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**Quantum Theory** __How to Cite This Article__ **Quantum Theory, **a theory that provides a consistent basis for analyzing matter, motion, and radiation. Quantum theory and the theory of relativity now govern all physics. Classical theory, which was superseded by quantum theory in 1926, survives as an approximation. It is incorrect for understanding the behavior of particles the size of atoms and molecules, but it is highly reliable in describing physical properties of bulk matter. Unlike classical physics, quantum theory allows such quantities as radiant energy to exist only in discrete amounts, which are called *quanta.* It often is used to calculate only probabilities of observing results, whereas in classical theory results were presumed to be exactly measurable. Also, quantum theory treats matter, energy, and forms of radiation on the same basis, whereas they were considered separate and different in classical physics. **Applications.** To explain the behavior of particles the size of atoms, quantum theory is required. But not all applications are submicroscopic. The structure and evolution of stars, for instance, are explicable only by quantum theory. Quantum theory and its experimental results provide explanations and tools for science and engineering. In chemistry, for example, it explains the periodic table of the elements, the bonding of atoms in molecules, and the nature of reaction rates, and it provides nuclear isotopes for radiochemistry. In biology and medicine it leads to an understanding of genetics and mutations and it provides isotopes and radiation sources for diagnosis and therapy. In archaeology, paleontology, and geology it provides ways, such as carbon dating, to learn the age of materials. In physics and technology it provides an understanding of semiconductors, solid-state diodes and transistors, lasers, and atomic clocks. It also explains such energy sources as nuclear reactors, photocells for solar energy, and nuclear-fusion sources. **Difficulties.** Despite many years of successful application, quantum theory has four difficulties. (1) From 1926, some physicists, including Einstein, have demurred at probabilistic predictions. Attempts are still being made to find more basic variables that would yield deterministic results. (2) Electromagnetic interactions binding atoms, molecules, and matter in bulk are known in exact mathematical form, but the strong interactions between hadrons and the weak interactions between leptons are not as well understood. (3) When high energies are involved, relativistic quantum mechanics must be used. Thereupon much of the broadly organized nonrelativistic theory ceases to apply. In a sense, each relativistic problem is a special case. (4) In high-energy particle physics, better experiments with larger accelerators are steadily discovering new "elementary" particles. Quantum theory has explained many objects such as nuclei in terms of a few simple particles, but the newly found elementary particles still elude clarification.(Strauss & Xiang, 2006; Waugh, Hall, McNeil, Key, & Matear, 2006) **Work Leading to the Quantum Theory** The work of various physicists, including Max Planck, Albert Einstein, Niels Bohr, and Arthur Compton, led to the development of quantum theory. **Planck's Quantum Hypothesis.** Quantum theory and its use in science and technology are dominated by Planck's constant *h,* 6.62 × 10^{–27} erg-second. The constant *h* was first introduced by Max Planck in 1901 to explain the distribution in frequency of the radiant energy in a cavity of a body as a function of the absolute temperature *T* of the body. According to the classical theory of light, the sources of such radiation are equivalent to a system of harmonic oscillators of all frequencies and of any energies. The energy *E*_{}d in the small frequency range between and + d is *E**d* = (^{8}^{V}/*c*^{3}) (*kT*)*V*^{2}*d*, where *V* is the volume of the cavity, *c* is the speed of light, *k* is Boltzmann's constant, and *kT* is the average energy per oscillator. The rest of the equation gives the number of oscillators to be expected with frequencies between and + d. The derivation of the equation was faultless, but the equation was unacceptable, notably because it predicted an infinite total amount of radiant energy at an infinite frequency. Planck found that he could derive the correct law of distribution of the radiant energy by making two assumptions: (1) each oscillator can possess only discrete amounts of energy, *nh,* where *n* is an integer and is the frequency of the oscillator in question; and (2) the probability that an oscillator has the energy *nh* is proportional to *e*^{–}^{nh}^{/}^{kT}. Planck's law of distribution is *E*_{}*d* = [(^{8}^{V}/*c*^{3}) ^{(}^{h}^{3)}/(*e*^{h}^{/}^{kT} – 1)]*d*, which agrees with the experimental observations in all respects. His second assumption was not foreign to statistical mechanics, but his first—that energy is quantized, or exists in discrete quanta—was radically new. Since all the other quantities in Planck's equation are measurable, the value of *h* could be determined. **The Photoelectric Effect.