 quantum mechanics

—quantummechanical, adj.a theory of the mechanics of atoms, molecules, and other physical systems that are subject to the uncertainty principle. Abbr.: QM Cf. nonrelativistic quantum mechanics, relativistic quantum mechanics.[192025]
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Branch of mathematical physics that deals with atomic and subatomic systems.It is concerned with phenomena that are so smallscale that they cannot be described in classical terms, and it is formulated entirely in terms of statistical probabilities. Considered one of the great ideas of the 20th century, quantum mechanics was developed mainly by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, and Max Born and led to a drastic reappraisal of the concept of objective reality. It explained the structure of atoms, atomic nuclei (see nucleus), and molecules; the behaviour of subatomic particles; the nature of chemical bonds (see bonding); the properties of crystalline solids (see crystal); nuclear energy; and the forces that stabilize collapsed stars. It also led directly to the development of the laser, the electron microscope, and the transistor.* * *
▪ physicsIntroductionscience dealing with the behaviour of matter and light on the atomic (atom) and subatomic (subatomic particle) scale. It attempts to describe and account for the properties of molecules and atoms and their constituents—electrons (electron), protons, neutrons, and other more esoteric particles such as quarks and gluons. These properties include the interactions of the particles with one another and with electromagnetic radiation (i.e., light, Xrays, and gamma rays).The behaviour of matter and radiation on the atomic scale often seems peculiar, and the consequences of quantum theory are accordingly difficult to understand and to believe. Its concepts frequently conflict with commonsense notions derived from observations of the everyday world. There is no reason, however, why the behaviour of the atomic world should conform to that of the familiar, largescale world. It is important to realize that quantum mechanics is a branch of physics and that the business of physics is to describe and account for the way the world—on both the large and the small scale—actually is and not how one imagines it or would like it to be.The study of quantum mechanics is rewarding for several reasons. First, it illustrates the essential methodology of physics. Second, it has been enormously successful in giving correct results in practically every situation to which it has been applied. There is, however, an intriguing paradox. In spite of the overwhelming practical success of quantum mechanics, the foundations of the subject contain unresolved problems—in particular, problems concerning the nature of measurement. An essential feature of quantum mechanics is that it is generally impossible, even in principle, to measure a system without disturbing it; the detailed nature of this disturbance and the exact point at which it occurs are obscure and controversial. Thus, quantum mechanics has attracted some of the ablest scientists of the 20th century, and they have erected what is perhaps the finest intellectual edifice of the period.Historical basis of quantum theoryBasic considerationsAt a fundamental level, both radiation and matter have characteristics of particles (waveparticle duality) and waves (wave motion). The gradual recognition by scientists that radiation has particlelike properties and that matter has wavelike properties provided the impetus for the development of quantum mechanics. Influenced by Newton, most physicists of the 18th century believed that light consisted of particles, which they called corpuscles. From about 1800, evidence began to accumulate for a wave theory of light. At about this time Thomas Young showed that, if monochromatic light passes through a pair of slits, the two emerging beams interfere, so that a fringe pattern of alternately bright and dark bands appears on a screen. The bands are readily explained by a wave theory of light. According to the theory, a bright band is produced when the crests (and troughs) of the waves from the two slits arrive together at the screen; a dark band is produced when the crest of one wave arrives at the same time as the trough of the other, and the effects of the two light beams cancel. Beginning in 1815, a series of experiments by AugustinJean Fresnel of France and others showed that, when a parallel beam of light passes through a single slit, the emerging beam is no longer parallel but starts to diverge; this phenomenon is known as diffraction. Given the wavelength of the light and the geometry of the apparatus (i.e., the separation and widths of the slits and the distance from the slits to the screen), one can use the wave theory to calculate the expected pattern in each case; the theory agrees precisely with the experimental data.Early developmentsPlanck's radiation lawBy the end of the 19th century, physicists almost universally accepted the wave theory of light. However, though the ideas of classical physics explain interference and diffraction phenomena relating to the propagation of light, they do not account for the absorption and emission of light. All bodies radiate electromagnetic energy as heat; in fact, a body emits radiation at all wavelengths. The energy radiated at different wavelengths is a maximum at a wavelength that depends on the temperature of the body; the hotter the body, the shorter the wavelength for maximum radiation. Attempts to calculate the energy distribution for the radiation from a blackbody using classical ideas were unsuccessful. (A blackbody is a hypothetical ideal body or surface that absorbs and reemits all radiant energy falling on it.) One formula, proposed by Wilhelm Wien of Germany, did not agree with observations at long wavelengths, and another, proposed by Lord Rayleigh (John William Strutt) of England, disagreed with those at short wavelengths.In 1900 the German theoretical physicist Max Planck (Planck, Max) made a bold suggestion. He assumed that the radiation energy is emitted, not continuously, but rather in discrete packets called quanta. The energy E of the quantum is related to the frequency ν by E = hν. The quantity h (Planck's constant), now known as the Planck constant, is a universal constant with the approximate value of 6.626075 × 10^{−34} joule∙second. Planck showed that the calculated energy spectrum then agreed with observation over the entire wavelength range.In 1905 Einstein extended Planck's hypothesis to explain the photoelectric effect, which is the emission of electrons by a metal surface when it is irradiated by light or moreenergetic photons. The kinetic energy of the emitted electrons depends on the frequency ν of the radiation, not on its intensity; for a given metal, there is a threshold frequency ν_{0} below which no electrons are emitted. Furthermore, emission takes place as soon as the light shines on the surface; there is no detectable delay. Einstein showed that these results can be explained by two assumptions: (1) that light is composed of corpuscles or photons (photon), the energy of which is given by Planck's relationship, and (2) that an atom in the metal can absorb either a whole photon or nothing. Part of the energy of the absorbed photon frees an electron, which requires a fixed energy W, known as the work function (electronic work function) of the metal; the rest is converted into the kinetic energy m_{e}u^{2}/2 of the emitted electron (m_{e} is the mass of the electron and u is its velocity). Thus, the energy relation isIf ν is less than ν_{0}, where hν_{0} = W, no electrons are emitted. Not all the experimental results mentioned above were known in 1905, but all Einstein's predictions have been verified since.Bohr's theory (Bohr atomic model) of the atomA major contribution to the subject was made by Niels Bohr (Bohr, Niels) of Denmark, who applied the quantum hypothesis to atomic spectra (spectrum) in 1913. The spectra of light emitted by gaseous atoms had been studied extensively since the mid19th century. It was found that radiation from gaseous atoms at low pressure consists of a set of discrete wavelengths (wavelength). This is quite unlike the radiation from a solid, which is distributed over a continuous range of wavelengths. The set of discrete wavelengths from gaseous atoms is known as a line spectrum, because the radiation (light) emitted consists of a series of sharp lines. The wavelengths of the lines are characteristic of the element and may form extremely complex patterns. The simplest spectra are those of atomic hydrogen and the alkali atoms (e.g., lithium, sodium, and potassium). For hydrogen, the wavelengths λ are given by the empirical formulawhere m and n are positive integers with n > m and R_{∞}, known as the Rydberg constant, has the value 1.097373177 × 10^{7} per metre. For a given value of m, the lines for varying n form a series. The lines for m = 1, the Lyman series, lie in the ultraviolet part of the spectrum; those for m = 2, the Balmer series, lie in the visible spectrum; and those for m = 3, the Paschen series, lie in the infrared.Bohr started with a model suggested by the New Zealandborn British physicist Ernest Rutherford (Rutherford, Ernest, Baron Rutherford of Nelson, of Cambridge). The model was based on the experiments of Hans Geiger (Geiger, Hans) and Ernest Marsden, who in 1909 bombarded gold atoms with massive, fastmoving alpha particles; when some of these particles were deflected backward, Rutherford (Rutherford atomic model) concluded that the atom has a massive, charged nucleus. In Rutherford's model, the atom resembles a miniature solar system with the nucleus acting as the Sun and the electrons as the circulating planets. Bohr made three assumptions. First, he postulated that, in contrast to classical mechanics, where an infinite number of orbits is possible, an electron can be in only one of a discrete set of orbits, which he termed stationary states (energy state). Second, he postulated that the only orbits allowed are those for which the angular momentum of the electron is a whole number n times ℏ (ℏ = h/2π). Third, Bohr assumed that Newton's laws of motion, so successful in calculating the paths of the planets around the Sun, also applied to electrons orbiting the nucleus. The force on the electron (the analogue of the gravitational force between the Sun and a planet) is the electrostatic attraction between the positively charged nucleus and the negatively charged electron. With these simple assumptions, he showed that the energy of the orbit has the formwhere