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    Quantum Mechanics with Applications
    Quantum Mechanics with Applications
    Quantum Mechanics with Applications
    Ebook522 pages5 hours

    Quantum Mechanics with Applications

    By David B Beard, George B Beard and Kevin B Beard

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    This introductory text emphasizes Feynman's development of path integrals and its application to wave theory for particles. Suitable for undergraduate and graduate students of physics, the well-written, clear, and rigorous text was written by two of the nation's leading authorities on quantum physics. A solid foundation in quantum mechanics and atomic physics is assumed.
    Early chapters provide background in the mathematical treatment and particular properties of ordinary wave motion that also apply to particle motion. The close relation of quantum theory to physical optics is stressed. Subsequent sections emphasize the physical consequences of a wave theory of material properties, and they offer extensive applications in atomic physics, nuclear physics, solid state physics, and diatomic molecules. Four helpful Appendixes supplement the text.
    LanguageEnglish
    PublisherDover Publications
    Release dateAug 20, 2014
    ISBN9780486798219
    Quantum Mechanics with Applications

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      Quantum Mechanics with Applications - David B Beard

      Applications

      I

      BLACK-BODY RADIATION

      Historically, the quantum theory first arose from studies of radiation, and it is useful to introduce the subject along historical lines in which revelant aspects of electromagnetic waves may be reviewed. Quantum mechanics has, with good cause, also been called wave mechanics since it emphasizes and treats wavelike behavior of particles. Hence, it is appropriate to begin our study by an examination of familiar classical radiation theory. By this means we will be able to investigate relevant consequences of wave motion and develop for later advantage the mathematical tools frequently used in classical treatments of electromagnetic waves. As an introduction to waves in three dimensions the reader may wish to review the theory of waves confined to one dimension in a string, presented in Appendix A.

      1.1 DERIVATION OF THE ELECTROMAGNETIC WAVES IN A CAVITY

      Suppose a light wave to be incident on a small hole in the side of a box which is otherwise completely light-tight so that radiation enters or leaves the box only through the small hole. Such a box is called a black body since the hole completely absorbs all light incident on it. The incident light wave will be lost inside the box after repeated reflections from the walls and will come into thermal equilibrium with the walls. Thus a light wave escaping outside the box through the hole will have been in thermal equilibrium with the interior of the box. The radiation in equilibrium within the box consists of electromagnetic waves or, in the parlance of classical radiation theory, oscillations of the ether, the vibrating medium within the box by which the waves are propagated and sustained. In order to estimate the radiation to be expected from the pinhole, it is necessary first to calculate the possible radiation frequencies and the energy of each separate electromagnetic wave. The energy or light intensity emitted from the opening at a particular frequency is proportional to the total energy of all the waves having that frequency contained within the box. Hence, the intensity of the electromagnetic radiation within the box must first be estimated by means of Maxwell's equations for the electric and magnetic field vectors of the waves in the empty interior of the box. Expressed in Gaussian units, Maxwell's equations are

      Taking the curl of both sides of Eq. (1.2), interchanging the order of the space and time derivatives, and substituting Eqs. (1.1, 1.2) in the result, by the the rules of vector analysis we obtain three equations for the three components of the electric field vector, summarized in

      This is a very familiar differential equation in physics, occurring for such diverse motions as a vibrating string, pressure variations in an organ pipe, flow of electricity in a cable, or the waves on a drum head. It solution describes wave motion, and it is called the wave equation.

      In Cartesian coordinates Eq. (1.5) has the particular form or representation

      Equation (1.5a) may be solved for one component Ex by assuming Ex to be written as a product of functions of one variable each. (This is known as assuming the variables separable.) Thus, let Ex = X(x) Y(y)Z(z)T(t). This procedure will be justified if a solution is obtained; not all partial differential equations have solutions which can be obtained in this way. Dividing Eq. (1.5a) for the x component of the field by Ex, one finds in Cartesian coordinates

      The left-hand side is completely independent of the time, while the right-hand side is independent of space. Therefore, both sides of the equation are equal to a constant, independent of both time and space. Let the constant be written as − ω²/c². Then the time-dependent part of (1.6) becomes

      whose solution is

      where D1 and D2 are arbitrary constants for the time-dependent part of Ex.

      Similarly,

      is equal to a constant, − k′ ², and thus the solution to the space-dependent part of (1.6) for Ex is

      where l′ and m′ are additional constants introduced for the Y, Z terms. Identical solutions may be obtained for Ey and EZ.