** As discovered by Wilhelm Hallwachs and Heinrich Hertz in the 1880s and analyzed further by later experimenters, incident light can eject electrons from metals. The velocities of these photoelectrons are independent of the intensity of the light but increase with the frequency of the light. Also, the number of electrons ejected per unit of time is proportional to the intensity of the light. In 1905, Albert Einstein proposed a model of this phenomenon in which the light acts as though it remained concentrated in bundles, or quanta, of energy *h*, as in Planck's postulate, with the frequency of the light. Each of these bundles can be absorbed, but only as a whole and by an individual electron. In this way the electron is given an additional kinetic energy of amount *h*. In passing through the surface barrier of the metal, the electron loses from this energy a portion that can be designated as *h*_{0}. The kinetic energy *KE* with which the electron emerges is then given by KE = (^{1}/2)mv^{2} = h ( – _{0}), where *m* is the mass and *v* the velocity of the electron. This gives the maximum energy of ejection, since electrons can also lose some energy inside the metal before reaching the surface. The equation indicates that unless is greater than _{0} the electrons cannot escape at all, so that there exists a low-frequency limit for the ejection of any electrons. The equation contains no reference to the intensity of the light but gives the energy of the ejected electrons in terms of the frequency only. This expression, when later modified to take account of the various energies possessed by the electrons before they absorb the light, agreed with the results of the experiments in detail. The equation itself, however, is completely paradoxical from the classical point of view, which regards light as an electromagnetic wave and the electrons as charged material particles. **Specific Heats of Crystals.** Classical molecular theory predicts a constant molar specific heat for crystalline solids at all temperatures. Experimentally, however, specific heats drop toward zero as the absolute temperature decreases. In 1907, Einstein postulated that the energy of atoms oscillating in the crystal lattice is also quantized in the Planck manner. Later improvements by Peter J. W. Debye and others produced excellent theoretical agreement with experiment. **Atomic Spectra and Energy Levels.** The distinct patterns of lines in atomic spectra were not explained by classical physics, although there were a few empirical formulas. In 1913, Niels Bohr gave an explanation by adding quantum concepts to the planetary model of the atom postulated by Ernest Rutherford. The atom normally remains in one of various energy levels, characterized by particular electron orbits about the nucleus. Contrary to classical electromagnetic theory, it does not emit radiation. Only in "jumping" from some level of energy *E*_{a} to another level of energy *E*_{b} does the atom absorb or emit a Planck quantum of radiation with energy h: h = E_{a} – E_{b} . Electron bombardment of atoms by James Franck and Gustav Hertz in 1914 proved the reality of discrete energy levels. Bohr quantitatively explained the hydrogen atom and spectrum by restricting the angular momenta of the single-electron orbits to the amount ^{nh}/2, where *n* is an integer. He inferred this rule by introducing the correspondence principle: for a very large quantum number *n,* quantum theory yields classical laws.(Ito & Sasai, 2006; Smeets & Medema, 2006) **The Compton Effect.** In 1923, Arthur H. Compton showed that when X rays are scattered from matter their wavelength is increased by the amount – ' = (^{h}/*mc*)(1 – *cos* ), where is the wavelength before scattering, ' is the wavelength after scattering, is the angle between the incident and the scattered beam of X rays, *m* is the electron mass, and *c* is the speed of light. His derivation of the equation introduced the concept of photons—"particles" of radiation with the Planck energy *h* and the momentum (suggested by relativity theory) *h*^{}/*c* = ^{h}/. In collisions of photons with particles of matter, energy and momentum are conserved. **Electron Diffraction.** Clinton J. Davisson and Lester H. Germer (1927), and independently George P. Thomson (1928), showed that a beam of electrons is scattered from a crystal at the same angles as a beam of X rays of wavelength = ^{h}/*mv*, where *v* is the velocity and *m* the mass of the electron. Such a beam shows the phenomenon of diffraction characteristic of a wave. This cannot be explained in terms of the classical picture of electrons as charged particles. Under these circumstances, a beam of electrons behaves as a train of waves whose wavelength can be given in terms of the mass and velocity of the electrons, quantities that do not refer to waves at all. **Modern Theory** By 1923 quantum theory was entrenched in physics, but usually as a contradictory addition to classical laws. The dual role of radiation—sometimes particles, sometimes waves—was perplexing. Then Louis V. de Broglie postulated the same duality for matter. This led to Erwin Schrödinger's *wave mechanics* (1926), based on a fundamental wave equation, which superseded classical laws in atomic physics. Equivalent, though differently phrased, were Werner Heisenberg's *matrix mechanics* (1925) and Paul A. M. Dirac's method of noncommutative algebra (1926). The new theory, *quantum mechanics,* flowered rapidly. By 1930 the successes of earlier quantum theory were repeated, its failures were corrected, and the wave-particle duality was explained. Dirac's relativistic wave equation (1928) explained electron spin (intrinsic angular momentum), confirming earlier deductions of its existence. The laws of atomic structure and the interactions of atoms and radiation were formulated. Notable contributors included Wolfgang Pauli, Max Born, and Pascual Jordan. New discoveries in physics—the neutron (1932), the positron (1932), the mu meson (1938), and the pi meson (1947)—facilitated application of quantum mechanics to nuclear structure and radiation. Quantum electrodynamics and field theory examined the joint behavior of assemblies of particles and radiation, and the nature of elementary particles. Meanwhile, quantum mechanics was used to explain the properties of molecules and of matter in bulk. The Heisenberg-Dirac forms of quantum mechanics are powerful