. In determining whether a given body will float in a given fluid, both weight and volume must be considered; that is, the relative density, or weight per unit of volume, of the body compared to the fluid determines the buoyant force. If the body is less dense than the fluid, it will float or, in the case of a balloon, it will rise. If the body is denser than the fluid, it will sink. Relative density also determines the proportion of a floating body that will be submerged in a fluid. If the body is two thirds as dense as the fluid, then two thirds of its volume will be submerged, displacing in the process a volume of fluid whose weight is equal to the entire weight of the body. In the case of a submerged body, the apparent weight of the body is equal to its weight in air less the weight of an equal volume of fluid. The fluid most often encountered in applications of Archimedes' principle is water, and the specific gravity of a substance is a convenient measure of its relative density compared to water. In calculating the buoyant force on a body, however, one must also take into account the shape and position of the body. A steel rowboat placed on end into the water will sink because the density of steel is much greater than that of water. However, in its normal, keel-down position, the effective volume of the boat includes all the air inside it, so that its average density is then less than that of water, and as a result it will float.

Alembert's principle: see D'Alembert's principle.

Criterion of meaningfulness associated with logical positivism and the Vienna Circle. Moritz Schlick's formulation “The meaning of a [declarative sentence] is the method of its verification” was close to the view held in pragmatism, and later in operationalism, that an assertion has factual meaning only if there is a difference in principle, open to test by observation, between the affirmation and the denial of the assertion. Thus, the statements of ethics, metaphysics, religion, and aesthetics were held to be meaningless. The verifiability criterion of meaningfulness was in part inspired by Albert Einstein's abandonment of the ether hypothesis and the notion of absolute simultaneity.

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or Heisenberg uncertainty principle or indeterminacy principle

Principle that states that the position and velocity of an object cannot both be measured exactly at the same time, and that the concepts of exact position and exact velocity together have no meaning in nature. Articulated by Werner Heisenberg in 1927, it applies only at the small scales of atoms and subatomic particles and is not noticeable for macroscopic objects, such as moving vehicles. Any attempt to measure the velocity of a subatomic particle precisely will displace the particle in an unpredictable way, thus invalidating any simultaneous measurement of its position. This displacement is a result of the wave nature of particles (see wave-particle duality). The principle also applies to other related pairs of variables, such as energy and time.

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Assertion proposed by Wolfgang Pauli that no two electrons in an atom can be in the same state or configuration at the same time. It accounts for the observed patterns of light emission from atoms. The principle has since been generalized to include the whole class of particles called fermions. The spin of such particles is always an odd whole-number multiple of 12. For example, electrons have spin 12, and can occupy two distinct states with opposite spin directions. The Pauli exclusion principle indicates, therefore, that only two electrons are allowed in each atomic energy state, leading to the successive buildup of orbitals around the nucleus. This prevents matter from collapsing to an extremely dense state.

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Fundamental principle of physics that in its weak form states that gravitational (see gravitation) and inertial (see inertia) masses are the same. Albert Einstein's stronger version states that gravitation and acceleration are indistinguishable. It implies that the effect of gravity is removed in a suitably accelerated reference frame, such as an elevator with its cable cut, in which a person would experience free fall.

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Hypothesis that the inertial forces acting on a body in accelerated motion are determined by the quantity and distribution of matter in the universe. Albert Einstein found its suggested connection between geometry and matter helpful in formulating his theory of general relativity; unaware that George Berkeley had proposed similar views in the 18th century, he attributed the idea to Ernst Mach. He abandoned the principle when he realized that inertia is assumed in the geodesic equation of motion (see geodesy) and need not depend on the existence of matter elsewhere in the universe.

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or Bernoulli's theorem

Principle that relates pressure, velocity, and height for a nonviscous fluid with steady flow. A consequence is that, for horizontal flow, as the speed of a fluid increases, the pressure it exerts decreases. Derived by Daniel Bernoulli (see Bernoulli family), the principle explains the lift of an airplane in motion. As the speed of the plane increases, air flows faster over the curved top of the wing than underneath. The upward pressure exerted by the air under the wing is thus greater than the pressure exerted downward above the wing, resulting in a net upward force, or lift. Race cars use the principle to keep their wheels pressed to the ground as they accelerate. A race car's spoiler—shaped like an upside-down wing, with the curved surface at the bottom—produces a net downward force.

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Law of buoyancy, discovered by Archimedes, which states that any object that is completely or partially submerged in a fluid at rest is acted on by an upward, or buoyant, force. The magnitude of this force is equal to the weight of the fluid displaced by the object. The volume of fluid displaced is equal to the volume of the portion of the object submerged.

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