      Since the tangential components of E must be continuous at the walls, E at the sides of the box cannot be arbitrary but is fixed by the physical properties of the wall material. In a box with infinitely conducting walls, for example, the tangential components of E at the walls must be zero (that is, Ex must be zero in the four walls of the box perpendicular to the y or z axes). Hence, in a rectangular infinitely conducting box of dimensions a, b and d, Y(0) = 0 and therefore B2l′x = 0, and Y(b) = 0 and therefore sin l′b = 0, which means that l′b = πl where l is an integer. Thus, two of Eqs. (1.9) become

      Similar results are obtained for the other two components of E. Let B1l′x, C 1m′x, A1k″y, C1m″y, A1k′″z, and B1l′″z equal unity, which may be done with no loss of generality by readjusting the values of the remaining undetermined constants. Then the total solution for the space-dependent part of E may be written as

      The total solution for E must also satisfy Eq. (1.4), from which we obtain (after dividing out the time-dependent part)

      Since this equation must hold for any values of x, y, and z, the coefficients of each term must each individually be equal to zero, and therefore only the A2k′xK′, B2l″yl″, and C2m″′zm″′ can be nonzero (although their sum must be zero) provided that k′″ = k″ = k′, l″′ = l″ = l′, and m′″ = m″ = m′. Thus the spatial dependence of E is given by

      where

      Equations (1.8,1.10) together describe a sinusoidal wave of angular frequency ω(ω = 2πυ where υ is the number of cycles per second and ω the number of radians per second), and velocity c. Equations (1.10) alone would describe a standing wave such as the particular example illustrated in Figure 1.1 for Cklm = C410,

      By substituting Eqs. (1.7) and (1.10) into Eq. (1.5), one finds that

      For a particular frequency ω, all integral values of k, l, and m are possible (k, l, m) which satisfy Eq. (1.11). For a cube, for example, besides the mode (4, 1, 0) for Ez illustrated in Figure 1.1, there are four other modes for Ez(The modes (4, 0, 1), (1, 0, 4), (0, 4, 1), and (0, 1, 4) are not included since Ez ≡ 0 for these modes.)

      FIGURE 1.1 Graphs of Ez as function of x and y for the mode k = 4, l = 1, and m = 0.

      1.2 NUMBER OF MODES HAVING THE SAME FREQUENCY

      In order to obtain the total energy of radiation of a particular frequency ω contained within and emitted from the box, it is necessary to find the number of different modes of oscillation all having the same frequency ω. The number of modes having a frequency equal to or less than ω is found by summing all integers l, k, and m which make the left-hand side of Eq. (1.11) less than or equal to the right-hand side. For large values of ω such that the wavelength 2πc/ω is orders of magnitude smaller than the box dimensions a, b, and d, the summation may be replaced by a triple integral over k, l, and m, that is, a volume integral over a k, l, m coordinate space. Equation (1.11) is the equation for an ellipsoid in which k, l, and m are the coordinates, and the semi-axes are (aω/πc), (bω/πc), and (dω/πc) respectively (Fig. 1.2). The number of modes having a frequency equal to or less than ω is the number of different combinations of positive integers k, l, and m such that the left-hand side of Eq. (1.11) is less than or equal to ω²/c². This number is given by the volume of the positive octant of the ellipsoid,

      not the total volume of the ellipsoid since k, l, and m must all be positive integers. The three amplitudes Aklm, Bklm, Cklm may have any value provided that Aklmk′ + Bklml′ + Cklmm′ = 0. Therefore there are two independent directions of vibration for each set of k, l, m, which doubles the number of possible modes. Hence, the total number of different modes of oscillation up to and including a frequency ω will be Vω³/3π²c³, where V

      FIGURE 1.2 Illustration of the octant of the ellipsoid expressed in Eq. (1.11) containing all those values of k, l, m which make the left-hand side of Eq. (1.11) less than or equal to the right-hand side. Each mode of oscillation is represented by a single point on the diagram at an integral value of k, l, and m. The dimensions of the graph have been so chosen that each point is equivalent to a unit volume of the ellipsoid.

      is the volume of the cavity. The total possible number of distinct electromagnetic waves or modes of oscillation per unit volume within the cavity, whose frequency is less than or equal to ω, is therefore ω³/3π²c³. The number of vibrational modes per unit volume having a frequency between ω and ω + dω is found by differentiating that expression:

      or, in cycles per second,

      As long as the total energy in the box is conserved, two of the amplitudes Aklm, Bklm, Cklm of each single oscillation are in no way restricted. The validity of this important assumption will be questioned later in this chapter.