and compact. They retain familiar concepts, symbols, and formal laws from older physics, but alter some definitions and the mathematical rules for manipulating physical symbols. Schrödinger's wave mechanics is usually more convenient for detailed calculations. Its principal characteristic is its representation of the state of a mechanical system by a wave function. In wave mechanics, the state of a system is represented by a continuous function of the coordinates, known as the wave function. This function represents a position of the particle only roughly defined, and a momentum likewise only roughly defined. Associated with this representation is the fact that it is impossible to specify the position and the momentum of the particle at the same time. These things exist only insofar as they can be specified by a wave function in terms of a probability distribution. The probability distribution of the coordinate is given by the square of the absolute value of the wave function, and the probability distribution of momentum is given by the square of the absolute value of the Fourier transform of the wave function. The probability distribution gives the probability that a physical quantity has some one of its allowed quantized values. A single measurement of the quantity can yield only some one of its allowed values, but if many measurements are made under identical conditions the results will be distributed according to the calculated probability distribution. **Uncertainty**** Principle.** It follows from the method of representing the state, and of computing the probability distributions, that if the distribution for a coordinate is closely concentrated around a most probable value, the distribution for the momentum will be very widely spread. If *x* and *p* are the root-mean-square deviations of the coordinate *x* and corresponding momentum *p* from their mean values, it can be shown that x · p > or = ^{h}/4 . This is Heisenberg's **uncertainty*** principle,* or *principle of indetermination.* It can be regarded as a consequence of quantum mechanics, or as a condition that any satisfactory theory must satisfy. A conceptual experiment with the gamma-ray microscope is a famous argument leading to the **uncertainty** principle. To determine the position of a particle, let it be observed with a lens; to minimize the impact of light on the particle, let only one quantum of light be scattered. This quantum can produce a spot on a photographic plate. From the position of this spot one can infer the original position of the particle, by the laws of physical optics, with an **uncertainty** *x* ^{}/*sin* where is the aperture of the lens and is the wavelength of the light used. The **uncertainty** can be made as small as desired by using light of very short wavelength. But the scattering of one quantum of light will cause a change in the momentum of the particle (Compton effect). If the direction of scattering were known, this change in momentum could be calculated and allowed for; but it is not, since the scattered quantum might have gone through any part of the lens. The analysis shows that the **uncertainty** about the amount of momentum given to the particle is *p* ^{hsin}^{ }/. The presence of in the denominator shows that the condition for an accurate measurement of position—small —is just contrary to the condition for a small **uncertainty** in the momentum change. The product of these two expressions gives the **uncertainty** principle, independently of the wavelength of light used. This example might lead one to conclude that the indetermination is instrumental and due to inadequate methods of making precise measurements. Bohr, however, emphasized that the **uncertainty** is inescapable, because it is due to the incompatibility, in definition, of coordinate and momentum. The two things are so defined that they cannot exist, except approximately, at the same time. Similar **uncertainty** relations connect some other pairs of variables, such as energy and time. It is this kind of radical change in concept, necessitated by experimental results, that makes quantum mechanics difficult to understand. Nevertheless it permeates the whole of physics and permits a unified and quantitative treatment of a wide range of experimental facts. It is also of much philosophical interest because of its effect on ordinary ideas of determinism. **A. O. Williams, Jr.** Brown University **Bibliography** **Baggott, Jim, ***The Meaning of Quantum Chemistry: A Guide for Students of Chemistry and Physics *(Oxford 1992). **Bohm, David, and Basil J. 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Press 1994). **Silverman, Mark P., ***More Than One Mystery: Explorations in Quantum Interference *(Springer-Verlag 1995). **Slater, John, ***Quantum Theory of Matter *(Krieger 1977). **Suppes, Patrick, **ed., *Logic and Probability in Quantum Mechanics *(Kluwer 1975). **Von Neumann, John, ***Mathematical Foundations of Quantum Mechanics *(Princeton Univ. Press 1955). **Wiedner, Sol, ***The Foundations of Quantum Theory *(Academic Press 1972). **Wolf, Fred A., ***Taking the Quantum Leap *(Harper 1989).
__Top of Page__ | **How to cite this article:** **MLA (Modern Language Association) style:** Williams, Jr., A. O. "Quantum Theory." *Encyclopedia Americana*. 2006. Grolier Online. 6 Nov. 2006 . **Chicago Manual of Style:** Williams, Jr., A. O. "Quantum Theory." *Encyclopedia Americana*. Grolier Online http://ea.grolier.com/cgi-bin/article?assetid=0325750-00 (accessed November 6, 2006). **APA (American Psychological Association) style:** Williams, Jr., A. O. (2006). Quantum Theory. *Encyclopedia Americana*. Retrieved November 6, 2006, from Grolier Online http://ea.grolier.com/cgi-bin/article?assetid=0325750-00 |
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