E_{0} is a constant that may be expressed by a combination of the known constants e, m_{e}, and ℏ. While in a stationary state, the atom does not give off energy as light; however, when an electron makes a transition from a state with energy E_{n} to one with lower energy E_{m}, a quantum of energy is radiated with frequency ν, given by the equationInserting the expression for E_{n} into this equation and using the relation λν = c, where c is the speed of light, Bohr derived the formula for the wavelengths of the lines in the hydrogen spectrum, with the correct value of the Rydberg constant.Bohr's theory was a brilliant step forward. Its two most important features have survived in presentday quantum mechanics. They are (1) the existence of stationary, nonradiating states and (2) the relationship of radiation frequency to the energy difference between the initial and final states in a transition. Prior to Bohr, physicists had thought that the radiation frequency would be the same as the electron's frequency of rotation in an orbit.scattering of Xrays (Xray)Soon scientists were faced with the fact that another form of radiation, Xrays, also exhibits both wave and particle properties. Max von Laue (Laue, Max von) of Germany had shown in 1912 that crystals can be used as threedimensional diffraction gratings for Xrays; his technique constituted the fundamental evidence for the wavelike nature of Xrays. The atoms of a crystal, which are arranged in a regular lattice, scatter the Xrays. For certain directions of scattering, all the crests of the Xrays coincide. (The scattered Xrays are said to be in phase and to give constructive interference.) For these directions, the scattered Xray beam is very intense. Clearly, this phenomenon demonstrates wave behaviour. In fact, given the interatomic distances in the crystal and the directions of constructive interference, the wavelength of the waves can be calculated.In 1922 the American physicist Arthur Holly Compton (Compton, Arthur Holly) showed that Xrays scatter (Compton effect) from electrons as if they are particles. Compton performed a series of experiments on the scattering of monochromatic, highenergy Xrays by graphite. He found that part of the scattered radiation had the same wavelength λ_{0} as the incident Xrays but that there was an additional component with a longer wavelength λ. To interpret his results, Compton regarded the Xray photon as a particle that collides and bounces off an electron in the graphite target as though the photon and the electron were a pair of (dissimilar) billiard balls. Application of the laws of conservation of energy and momentum to the collision leads to a specific relation between the amount of energy transferred to the electron and the angle of scattering. For Xrays scattered through an angle θ, the wavelengths λ and λ_{0} are related by the equationThe experimental correctness of Compton's formula is direct evidence for the corpuscular behaviour of radiation.Broglie's wave (de Broglie wave) hypothesisFaced with evidence that electromagnetic radiation has both particle and wave characteristics, LouisVictor de Broglie (Broglie, LouisVictor, 7e duc de) of France suggested a great unifying hypothesis in 1924. Broglie proposed that matter has wave, as well as particle, properties. He suggested that material particles can behave as waves and that their wavelength λ is related to the linear momentum p of the particle by λ = h/p.In 1927 Clinton Davisson (Davisson, Clinton Joseph) and Lester Germer (Germer, Lester Halbert) of the United States confirmed Broglie's hypothesis for electrons. Using a crystal of nickel, they diffracted a beam of monoenergetic electrons and showed that the wavelength of the waves is related to the momentum of the electrons by the Broglie equation. Since Davisson and Germer's investigation, similar experiments have been performed with atoms, molecules, neutrons, protons, and many other particles. All behave like waves with the same wavelengthmomentum relationship.Basic concepts and methodsBohr's theory, which assumed that electrons moved in circular orbits, was extended by the German physicist Arnold Sommerfeld (Sommerfeld, Arnold) and others to include elliptic orbits and other refinements. Attempts were made to apply the theory to more complicated systems than the hydrogen atom. However, the ad hoc mixture of classical and quantum ideas made the theory and calculations increasingly unsatisfactory. Then in the 12 months starting in July 1925, a period of creativity without parallel in the history of physics, there appeared a series of papers by German scientists that set the subject on a firm conceptual foundation. The papers took two approaches: (1) matrix mechanics, proposed by Werner Heisenberg (Heisenberg, Werner), Max Born, and Pascual Jordan, and (2) wave mechanics, put forward by Erwin Schrödinger (Schrödinger, Erwin). The protagonists were not always polite to each other. Heisenberg found the physical ideas of Schrödinger's theory “disgusting,” and Schrödinger was “discouraged and repelled” by the lack of visualization in Heisenberg's method. However, Schrödinger, not allowing his emotions to interfere with his scientific endeavours, showed that, in spite of apparent dissimilarities, the two theories are equivalent mathematically. The present discussion follows Schrödinger's wave mechanics because it is less abstract and easier to understand than Heisenberg's matrix mechanics.Schrödinger's wave mechanicsSchrödinger expressed Broglie's hypothesis concerning the wave behaviour of matter in a mathematical form (Schrödinger equation) that is adaptable to a variety of physical problems without additional arbitrary assumptions. He was guided by a mathematical formulation of optics, in which the straightline propagation of light rays can be derived from wave motion when the wavelength is small compared to the dimensions of the apparatus employed. In the same way, Schrödinger set out to find a wave equation for matter that would give particlelike propagation when the wavelength becomes comparatively small. According to classical mechanics, if a particle of mass m_{e} is subjected to a force such that its potential energy is V(x, y, z) at position x, y, z, then the sum of V(x, y, z) and the kinetic energy p^{2}/2m_{e} is equal to a constant, the total energy E of the particle. Thus,It is assumed that the particle is bound—i.e., confined by the potential to a certain region in space because its energy E is insufficient for it to escape. Since the potential varies with position, two other quantities do also: the momentum and, hence, by extension from the Broglie relation, the wavelength of the wave. Postulating a wave function Ψ(x, y, z) that varies with position, Schrödinger replaced p in the above energy equation with a differential operator that embodied the Broglie relation. He then showed that Ψ satisfies the partial differential equationThis is the (timeindependent) Schrödinger wave equation, which established quantum mechanics in a widely applicable form. An important advantage of Schrödinger's theory is that no further arbitrary quantum conditions need be postulated. The required quantum results follow from certain reasonable restrictions placed on the wave function—for example, that it should not become infinitely large at large distances from the centre of the potential.Schrödinger applied his equation to the hydrogen atom, for which the potential function, given by classical electrostatics, is proportional to −e^{2}/r, where −e is the charge on the electron. The nucleus (a proton of charge e) is situated at the origin, and r is the distance from the origin to the position of the electron. Schrödinger solved the equation for this particular potential with straightforward, though not elementary, mathematics. Only certain discrete values of E lead to acceptable functions Ψ. These functions are characterized by a trio of integers n, l, m, termed quantum numbers (quantum number). The values of E depend only on the integers n (1, 2, 3, etc.) and are identical with those given by the Bohr theory. The quantum numbers l and m are related to the angular momentum of the electron; √(l(l + 1)) ℏ is the magnitude of the angular momentum, and mℏ is its component along some physical direction.The square of the wave function, Ψ^{2}, has a physical interpretation. Schrödinger originally supposed that the electron was spread out in space and that its density at point x, y, z was given by the value of Ψ^{2} at that point. Almost immediately Born (Born, Max) proposed what is now the accepted interpretation—namely, that Ψ^{2} gives the probability of finding the electron at x, y, z. The distinction between the two interpretations is important. If Ψ^{2} is small at a particular position, the original interpretation implies that a small fraction of an electron will always be detected there. In Born's interpretation, nothing will be detected there most of the time, but, when something is observed, it will be a whole electron. Thus, the concept of the electron as a point particle moving in a welldefined path around the nucleus is replaced in wave mechanics by clouds that describe the probable locations of electrons in different states.Electron spin and antiparticles (antiparticle)In 1928 the English physicist Paul A.M. Dirac (Dirac, P.A.M.) produced a wave equation for the electron that combined relativity with quantum mechanics (relativistic mechanics). Schrödinger's wave equation does not satisfy the requirements of the special theory of relativity because it is based on a nonrelativistic expression for the kinetic energy (p^{2}/2m_{e}). Dirac showed that an electron has an additional quantum number m_{s}. Unlike the first three quantum numbers, m_{s} is not a whole integer and can have only the values +^{1}/_{2} and −^{1}/_{2}. It corresponds to an additional form of angular momentum ascribed to a spinning motion. (The angular momentum mentioned above is due to the orbital motion of the electron, not its spin.) The concept of spin angular momentum was introduced in 1925 by Samuel A. Goudsmit (Goudsmit, Samuel Abraham) and George E. Uhlenbeck (Uhlenbeck, George Eugene), two graduate students at the University of Leiden, Neth., to explain the magnetic moment measurements made by Otto Stern and Walther Gerlach of Germany several years earlier. The magnetic moment of a particle is closely related to its angular momentum; if the angular momentum is zero, so is the magnetic moment. Yet Stern and Gerlach had observed a magnetic moment for electrons in silver atoms, which were known to have zero orbital angular momentum. Goudsmit