      1.3 THE RADIATION ENERGY DENSITY

      The energy in each mode of oscillation is proportional to the square of the amplitude of the particular oscillation. To obtain the energy density inside the box as a function of frequency, Eq. (1.12) must be multiplied by the average or mean energy of each mode of oscillation. To find the average energy of a wave we may note that the electromagnetic vibrations are analogous to mechanical oscillations. By introducing normal coordinates to replace each separate degree of freedom or mode of oscillation one may identify each wave with a corresponding mechanical oscillator having the same frequency. Statistical mechanics is unfortunately outside the scope of this course, but from his classic studies of heat Maxwell was able to show that any particular part of a classical system, such as a single oscillator in a system of oscillators, has a probability of possessing an energy E between E and E + dE proportional to e−E/kTdE, where k is the experimentally determined Boltzmann constant, 1.38 · 10−16 erg/°K, and T is the absolute temperature of the system. The constant of proportionality is determined by requiring that the total probability that this part of the system has any energy at all must be unity. Hence the probability that any given mode of oscillation will have an energy E between E and E + dE, p(E)dE, will be given by

      The average or expected energy of each mode of oscillation is obtained by averaging the energy over the probability of having that particular energy :

      Hence the energy density u(ω) of the modes of oscillation having a frequency between ω and ω + dω is the product of Eqs. (1.12) and (1.14):

      Equation (1.15) was first derived by Rayleigh and Jeans and is called the Rayleigh-Jeans law. Unfortunately, as ω increases, the energy density also increases without limit to infinity, leading to an absurd result for infinite frequencies. As shown in Figure 1.3, Eq. (1.15) does indeed fit experimental observations made for very small frequencies or long wavelengths. Observations at high frequencies, however, were empirically described by Wien to conform to

      where ħ is an experimentally determined constant (1.05 · 10−27 erg-sec).

      Every step of the derivation of Eq. (1.15), the Rayleigh-Jeans law, resulted from what at the time were well-understood and experimentally confirmed physical principles. Yet it did not fit all of the experimental observations. The trouble lay in assuming that all the modes of oscillation have the average

      FIGURE 1.3 Energy density of the modes of oscillation within a box as a function of frequency. Experimental observations are given by the solid line, the low and high frequency approximations by dotted lines.

      energy kT. Actually, somehow the high-frequency modes are frozen out so that their average energy is zero. The solution thus lay in questioning the assumption that the amplitude (and therefore the energy) of each mode of oscillation was unrestricted. If we assume that only discrete values of wave energies are possible, then the average energy of each wave or mechanical oscillator will be different from the constant classical value kT, as we will now show. This is as though one were to assume, for example, that ocean waves enclosed in a breakwater could have heights of the square root of one, two, or three feet but not of one and a half or two and three quarters feet or any other in-between value. If the energy of each wave can be only some integral multiple of ћω. that is,

      then Eq. (1.13) must be replaced by a different expression for the probability that any given mode of oscillation will have a unique energy E. First note that

      where x = ћω/kT and y = e− x. Then in place of Eq. (1.13) we have for the number of discrete states p(E) having an energy E = nћω,

      The average energy of each mode of oscillation becomes

      This result for the summation can be easily seen if we take the derivative with respect to x of both sides of Eq. (1.18).

      is seen to be approximately ћωe−ћω/kT, which does approach zero as ω increases. This high-frequency result for the average energy is readily understood from the first two terms in the summation, 0 + ћωe−ћω/kT, which illustrate that if the first level above n = 0 has energy much greater than kT Multiplying Eq. (1.12), based on Maxwell’s theory of electricity and magnetism, by our newly adopted value for the average energy of each mode of oscillation, Eq. (1.20), we find for the energy density of waves having a frequency ω within a range dω,

      which is the Planck formula for the energy distribution for the modes of oscillation within a black body. Throughout this book a bar through a letter means that the physical constant or quantity the letter represents has been divided by 2π; thus ћ = h/2π where h is a natural constant, Planck’s constant, experimentally observed to be 6.6252 ± 0.0005 × 10−27 erg-sec. If one expresses the frequency in cycles per second v instead of in radians per second ω, the energy of a wave would be written E = nhv instead of as in Eq. (1.17).

