Quantum mechanics(英文)

Basic Concepts And Methods

Bohr’s theory, which assumed that electrons moved in circular orbits, was extended by the German physicist Arnold Sommerfeld 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 started 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, Max Born, and Pascual Jordan, and (2) wave mechanics, put forward by Erwin Schrödinger. 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 applied his equation to the hydrogen atom, for which the potential function, given by classical electrostatics, is proportional to −e²/r, where −e is the charge on the electron. The nucleus (a protonof 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. 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; Square root of√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, Ψ², 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 Ψ²at that point. Almost immediately Born proposed what is now the accepted interpretation—namely, that Ψ² gives the probability of finding the electron at x, y, z. The distinction between the two interpretations is important. If Ψ² 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 well-defined 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

In 1928 the English physicist Paul A.M. Dirac produced a wave equation for the electron that combined relativity with quantum 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 (p²/2m_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. Goudsmitand George E. Uhlenbeck, 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 electron-spin 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. 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 atoms

Because 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 large-scale 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 Ψ² 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), it can be shown that particles with half-integral spin (¹/₂, ³/₂, etc.) have antisymmetric wave functions. They are called fermions after the Italian-born 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 after the Indian mathematician and physicist Satyendra Nath Bose, 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, first proposed in 1925 by the Austrian physicist Wolfgang Pauli. 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, an insulator, or a semiconductor (see atom; matter).

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, the Russian physicist Vladimir Fock, and others in the 1920s and 1930s 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 electron-electron correlations and the magnetic forces.

Time-dependent Schrödinger equation

At the same time that Schrödinger proposed his time-independent equation to describe the stationary states, he also proposed a time-dependent equation to describe how a system changes from one state to another. By replacing the energy E in Schrödinger’s equation with a time-derivative operator, he generalized his wave equation to determine the time variation of the wave function as well as its spatial variation. The time-dependent Schrödinger equation reads

The 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 form

Tunneling

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 one-dimensional 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 < V₀ (the maximum height of the potential barrier), the particle remains in the well forever; if E > V₀, 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 V₀, although the probability of escape is small unless E is close to V₀. 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 in which a nucleus emits an alpha particle (a helium nucleus). According to the quantum explanation given independently by George Gamow and by Ronald W. Gurney and Edward Condon 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 alpha-particle emitters, the value of E varies from about 2 to 8 million electron volts, or MeV (1 MeV = 10⁶ electron volts). Thus, the value of E varies only by a factor of 4, whereas the range of τ is from about 10¹¹ years down to about 10⁻⁶second, a factor of 10²⁴. 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 approach

Although the two Schrödinger equations form an important part of quantum mechanics, it is possible to present the subject in a more general way. Dirac gave an elegant exposition of an axiomaticapproach 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 ψ₁, ψ₂, . . ., ψ_N, and corresponding measurement values a₁, a₂, . . ., 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 as

For a given Ψ, the quantities c₁, c₂, 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 a-value—i.e., a₁, a₂, or a₃, etc. No other value is possible. Second, before the measurement is made, the probability of obtaining the value a₁ is c₁², and that of obtaining the value a₂ is c₂², and so on. If the value obtained is, say, a₅, the theory asserts that after the measurement the state of the system is no longer the original Ψ but has changed to ψ₅, the state corresponding to a₅.

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 ψ₁, sometimes to ψ₂, 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 = ψ₃. Then c₃ = 1 and all the other cs are zero. This means that, before the measurement is made, the probability of obtaining the value a₃ 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 ψ₃, 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 time-dependent 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 ψ₁, ψ₂, . . ., 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 observables

The 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 Stern-Gerlach 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_squantum 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 = −¹/₂are deflected downward and emerge as beam 2. If the direction of the magnetic field is the x-axis, the apparatus measures S_x, which is the x-component 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 x-axis with opposite senses of rotation.

The y-component 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 y-axis. 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 x-axis is a superposition of the two states with spin about the y-axis. 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 y-axis 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 y-components 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 x-component of angular momentum is known but the y-component 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 y-component of angular momentum is known but the x-component 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 principle

The 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 half-integer, 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 x-component 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 θ₀, given by sin θ₀ = 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, θ₀ increases—i.e., the diffracted light is more spread out. Thus, θ₀ measures the spread of the beam.

The experiment can be repeated with a stream of electrons instead of a beam of light. According to de 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 y-direction in Figure 4A), their (linear) momentum is p = m_eu. Consider p_x, the component of momentum in the x-direction. After the electrons have passed through the aperture, the spread in their directions results in an uncertainty in p_xby an amount

where λ is the wavelength of the electrons and, according to the de 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 x-position 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 by

This is the well-known 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 10⁶ metre per second.

The above reasoning leading to the uncertainty principle is based on the wave-particle duality of the electron. When Heisenberg first propounded the principle in 1927 his reasoning was based, however, on the wave-particle 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 gamma-ray photon bounces off the electron, giving the Compton effect (see above Early developments: Scattering of X-rays). 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 by

This 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.

Quantum electrodynamics

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 aboveElectron spin and antiparticles). 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 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_einvolves 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 are

Since a itself represents a small correction term, the magnetic dipole moment of the electron is measured with an accuracy of about one part in 10¹¹. 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 10¹⁰.