and Uhlenbeck proposed that the observed magnetic moment was attributable to spin angular momentum.The electronspin hypothesis not only provided an explanation for the observed magnetic moment but also accounted for many other effects in atomic spectroscopy, including changes in spectral lines in the presence of a magnetic field (Zeeman effect), doublet lines in alkali spectra, and fine structure (close doublets and triplets) in the hydrogen spectrum.The Dirac equation also predicted additional states of the electron that had not yet been observed. Experimental confirmation was provided in 1932 by the discovery of the positron by the American physicist Carl David Anderson (Anderson, Carl David). Every particle described by the Dirac equation has to have a corresponding antiparticle, which differs only in charge. The positron is just such an antiparticle of the negatively charged electron, having the same mass as the latter but a positive charge.Identical particles and multielectron atomsBecause electrons are identical to (i.e., indistinguishable from) each other, the wave function of an atom with more than one electron must satisfy special conditions. The problem of identical particles does not arise in classical physics, where the objects are largescale and can always be distinguished, at least in principle. There is no way, however, to differentiate two electrons in the same atom, and the form of the wave function must reflect this fact. The overall wave function Ψ of a system of identical particles depends on the coordinates of all the particles. If the coordinates of two of the particles are interchanged, the wave function must remain unaltered or, at most, undergo a change of sign; the change of sign is permitted because it is Ψ^{2} that occurs in the physical interpretation of the wave function. If the sign of Ψ remains unchanged, the wave function is said to be symmetric with respect to interchange; if the sign changes, the function is antisymmetric.The symmetry of the wave function for identical particles is closely related to the spin of the particles. In quantum field theory (see below Quantum electrodynamics (quantum mechanics)), it can be shown that particles with halfintegral spin (^{1}/_{2}, ^{3}/_{2}, etc.) have antisymmetric wave functions. They are called fermions (fermion) after the Italianborn physicist Enrico Fermi. Examples of fermions are electrons, protons, and neutrons, all of which have spin ^{1}/_{2}. Particles with zero or integral spin (e.g., mesons, photons) have symmetric wave functions and are called bosons (boson) after the Indian mathematician and physicist Satyendra Nath Bose (Bose, Satyendra Nath), who first applied the ideas of symmetry to photons in 1924–25.The requirement of antisymmetric wave functions for fermions leads to a fundamental result, known as the exclusion principle (Pauli exclusion principle), first proposed in 1925 by the Austrian physicist Wolfgang Pauli (Pauli, Wolfgang). The exclusion principle states that two fermions in the same system cannot be in the same quantum state. If they were, interchanging the two sets of coordinates would not change the wave function at all, which contradicts the result that the wave function must change sign. Thus, two electrons in the same atom cannot have an identical set of values for the four quantum numbers n, l, m, m_{s}. The exclusion principle forms the basis of many properties of matter, including the periodic classification of the elements, the nature of chemical bonds, and the behaviour of electrons in solids; the last determines in turn whether a solid is a metal, insulator, or semiconductor (see atom and matter (phase)).The Schrödinger equation cannot be solved precisely for atoms with more than one electron. The principles of the calculation are well understood, but the problems are complicated by the number of particles and the variety of forces involved. The forces include the electrostatic forces between the nucleus and the electrons and between the electrons themselves, as well as weaker magnetic forces arising from the spin and orbital motions of the electrons. Despite these difficulties, approximation methods introduced by the English physicist Douglas R. Hartree and others in the 1920s have achieved considerable success. Such schemes start by assuming that each electron moves independently in an average electric field because of the nucleus and the other electrons—i.e., correlations between the positions of the electrons are ignored. Each electron has its own wave function, called an orbital. The overall wave function for all the electrons in the atom satisfies the exclusion principle. Corrections to the calculated energies are then made, which depend on the strengths of the electronelectron correlations and the magnetic forces.Timedependent Schrödinger equationAt the same time that Schrödinger proposed his timeindependent equation to describe the stationary states, he also proposed a timedependent equation to describe how a system changes from one state to another. By replacing the energy E in Schrödinger's equation with a timederivative operator, he generalized his wave equation to determine the time variation of the wave function as well as its spatial variation. The timedependent Schrödinger equation readsThe quantity i is the square root of −1. The function Ψ varies with time t as well as with position x, y, z. For a system with constant energy, E, Ψ has the formwhere exp stands for the exponential function, and the timedependent Schrödinger equation reduces to the timeindependent form.The probability of a transition between one atomic stationary state and some other state can be calculated with the aid of the timedependent Schrödinger equation. For example, an atom may change spontaneously from one state to another state with less energy, emitting the difference in energy as a photon with a frequency given by the Bohr relation. If electromagnetic radiation is applied to a set of atoms and if the frequency of the radiation matches the energy difference between two stationary states, transitions can be stimulated. In a stimulated transition, the energy of the atom may increase—i.e., the atom may absorb a photon from the radiation—or the energy of the atom may decrease, with the emission of a photon, which adds to the energy of the radiation. Such stimulated emission processes form the basic mechanism for the operation of lasers (laser). The probability of a transition from one state to another depends on the values of the l, m, m_{s} quantum numbers of the initial and final states. For most values, the transition probability is effectively zero. However, for certain changes in the quantum numbers, summarized as selection rules, there is a finite probability. For example, according to one important selection rule, the l value changes by unity because photons have a spin of 1. The selection rules for radiation relate to the angular momentum properties of the stationary states. The absorbed or emitted photon has its own angular momentum, and the selection rules reflect the conservation of angular momentum between the atoms and the radiation.The phenomenon of tunneling, which has no counterpart in classical physics, is an important consequence of quantum mechanics. Consider a particle with energy E in the inner region of a onedimensional potential well V(x), as shown in Figure 1—>. (A potential well is a potential that has a lower value in a certain region of space than in the neighbouring regions.) In classical mechanics, if E < V_{0} (the maximum height of the potential barrier), the particle remains in the well forever; if E > V_{0}, the particle escapes. In quantum mechanics, the situation is not so simple. The particle can escape even if its energy E is below the height of the barrier V_{0}, although the probability of escape is small unless E is close to V_{0}. In that case, the particle may tunnel through the potential barrier and emerge with the same energy E.The phenomenon of tunneling has many important applications. For example, it describes a type of radioactive decay (alpha decay) in which a nucleus emits an alpha particle (a helium nucleus). According to the quantum explanation given independently by George Gamow (Gamow, George) and by Ronald W. Gurney and Edward Condon (Condon, Edward U) in 1928, the alpha particle is confined before the decay by a potential of the shape shown in Figure 1—>. For a given nuclear species, it is possible to measure the energy E of the emitted alpha particle and the average lifetime τ of the nucleus before decay. The lifetime of the nucleus is a measure of the probability of tunneling through the barrier—the shorter the lifetime, the higher the probability. With plausible assumptions about the general form of the potential function, it is possible to calculate a relationship between τ and E that is applicable to all alpha emitters. This theory, which is borne out by experiment, shows that the probability of tunneling, and hence the value of τ, is extremely sensitive to the value of E. For all known alphaparticle emitters, the value of E varies from about 2 to 8 million electron volts, or MeV (1 MeV = 10^{6} electron volts). Thus, the value of E varies only by a factor of 4, whereas the range of τ is from about 10^{11} years down to about 10^{−6} second, a factor of 10^{24}. It would be difficult to account for this sensitivity of τ to the value of E by any theory other than quantum mechanical tunneling.Axiomatic approachAlthough the two Schrödinger equations form an important part of quantum mechanics, it is possible to present the subject in a more general way. Dirac (Dirac, P.A.M.) gave an elegant exposition of an axiomatic approach based on observables and states in a classic textbook entitled The Principles of Quantum Mechanics. (The book, published in 1930, is still in print.) An observable is anything that can be measured—energy, position, a component of angular momentum, and so forth. Every observable has a set of states, each state being represented by an algebraic function. With each state is associated a number that gives the result of a measurement of the observable. Consider an observable with N states, denoted by ψ_{1}, ψ_{2}, . . . , ψ_{N}, and corresponding measurement values a_{1}, a_{2}, . . . , a_{N}. A physical system—e.g., an atom in a particular state—is represented by a wave function Ψ, which can be expressed as a linear combination, or mixture, of the states of the observable. Thus, the Ψ may be written asFor a given Ψ, the quantities c_{1}, c_{2}, etc., are a set of numbers that can be calculated. In general, the numbers are complex, but, in the present discussion, they are assumed to be real numbers.The theory postulates, first, that the result of a measurement must be an avalue—i.e., a_{1}, a_{2}, or a_{3}, etc. No other value is possible. Second, before the measurement is made, the probability of obtaining the value a_{1} is c_{1}^{2}, and that of obtaining the value a_{2} is c_{2}^{2}, and so on. If the value obtained is, say, a_{5}, the theory asserts that after the measurement the state of the system is no longer the original Ψ but has changed to ψ_{5}, the state corresponding to a_{5}.A number of consequences follow from these assertions. First, the result of a measurement cannot be predicted with certainty. Only the probability of a particular result can be predicted, even though the initial state (represented by the function Ψ) is known exactly. Second, identical measurements made on a large number of identical systems, all in the identical state Ψ, will produce different values for the measurements. This is, of course, quite contrary to classical physics and common sense, which say that the same measurement on the same object in the same state must produce the same result. Moreover, according to the theory, not only does the act of measurement change the state of the system, but it does so in an indeterminate way. Sometimes it changes the state to ψ_{1}, sometimes to ψ_{2}, and so forth.There is an important exception to the above statements. Suppose that, before the measurement is made, the state Ψ happens to be one of the ψs, say Ψ = ψ_{3}. Then c_{3} = 1 and all the other cs are zero. This means that, before the measurement is made, the probability of obtaining the value a_{3} is unity and the probability of obtaining any other value of a is zero. In other words, in this particular case, the result of the measurement can be predicted with certainty. Moreover, after the measurement is made, the state will be ψ_{3}, the same as it was before. Thus, in this particular case, measurement does not disturb the system. Whatever the initial state of the system, two measurements made in rapid succession (so that the change in the wave function given by the timedependent Schrödinger equation is negligible) produce the same result.The value of one observable can be determined by a single measurement. The value of two observables for a given system may be known at the same time, provided that the two observables have the same set of state functions ψ_{1}, ψ_{2}, . . . , ψ_{N}. In this case, measuring the first observable results in a state function that is one of the ψs. Because this is also a state function of the second observable, the result of measuring the latter can be predicted with certainty. Thus the values of both observables are known. (Although the ψs are the same for the two observables, the two sets of a values are, in general, different.) The two observables can be measured repeatedly in any sequence. After the first measurement, none of the measurements disturbs the system, and a unique pair of values for the two observables is obtained.Incompatible observablesThe measurement of two observables with different sets of state functions is a quite different situation. Measurement of one observable gives a certain result. The state function after the measurement is, as always, one of the states of that observable; however, it is not a state function for the second observable. Measuring the second observable disturbs the system, and the state of the system is no longer one of the states of the first observable. In general, measuring the first observable again does not produce the same result as the first time. To sum up, both quantities cannot be known at the same time, and the two observables are said to be incompatible.A specific example of this behaviour is the measurement of the component of angular momentum along two mutually perpendicular directions. The SternGerlach experiment mentioned above involved measuring the angular momentum of a silver atom in the ground state. In reconstructing this experiment, a beam of silver atoms is passed between the poles of a magnet. The poles are shaped so that the magnetic field varies greatly in strength over a very small distance (Figure 2—>). The apparatus determines the m_{s} quantum number, which can be +^{1}/_{2} or −^{1}/_{2}. No other values are obtained. Thus in this case the observable has only two states—i.e., N = 2. The inhomogeneous magnetic field produces a force on the silver atoms in a direction that depends on the spin state of the atoms. The result is shown schematically in Figure 3—>. A beam of silver atoms is passed through magnet A. The atoms in the state with m_{s} = +^{1}/_{2} are deflected upward and emerge as beam 1, while those with m_{s} = −^{1}/_{2} are deflected downward and emerge as beam 2. If the direction of the magnetic field is the xaxis, the apparatus measures S_{x}, which is the xcomponent of spin angular momentum. The atoms in beam 1 have S_{x} = +ℏ/2 while those in beam 2 have S_{x} = −ℏ/2. In a classical picture, these two states represent atoms spinning about the direction of the xaxis with opposite senses of rotation.The ycomponent of spin angular momentum S_{y} also can have only the values +ℏ/2 and −ℏ/2; however, the two states of S_{y} are not the same as for S_{x}. In fact, each of the states of S_{x} is an equal mixture of the states for S_{y}, and conversely. Again, the two S_{y} states may be pictured as representing atoms with opposite senses of rotation about the yaxis. These classical pictures of quantum states are helpful, but only up to a certain point. For example, quantum theory says that each of the states corresponding to spin about the xaxis is a superposition of the two states with spin about the yaxis. There is no way to visualize this; it has absolutely no classical counterpart. One simply has to accept the result as a consequence of the axioms of the theory. Suppose that, as in Figure 3—>, the atoms in beam 1 are passed into a second magnet B, which has a magnetic field along the yaxis perpendicular to x. The atoms emerge from B and go in equal numbers through its two output channels. Classical theory says that the two magnets together have measured both the x and ycomponents of spin angular momentum and that the atoms in beam 3 have S_{x} = +ℏ/2, S_{y} = +ℏ/2, while those in beam 4 have S_{x} = +ℏ/2, S_{y} = −ℏ/2. However, classical theory is wrong, because if beam 3 is put through still another magnet C, with its magnetic field along x, the atoms divide equally into beams 5 and 6 instead of emerging as a single beam 5 (as they would if they had S_{x} = +ℏ/2). Thus, the correct statement is that the beam entering B has S_{x} = +ℏ/2 and is composed of an equal mixture of the states S_{y} = +ℏ/2 and S_{y} = −ℏ/2—i.e., the xcomponent of angular momentum is known but the ycomponent is not. Correspondingly, beam 3 leaving B has S_{y} = +ℏ/2 and is an equal mixture of the states S_{x} = +ℏ/2 and S_{x} = −ℏ/2; the ycomponent of angular momentum is known but the xcomponent is not. The information about S_{x} is lost because of the disturbance caused by magnet B in the measurement of S_{y}.Heisenberg uncertainty principleThe observables discussed so far have had discrete sets of experimental values. For example, the values of the energy of a bound system are always discrete, and angular momentum components have values that take the form mℏ, where m is either an integer or a halfinteger, positive or negative. On the other hand, the position of a particle or the linear momentum of a free particle can take continuous values in both quantum and classical theory. The mathematics of observables with a continuous spectrum of measured values is somewhat more complicated than for the discrete case but presents no problems of principle. An observable with a continuous spectrum of measured values has an infinite number of state functions. The state function Ψ of the system is still regarded as a combination of the state functions of the observable, but the sum in equation (10) must be replaced by an integral.Measurements can be made of position x of a particle and the xcomponent of its linear momentum, denoted by p_{x}. These two observables are incompatible because they have different state functions. The phenomenon of diffraction noted above illustrates the impossibility of measuring position and momentum simultaneously and precisely. If a parallel monochromatic light beam passes through a slit (Figure 4A—>), its intensity varies with direction, as shown in Figure 4B—>. The light has zero intensity in certain directions. Wave theory shows that the first zero occurs at an angle θ_{0}, given by sin θ_{0} = λ/b, where λ is the wavelength of the light and b is the width of the slit. If the width of the slit is reduced, θ_{0} increases—i.e., the diffracted light is more spread out. Thus, θ_{0} measures the spread of the beam.The experiment can be repeated with a stream of electrons instead of a beam of light. According to Broglie, electrons have wavelike properties; therefore, the beam of electrons emerging from the slit should widen and spread out like a beam of light waves. This has been observed in experiments. If the electrons have velocity u in the forward direction (i.e., the ydirection in Figure 4A—>), their (linear) momentum is p = m_{e}u. Consider p_{x}, the component of momentum in the xdirection. After the electrons have passed through the aperture, the spread in their directions results in an uncertainty in p_{x} by an amountwhere λ is the wavelength of the electrons and, according to the Broglie formula, equals h/p. Thus, Δp_{x} ≈ h/b. Exactly where an electron passed through the slit is unknown; it is only certain that an electron went through somewhere. Therefore, immediately after an electron goes through, the uncertainty in its xposition is Δx ≈ b/2. Thus the product of the uncertainties is of the order of ℏ. More exact analysis shows that the product has a lower limit, given byThis is the wellknown Heisenberg uncertainty principle for position and momentum. It states that there is a limit to the precision with which the position and the momentum of an object can be measured at the same time. Depending on the experimental conditions, either quantity can be measured as precisely as desired (at least in principle), but the more precisely one of the quantities is measured, the less precisely the other is known.The uncertainty principle is significant only on the atomic scale because of the small value of h in everyday units. If the position of a macroscopic object with a mass of, say one gram, is measured with a precision of 10^{−6} metre, the uncertainty principle states that its velocity cannot be measured to better than about 10^{−25} metre per second. Such a limitation is hardly worrisome. However, if an electron is located in an atom about 10^{−10} metre across, the principle gives a minimum uncertainty in the velocity of about 10^{6} metre per second.The above reasoning leading to the uncertainty principle is based on the waveparticle duality of the electron. When Heisenberg first propounded the principle in 1927 his reasoning was based, however, on the waveparticle duality of the photon. He considered the process of measuring the position of an electron by observing it in a microscope. Diffraction effects due to the wave nature of light result in a blurring of the image; the resulting uncertainty in the position of the electron is approximately equal to the wavelength of the light. To reduce this uncertainty, it is necessary to use light of shorter wavelength—e.g., gamma rays. However, in producing an image of the electron, the gammaray photon bounces off the electron, giving the Compton effect (see above: Early developments: Scattering of Xrays (quantum mechanics)). As a result of the collision, the electron recoils in a statistically random way. The resulting uncertainty in the momentum of the electron is proportional to the momentum of the photon, which is inversely proportional to the wavelength of the photon. So it is again the case that increased precision in knowledge of the position of the electron is gained only at the expense of decreased precision in knowledge of its momentum. A detailed calculation of the process yields the same result as before (equation [12]). Heisenberg's reasoning brings out clearly the fact that the smaller the particle being observed, the more significant is the uncertainty principle. When a large body is observed, photons still bounce off it and change its momentum, but, considered as a fraction of the initial momentum of the body, the change is insignificant.The Schrödinger and Dirac theories give a precise value for the energy of each stationary state, but in reality the states do not have a precise energy. The only exception is in the ground (lowest energy) state. Instead, the energies of the states are spread over a small range. The spread arises from the fact that, because the electron can make a transition to another state, the initial state has a finite lifetime. The transition is a random process, and so different atoms in the same state have different lifetimes. If the mean lifetime is denoted as τ, the theory shows that the energy of the initial state has a spread of energy ΔE, given byThis energy spread is manifested in a spread in the frequencies of emitted radiation. Therefore, the spectral lines are not infinitely sharp. (Some experimental factors can also broaden a line, but their effects can be reduced; however, the present effect, known as natural broadening, is fundamental and cannot be reduced.) Equation (13) is another type of Heisenberg uncertainty relation; generally, if a measurement with duration τ is made of the energy in a system, the measurement disturbs the system, causing the energy to be uncertain by an amount ΔE, the magnitude of which is given by the above equation.The application of quantum theory to the interaction between electrons and radiation requires a quantum treatment of Maxwell's field equations, which are the foundations of electromagnetism, and the relativistic theory of the electron formulated by Dirac (see above Electron spin and antiparticles (quantum mechanics)). The resulting quantum field theory is known as quantum electrodynamics, or QED.QED accounts for the behaviour and interactions of electrons, positrons, and photons. It deals with processes involving the creation of material particles from electromagnetic energy and with the converse processes in which a material particle and its antiparticle annihilate each other and produce energy. Initially the theory was beset with formidable mathematical difficulties, because the calculated values of quantities such as the charge and mass of the electron proved to be infinite. However, an ingenious set of techniques developed (in the late 1940s) by Hans Bethe, Julian S. Schwinger, Tomonaga Shin'ichirō, Richard P. Feynman, and others dealt systematically with the infinities to obtain finite values of the physical quantities. Their method is known as renormalization. The theory has provided some remarkably accurate predictions.According to the Dirac theory, two particular states in hydrogen with different quantum numbers have the same energy. QED, however, predicts a small difference in their energies; the difference may be determined by measuring the frequency of the electromagnetic radiation that produces transitions (transition) between the two states. This effect was first measured by Willis E. Lamb, Jr., and Robert Retherford in 1947. Its physical origin lies in the interaction of the electron with the random fluctuations in the surrounding electromagnetic field. These fluctuations, which exist even in the absence of an applied field, are a quantum phenomenon. The accuracy of experiment and theory in this area may be gauged by two recent values for the separation of the two states, expressed in terms of the frequency of the radiation that produces the transitions:An even more spectacular example of the success of QED is provided by the value for μ_{e}, the magnetic dipole moment of the free electron. Because the electron is spinning and has electric charge, it behaves like a tiny magnet, the strength of which is expressed by the value of μ_{e}. According to the Dirac theory, μ_{e} is exactly equal to μ_{B} = eℏ/2m_{e}, a quantity known as the Bohr magneton; however, QED predicts that μ_{e} = (1 + a)μ_{B}, where a is a small number, approximately ^{1}/_{860}. Again, the physical origin of the QED correction is the interaction of the electron with random oscillations in the surrounding electromagnetic field. The best experimental determination of μ_{e} involves measuring not the quantity itself but the small correction term μ_{e} − μ_{B}. This greatly enhances the sensitivity of the experiment. The most recent results for the value of a areSince a itself represents a small correction term, the magnetic dipole moment of the electron is measured with an accuracy of about one part in 10^{11}. One of the most precisely determined quantities in physics, the magnetic dipole moment of the electron can be calculated correctly from quantum theory to within about one part in 10^{10}.The interpretation of quantum mechanicsAlthough quantum mechanics has been applied to problems in physics with great success, some of its ideas seem strange. A few of their implications are considered here.The electron: wave or particle (waveparticle duality)?Young's aforementioned experiment in which a parallel beam of monochromatic light is passed through a pair of narrow parallel slits (Figure 5A—>) has an electron counterpart. In Young's original experiment, the intensity of the light varies with direction after passing through the slits (Figure 5B—>). The intensity oscillates because of interference between the light waves emerging from the two slits, the rate of oscillation depending on the wavelength of the light and the separation of the slits. The oscillation creates a fringe pattern of alternating light and dark bands that is modulated by the diffraction pattern from each slit. If one of the slits is covered, the interference fringes disappear, and only the diffraction pattern (shown as a broken line in Figure 5B—>) is observed.Young's experiment can be repeated with electrons all with the same momentum. The screen in the optical experiment is replaced by a closely spaced grid of electron detectors. There are many devices for detecting electrons; the most common are scintillators. When an electron passes through a scintillating material, such as sodium iodide, the material produces a light flash which gives a voltage pulse that can be amplified and recorded. The pattern of electrons recorded by each detector is the same as that predicted for waves (wave motion) with wavelengths given by the Broglie formula. Thus, the experiment provides conclusive evidence for the wave behaviour of electrons.If the experiment is repeated with a very weak source of electrons so that only one electron passes through the slits, a single detector registers the arrival of an electron. This is a welllocalized event characteristic of a particle. Each time the experiment is repeated, one electron passes through the slits and is detected. A graph plotted with detector position along one axis and the number of electrons along the other looks exactly like the oscillating interference pattern in Figure 5B—>. Thus, the intensity function in the figure is proportional to the probability of the electron moving in a particular direction after it has passed through the slits. Apart from its units, the function is identical to Ψ^{2}, where Ψ is the solution of the timeindependent Schrödinger equation for this particular experiment.If one of the slits is covered, the fringe pattern disappears and is replaced by the diffraction pattern for a single slit. Thus, both slits are needed to produce the fringe pattern. However, if the electron is a particle, it seems reasonable to suppose that it passed through only one of the slits. The apparatus can be modified to ascertain which slit by placing a thin wire loop around each slit. When an electron passes through a loop, it generates a small electric signal, showing which slit it passed through. However, the interference fringe pattern then disappears, and the singleslit diffraction pattern returns. Since both slits are needed for the interference pattern to appear and since it is impossible to know which slit the electron passed through without destroying that pattern, one is forced to the conclusion that the electron goes through both slits at the same time.In summary, the experiment shows both the wave and particle properties of the electron. The wave property predicts the probability of direction of travel before the electron is detected; on the other hand, the fact that the electron is detected in a particular place shows that it has particle properties. Therefore, the answer to the question whether the electron is a wave or a particle is that it is neither. It is an object exhibiting either wave or particle properties, depending on the type of measurement that is made on it. In other words, one cannot talk about the intrinsic properties of an electron; instead, one must consider the properties of the electron and measuring apparatus together.Hidden variablesA fundamental concept (uncertainty principle) in quantum mechanics is that of randomness, or indeterminacy. In general, the theory predicts only the probability (probability and statistics) of a certain result. Consider the case of radioactivity. Imagine a box of atoms with identical nuclei that can undergo decay with the emission of an alpha particle. In a given time interval, a certain fraction will decay. The theory may tell precisely what that fraction will be, but it cannot predict which particular nuclei will decay. The theory asserts that, at the beginning of the time interval, all the nuclei are in an identical state and that the decay is a completely random process. Even in classical physics, many processes appear random. For example, one says that, when a roulette wheel is spun, the ball will drop at random into one of the numbered compartments in the wheel. Based on this belief, the casino owner and the players give and accept identical odds against each number for each throw. However, the fact is that the winning number could be predicted if one noted the exact location of the wheel when the croupier released the ball, the initial speed of the wheel, and various other physical parameters. It is only ignorance of the initial conditions and the difficulty of doing the calculations that makes the outcome appear to be random. In quantum mechanics, on the other hand, the randomness is asserted to be absolutely fundamental. The theory says that, though one nucleus decayed and the other did not, they were previously in the identical state.Many eminent physicists, including Einstein (Einstein, Albert), have not accepted this indeterminacy (determinism). They have rejected the notion that the nuclei were initially in the identical state. Instead, they postulated that there must be some other property—presently unknown, but existing nonetheless—that is different for the two nuclei. This type of unknown property is termed a hidden variable; if it existed, it would restore determinacy to physics. If the initial values of the hidden variables were known, it would be possible to predict which nuclei would decay. Such a theory would, of course, also have to account for the wealth of experimental data which conventional quantum mechanics explains from a few simple assumptions. Attempts have been made by Broglie, David Bohm, and others to construct theories based on hidden variables, but the theories are very complicated and contrived. For example, the electron would definitely have to go through only one slit in the twoslit experiment. To explain that interference occurs only when the other slit is open, it is necessary to postulate a special force on the electron which exists only when that slit is open. Such artificial additions make hidden variable theories unattractive, and there is little support for them among physicists.The orthodox view of quantum mechanics—and the one adopted in the present article—is known as the Copenhagen interpretation because its main protagonist, Niels Bohr, worked in that city. The Copenhagen view of understanding the physical world stresses the importance of basing theory on what can be observed and measured experimentally. It therefore rejects the idea of hidden variables as quantities that cannot be measured. The Copenhagen view is that the indeterminacy observed in nature is fundamental and does not reflect an inadequacy in present scientific knowledge. One should therefore accept the indeterminacy without trying to “explain” it and see what consequences come from it.Attempts have been made to link the existence of free will with the indeterminacy of quantum mechanics, but it is difficult to see how this feature of the theory makes free will more plausible. On the contrary, free will presumably implies rational thought and decision, whereas the essence of the indeterminism in quantum mechanics is that it is due to intrinsic randomness.Paradox of Einstein, Podolsky, and RosenIn 1935 Einstein (Einstein, Albert) and two other physicists in the United States, Boris Podolsky and Nathan Rosen, analyzed a thought experiment to measure position and momentum in a pair of interacting systems. Employing conventional quantum mechanics, they obtained some startling results, which led them to conclude that the theory does not give a complete description of physical reality. Their results, which are so peculiar as to seem paradoxical, are based on impeccable reasoning, but their conclusion that the theory is incomplete does not necessarily follow. Bohm (Bohm, David) simplified their experiment while retaining the central point of their reasoning; this discussion follows his account.The proton, like the electron, has spin ^{1}/_{2}; thus, no matter what direction is chosen for measuring the component of its spin angular momentum, the values are always +ℏ/2 or −ℏ/2. (The present discussion relates only to spin angular momentum, and the word spin is omitted from now on.) It is possible to obtain a system consisting of a pair of protons in close proximity and with total angular momentum equal to zero. Thus, if the value of one of the components of angular momentum for one of the protons is +ℏ/2 along any selected direction, the value for the component in the same direction for the other particle must be −ℏ/2. Suppose the two protons move in opposite directions until they are far apart. The total angular momentum of the system remains zero, and if the component of angular momentum along the same direction for each of the two particles is measured, the result is a pair of equal and opposite values. Therefore, after the quantity is measured for one of the protons, it can be predicted for the other proton; the second measurement is unnecessary. As previously noted, measuring a quantity changes the state of the system. Thus, if measuring S_{x} (the xcomponent of angular momentum) for proton 1 produces the value +ℏ/2, the state of proton 1 after measurement corresponds to S_{x} = +ℏ/2, and the state of proton 2 corresponds to S_{x} = −ℏ/2. Any direction, however, can be chosen for measuring the component of angular momentum. Whichever direction is selected, the state of proton 1 after measurement corresponds to a definite component of angular momentum about that direction. Furthermore, since proton 2 must have the opposite value for the same component, it follows that the measurement on proton 1 results in a definite state for proton 2 relative to the chosen direction, notwithstanding the fact that the two particles may be millions of kilometres apart and are not interacting with each other at the time. Einstein and his two collaborators thought that this conclusion was so obviously false that the quantum mechanical theory on which it was based must be incomplete. They concluded that the correct theory would contain some hidden variable feature that would restore the determinism of classical physics.A comparison of how quantum theory and classical theory describe angular momentum for particle pairs illustrates the essential difference between the two outlooks. In both theories, if a system of two particles has a total angular momentum of zero, then the angular momenta of the two particles are equal and opposite. If the components of angular momentum are measured along the same direction, the two values are numerically equal, one positive and the other negative. Thus, if one component is measured, the other can be predicted. The crucial difference between the two theories is that, in classical physics, the system under investigation is assumed to have possessed the quantity being measured beforehand. The measurement does not disturb the system; it merely reveals the preexisting state. It may be noted that, if a particle were actually to possess components of angular momentum prior to measurement, such quantities would constitute hidden variables.Does nature behave as quantum mechanics predicts? The answer comes from measuring the components of angular momenta for the two protons along different directions with an angle θ between them. A measurement on one proton can give only the result +ℏ/2 or −ℏ/2. The experiment consists of measuring correlations between the plus and minus values for pairs of protons with a fixed value of θ, and then repeating the measurements for different values of θ, as in Figure 6—>. The interpretation of the results rests on an important theorem by the British physicist John Stewart Bell. Bell began by assuming the existence of some form of hidden variable with a value that would determine whether the measured angular momentum gives a plus or minus result. He further assumed locality—namely, that measurement on one proton (i.e., the choice of the measurement direction) cannot affect the result of the measurement on the other proton. Both these assumptions agree with classical, commonsense ideas. He then showed quite generally that these two assumptions lead to a certain relationship, now known as Bell's inequality, for the correlation values mentioned above. Experiments have been conducted at several laboratories with photons instead of protons (the analysis is similar), and the results show fairly conclusively that Bell's inequality is violated. That is to say, the observed results agree with those of quantum mechanics and cannot be accounted for by a hidden variable (or deterministic) theory based on the concept of locality. One is forced to conclude that the two protons are a correlated pair and that a measurement on one affects the state of both, no matter how far apart they are. This may strike one as highly peculiar, but such is the way nature appears to be.It may be noted that the effect on the state of proton 2 following a measurement on proton 1 is believed to be instantaneous; the effect happens before a light signal initiated by the measuring event at proton 1 reaches proton 2. Alain Aspect and his coworkers in Paris demonstrated this result in 1982 with an ingenious experiment in which the correlation between the two angular momenta was measured, within a very short time interval, by a highfrequency switching device. The interval was less than the time taken for a light signal to travel from one particle to the other at the two measurement positions. Einstein's special theory of relativity states that no message can travel with a speed greater than that of light. Thus, there is no way that the information concerning the direction of the measurement on the first proton could reach the second proton before the measurement was made on it.Measurement in quantum mechanicsThe way quantum mechanics treats the process of measurement has caused considerable debate. Schrödinger's timedependent wave equation (equation [8]) is an exact recipe for determining the way the wave function varies with time for a given physical system in a given physical environment. According to the Schrödinger equation, the wave function varies in a strictly determinate way. On the other hand, in the axiomatic approach to quantum mechanics described above, a measurement changes the wave function abruptly and discontinuously. Before the measurement is made, the wave function Ψ is a mixture of the ψs as indicated in equation (10). The measurement changes Ψ from a mixture of ψs to a single ψ. This change, brought about by the process of measurement, is termed the collapse or reduction of the wave function. The collapse is a discontinuous change in Ψ; it is also unpredictable, because, starting with the same Ψ represented by the righthand side of equation (10), the end result can be any one of the individual ψs.The Schrödinger equation, which gives a smooth and predictable variation of Ψ, applies between the measurements. The measurement process itself, however, cannot be described by the Schrödinger equation; it is somehow a thing apart. This appears unsatisfactory, inasmuch as a measurement is a physical process and ought to be the subject of the Schrödinger equation just like any other physical process.The difficulty is related to the fact that quantum mechanics applies to microscopic systems containing one (or a few) electrons, protons, or photons. Measurements, however, are made with largescale objects (e.g., detectors, amplifiers, and meters) in the macroscopic world, which obeys the laws of classical physics. Thus, another way of formulating the question of what happens in a measurement is to ask how the microscopic quantum world relates and interacts with the macroscopic classical world. More narrowly, it can be asked how and at what point in the measurement process does the wave function collapse? So far, there are no satisfactory answers to these questions, although there are several schools of thought.One approach stresses the role of a conscious observer in the measurement process and suggests that the wave function collapses when the observer reads the measuring instrument. Bringing the conscious mind into the measurement problem seems to raise more questions than it answers, however.As discussed above, the Copenhagen interpretation of the measurement process is essentially pragmatic. It distinguishes between microscopic quantum systems and macroscopic measuring instruments. The initial object or event—e.g., the passage of an electron, photon, or atom—triggers the classical measuring device into giving a reading; somewhere along the chain of events, the result of the measurement becomes fixed (i.e., the wave function collapses). This does not answer the basic question but says, in effect, not to worry about it. This is probably the view of most practicing physicists.A third school of thought notes that an essential feature of the measuring process is irreversibility. This contrasts with the behaviour of the wave function when it varies according to the Schrödinger equation; in principle, any such variation in the wave function can be reversed by an appropriate experimental arrangement. However, once a classical measuring instrument has given a reading, the process is not reversible. It is possible that the key to the nature of the measurement process lies somewhere here. The Schrödinger equation is known to apply only to relatively simple systems. It is an enormous extrapolation to assume that the same equation applies to the large and complex system of a classical measuring device. It may be that the appropriate equation for such a system has features that produce irreversible effects (e.g., wavefunction collapse) which differ in kind from those for a simple system.One may also mention the socalled manyworlds interpretation, proposed by Hugh Everett III in 1957, which suggests that, when a measurement is made for a system in which the wave function is a mixture of states, the universe branches into a number of noninteracting universes. Each of the possible outcomes of the measurement occurs, but in a different universe. Thus, if S_{x} = ^{1}/_{2} is the result of a SternGerlach measurement on a silver atom (see above: Incompatible observables (quantum mechanics)), there is another universe identical to ours in every way (including clones of people), except that the result of the measurement is S_{x} = −^{1}/_{2}. Although this fanciful model solves some measurement problems, it has few adherents among physicists.Because the various ways of looking at the measurement process lead to the same experimental consequences, trying to distinguish between them on scientific grounds may be fruitless. One or another may be preferred on the grounds of plausibility, elegance, or economy of hypotheses, but these are matters of individual taste. Whether one day a satisfactory quantum theory of measurement will emerge, distinguished from the others by its verifiable predictions, remains an open question.Applications of quantum mechanicsAs has been noted, quantum mechanics has been enormously successful in explaining microscopic phenomena in all branches of physics. The three phenomena described in this section are examples that demonstrate the quintessence of the theory.Decay of the ^{} mesonThe K^{0} meson, discovered in 1953, is produced in highenergy collisions between nuclei and other particles. It has zero electric charge, and its mass is about onehalf the mass of the proton. It is unstable and, once formed, rapidly decays into either 2 or 3 pimesons. The average lifetime of the K^{0} is about 10^{−10} second.In spite of the fact that the K^{0} meson is uncharged, quantum theory predicts the existence of an antiparticle with the same mass, decay products, and average lifetime; the antiparticle is denoted by K^{0}. During the early 1950s, several physicists questioned the justification for postulating the existence of two particles with such similar properties. In 1955, however, Murray GellMann (GellMann, Murray) and Abraham Pais made an interesting prediction about the decay of the K^{0} meson. Their reasoning provides an excellent illustration of the quantum mechanical axiom that the wave function Ψ can be a superposition of states; in this case, there are two states, the K^{0} and K^{0} mesons themselves.A K^{0} meson may be represented formally by writing the wave function as Ψ = K^{0}; similarly Ψ = K^{0} represents a K^{0} meson. From the two states, K^{0} and K^{0}, the following two new states are constructed:From these two equations it follows thatThe reason for defining the two states K_{1} and K_{2} is that, according to quantum theory, when the K^{0} decays, it does not do so as an isolated particle; instead, it combines with its antiparticle to form the states K_{1} and K_{2}. The state K_{1} decays into two pimesons with a very short lifetime (about 10^{−10} second), while K_{2} decays into three pimesons with a longer lifetime (about 10^{−7} second).^{}The physical consequences of these results may be demonstrated in the following experiment. K^{0} particles are produced in a nuclear reaction at the point A (Figure 7—>). They move to the right in the figure and start to decay. At point A, the wave function is Ψ = K^{0}, which, from equation (16), can be expressed as the sum of K_{1} and K_{2}. As the particles move to the right, the K_{1} state begins to decay rapidly. If the particles reach point B in about 10^{−8} second, nearly all the K_{1} component has decayed, although hardly any of the K_{2} component has done so. Thus, at point B, the beam has changed from one of pure K^{0} to one of almost pure K_{2}, which equation (15) shows is an equal mixture of K^{0} and K^{0}. In other words, K^{0} particles appear in the beam simply because K_{1} and K_{2} decay at different rates. At point B, the beam enters a block of absorbing material. Both the K^{0} and K^{0} are absorbed by the nuclei in the block, but the K^{0} are absorbed more strongly. As a result, even though the beam is an equal mixture of K^{0} and K^{0} when it enters the absorber, it is almost pure K^{0} when it exits at point C. The beam thus begins and ends as K^{0}.GellMann and Pais predicted all this, and experiments subsequently verified it. The experimental observations are that the decay products are primarily two pimesons with a short decay time near A, three pimesons with longer decay time near B, and two pimesons again near C. (This account exaggerates the changes in the K_{1} and K_{2} components between A and B and in the K^{0} and K^{0} components between B and C; the argument, however, is unchanged.) The phenomenon of generating the K^{0} and regenerating the K_{1} decay is purely quantum. It rests on the quantum axiom of the superposition of states and has no classical counterpart.Cesium clockThe cesium clock is the most accurate type of clock yet developed. This device makes use of transitions (transition) between the spin states of the cesium nucleus and produces a frequency which is so regular that it has been adopted for establishing the time standard.Like electrons, many atomic nuclei have spin. The spin of these nuclei produces a set of small effects in the spectra, known as hyperfine structure. (The effects are small because, though the angular momentum of a spinning nucleus is of the same magnitude as that of an electron, its magnetic moment, which governs the energies of the atomic levels, is relatively small.) The nucleus of the cesium atom has spin quantum number ^{7}/_{2}. The total angular momentum of the lowest energy states of the cesium atom is obtained by combining the spin angular momentum of the nucleus with that of the single valence electron in the atom. (Only the valence electron contributes to the angular momentum because the angular momenta of all the other electrons total zero. Another simplifying feature is that the ground states have zero orbital momenta, so only spin angular momenta need to be considered.) When nuclear spin is taken into account, the total angular momentum of the atom is characterized by a quantum number, conventionally denoted by F, which for cesium is 4 or 3. These values come from the spin value ^{7}/_{2} for the nucleus and ^{1}/_{2} for the electron. If the nucleus and the electron are visualized as tiny spinning tops, the value F = 4 (^{7}/_{2} + ^{1}/_{2}) corresponds to the tops spinning in the same sense, and F = 3 (^{7}/_{2} − ^{1}/_{2}) corresponds to spins in opposite senses. The energy difference ΔE of the states with the two F values is a precise quantity. If electromagnetic radiation of frequency ν_{0}, whereis applied to a system of cesium atoms, transitions will occur between the two states. An apparatus that can detect the occurrence of transitions thus provides an extremely precise frequency standard. This is the principle of the cesium clock.The apparatus is shown schematically in Figure 8—>. A beam of cesium atoms emerges from an oven at a temperature of about 100° C. The atoms pass through an inhomogeneous magnet A, which deflects the atoms in state F = 4 downward and those in state F = 3 by an equal amount upward. The atoms pass through slit S and continue into a second inhomogeneous magnet B. Magnet B is arranged so that it deflects atoms with an unchanged state in the same direction that magnet A deflected them. The atoms follow the paths indicated by the broken lines in the figure and are lost to the beam. However, if an alternating electromagnetic field of frequency ν_{0} is applied to the beam as it traverses the centre region C, transitions between states will occur. Some atoms in state F = 4 will change to F = 3, and vice versa. For such atoms, the deflections in magnet B are reversed. The atoms follow the whole lines in the diagram and strike a tungsten wire, which gives electric signals in proportion to the number of cesium atoms striking the wire. As the frequency ν of the alternating field is varied, the signal has a sharp maximum for ν = ν_{0}. The length of the apparatus from the oven to the tungsten detector is about one metre.Each atomic state is characterized not only by the quantum number F but also by a second quantum number m_{F}. For F = 4, m_{F} can take integral values from 4 to −4. In the absence of a magnetic field, these states have the same energy. A magnetic field, however, causes a small change in energy proportional to the magnitude of the field and to the m_{F} value. Similarly, a magnetic field changes the energy for the F = 3 states according to the m_{F} value which, in this case, may vary from 3 to −3. The energy changes are indicated in Figure 9—>. In the cesium clock, a weak constant magnetic field is superposed on the alternating electromagnetic field in region C. The theory shows that the alternating field can bring about a transition only between pairs of states with m_{F} values that are the same or that differ by unity. However, as can be seen from the figure, the only transitions occurring at the frequency ν_{0} are those between the two states with m_{F} = 0. The apparatus is so sensitive that it can discriminate easily between such transitions and all the others.If the frequency of the oscillator drifts slightly so that it does not quite equal ν_{0}, the detector output drops. The change in signal strength produces a signal to the oscillator to bring the frequency back to the correct value. This feedback system keeps the oscillator frequency automatically locked to ν_{0}.The cesium clock is exceedingly stable. The frequency of the oscillator remains constant to about one part in 10^{13}. For this reason, the device has been used to redefine the second. This base unit of time in the SI system is defined as equal to 9,192,631,770 cycles of the radiation corresponding to the transition between the levels F = 4, m_{F} = 0 and F = 3, m_{F} = 0 of the ground state of the cesium133 atom. Prior to 1964, the second was defined in terms of the motion of the Earth. The latter, however, is not nearly as stable as the cesium clock. Specifically, the fractional variation of the Earth's rotation period is a few hundred times larger than that of the frequency of the cesium clock.A quantum voltage (volt) standardQuantum theory has been used to establish a voltage standard, and this standard has proven to be extraordinarily accurate and consistent from laboratory to laboratory.If two layers of superconducting material are separated by a thin insulating barrier, a supercurrent (i.e., a current of paired electrons) can pass from one superconductor to the other. This is another example of the tunneling process described earlier. Several effects based on this phenomenon were predicted in 1962 by the British physicist Brian D. Josephson (Josephson, Brian D.). Demonstrated experimentally soon afterwards, they are now referred to as the Josephson effects (Josephson effect).If a DC (directcurrent) voltage V is applied across the two superconductors, the energy of an electron pair changes by an amount of 2eV as it crosses the junction. As a result, the supercurrent oscillates with frequency ν given by the Planck relationship (E = hν). Thus,This oscillatory behaviour of the supercurrent is known as the AC (alternatingcurrent) Josephson effect. Measurement of V and ν permits a direct verification of the Planck relationship. Although the oscillating supercurrent has been detected directly, it is extremely weak. A more sensitive method of investigating equation (19) is to study effects resulting from the interaction of microwave radiation with the supercurrent.Several carefully conducted experiments have verified equation (19) to such a high degree of precision that it has been used to determine the value of 2e/h. This value can in fact be determined more precisely by the AC Josephson effect than by any other method. The result is so reliable that laboratories now employ the AC Josephson effect to set a voltage standard. The numerical relationship between V and ν isIn this way, measuring a frequency, which can be done with great precision, gives the value of the voltage. Before the Josephson method was used, the voltage standard in metrological laboratories devoted to the maintenance of physical units was based on highstability Weston cadmium cells. These cells, however, tend to drift and so caused inconsistencies between standards in different laboratories. The Josephson method has provided a standard giving agreement to within a few parts in 10^{8} for measurements made at different times and in different laboratories.The experiments described in the preceding two sections are only two examples of highprecision measurements in physics. The values of the fundamental constants, such as c, h, e, and m_{e}, are determined from a wide variety of experiments based on quantum phenomena. The results are so consistent that the values of the constants are thought to be known in most cases to better than one part in 10^{6}. Physicists may not know what they are doing when they make a measurement, but they do it extremely well.Gordon Leslie SquiresAdditional ReadingSeveral booklength studies have been written on the historical development of quantum mechanics; especially noteworthy are Olivier Darrigol, From CNumbers to QNumbers: The Classical Analogy in the History of Quantum Theory (1992); and Max Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed. (1989).Careful historical and philosophical studies of the work of many of the early architects of quantum theory may be found in Thomas S. Kuhn, BlackBody Theory and the Quantum Discontinuity, 1894–1912 (1978, reprinted 1987); Bruce R. Wheaton, The Tiger and the Shark: Empirical Roots of WaveParticle Dualism (1983, reissued 1991); Abraham Pais, “Subtle Is the Lord...”: The Science and Life of Albert Einstein (1982), and Niels Bohr's Times: In Physics, Philosophy, and Polity (1991); Arthur Fine, The Shaky Game: Einstein, Realism, and the Quantum Theory, 2nd ed. (1996); Max Dresden, H.A. Kramers: Between Tradition and Revolution (1987); David C. Cassidy, Uncertainty: The Life and Science of Werner Heisenberg (1992); Walter Moore, Schrödinger: Life and Thought (1989); and Dugald Murdoch, Niels Bohr's Philosophy of Physics (1987, reissued 1990). The birth of quantum theory in the period 1900–26, primarily within German university circles, is nicely contextualized by Christa Jungnickel and Russell McCormmach, Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein, 2 vol. (1986, reissued 1990). The transition from nonrelativistic quantum mechanics to renormalized quantum electrodynamics over the period 1926–49 is traced by Silvan S. Schweber, QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga (1994).There are a number of excellent texts on quantum mechanics at the undergraduate and graduate level. The following is a selection, beginning with the more elementary: A.P. French and Edwin F. Taylor, An Introduction to Quantum Physics (1978); Alastair I.M. Rae, Quantum Mechanics, 2nd ed. (1986); Richard L. Liboff, Introductory Quantum Mechanics, 2nd ed. (1992); Eugen Merzbacher, Quantum Mechanics, 2nd ed. (1970); J.J. Sakurai, Modern Quantum Mechanics, rev. ed. (1994); and Anthony Sudbery, Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians (1986), rather mathematical but including useful accounts and summaries of quantum metaphysics. Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, vol. 3, Quantum Mechanics (1965), is a personal and stimulating look at the subject. A good introduction to quantum electrodynamics is Richard P. Feynman, QED: The Strange Theory of Light and Matter (1985).J.C. Polkinghorne, The Quantum World (1984); John Gribbin, In Search of Schrödinger's Cat: Quantum Physics and Reality (1984); Heinz R. Pagels, The Cosmic Code: Quantum Physics as the Language of Nature (1982); and David Z. Albert, Quantum Mechanics and Experience (1992), are all highly readable and instructive books written at a popular level. Bernard d'Espagnat, Conceptual Foundations of Quantum Mechanics, 2nd ed. (1976), is a technical account of the fundamental conceptual problems involved. The proceedings of a conference, New Techniques and Ideas in Quantum Measurement Theory, ed. by Daniel M. Greenberger (1986), contain a wideranging set of papers that deal with both the experimental and theoretical aspects of the measurement problem.Applications are presented by H. Haken and H.C. Wolf, Atomic and Quantum Physics: An Introduction to the Fundamentals of Experiment and Theory, 2nd enlarged ed. (1987; originally published in German, 2nd rev. and enlarged ed., 1983); Emilio Segrè, Nuclei and Particles: An Introduction to Nuclear and Subnuclear Physics, 2nd rev. and enlarged ed. (1977, reissued 1980); Donald H. Perkins, Introduction to High Energy Physics, 3rd ed. (1987); Charles Kittel, Introduction to Solid State Physics, 6th ed. (1986); and Rodney Loudon, The Quantum Theory of Light, 2nd ed. (1983). B.W. Petley, The Fundamental Physical Constants and the Frontier of Measurement (1985), gives a good account of present knowledge of the fundamental constants.Gordon Leslie Squires Ed.* * *
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