      At low frequencies eћω/kT is approximately equal to 1 + ћω/kT, so that the right-hand side of Eq. (1.20) becomes simply the classical result kT. At high frequencies the one in the denominator of Eq. (1.21) may be neglected. Thus we should expect the Rayleigh-Jeans law to be valid at low frequencies and Wien’s law to be valid at high frequencies, which is in fact the case. Intermediate frequencies can be described only by the exact formula, Eq. (1.21).

      It should be emphasized that Planck’s use of E = nћω (Eq. 1.17) was a theoretical attempt to justify an empirical equation (1.21) which fitted the experimental observations remarkably well; there was no theoretical justification for using (1.17) other than that the experimental observations could be interpreted as requiring (1.17) to be satisfied by the individual electromagnetic waves in the box, or by the fictitious mechanical oscillators in normal coordinates corresponding to the electromagnetic waves in the box. It is easiest and most straightforward, however, to conclude for the present, subject to further examination later, that the energy transitions of all waves or oscillators occur in quantum jumps. Nondiscrete energy changes for any given frequency or oscillation are somehow forbidden.

      In deriving our results for the radiation inside a black body we assumed metallic walls. But it would have made no difference if the walls had been made of any other material. Black-body radiation is completely independent of the wall materials, and is observed and predicted to be always the same function of temperature provided only that all radiation incident on a pinhole in one wall is either lost through multiple reflections inside the box or, if incident on the pinhole from within the box, is transmitted freely to the exterior.

      1.4 SIGNIFICANT MATHEMATICAL PROCESSES USED IN THE DERIVATION

      There are several results incidental to the derivation of Eqs. (1.10) which should be emphasized because of their significance and utility to later developments:

      1. If Eq. (1.7) is substituted into Eq. (1.5) there results

      ∇² defines an operation to be carried out on E; the particular operation stated in Eq. (1.22) consists of taking partial derivatives of E stated explicitly in Eq. (1.6) in terms of Cartesian coordinates, but in other cases the operation may be multiplication by a constant or by a function of space or time, for example. ∇² is an operator whose operation on the wave amplitude or wave function E results, in this instance, in a constant (namely ω²/c²) times the wave function.

      2. Since the waves were confined in a box, the boundary conditions (i.e., conditions imposed on the wave function at the walls, which required the tangential component of E to be zero in the case of metallic walls) allowed only those solutions for which ω satisfies Eq. (1.11) where k, l, and m must be integers. This result is a phenomenon common to all confined waves; in vibrating violin strings or organ pipes, for example, it also happens that only those frequencies which satisfy the boundary conditions are permitted.

      3. The values of ω²/c² which satisfy the boundary conditions are called characteristic values or more commonly eigenvalues of the system. The solutions for E of Eq. (1.22) for particular eigenvalues ω²/c² are called characteristic functions or more commonly eigenfunctions. Thus, the operator ∇² operating on the particular eigenfunction Ez illustrated in Figure 1.1 has the eigenvalue 17(π/a)² in the case of the cube of side a. This particular eigenvalue 17(π/a)² has five different nonzero eigenfunctions corresponding to the different k, l, and m consistent with this eigenvalue. An eigenvalue of 16.7(π/a)² with its associated eigenfunctions, for example, is not possible for such a system since no combination of integral k, l, and m can give such an eigenvalue. (If we include the Ex and the Ey solutions, as we must, a total of nine different eigenfunctions are possible for ω²/c² = 17(ω/a)².)

      Problem 1.1 Find the four lowest eigenvalues and their corresponding eigenfunctions of a vibrating violin string 10 cm long if the tension on the string T is 10 newtons and the string has a mass per unit length µ of 0.2 kg/m. The wave equation for transverse waves in a string is

      where S is the perpendicular displacement of the string from equilibrium, x is the distance along the string, and t is the time. (Include the possibility that S may be taken in each of two perpendicular directions.)

      Problem 1.2 Graph the Rayleigh-Jeans law and the Planck black-body distribution as a function of frequency for the surface of the sun, whose temperature is 6000°K. Be careful in labeling the ordinate to give the units and magnitude of what is observed from a black-body surface. On the same graph paper plot the average energy of the electromagnetic waves on the solar surface as a function of their frequency. (Surface emission per unit area is proportional to the velocity of light c times the energy density per unit volume u(ω).)

      Problem 1.3 What are the minimum energies electromagnetic waves of wavelengths 5000 Å and 10 m may have?

      Problem 1.4 What are the eigenvalues and eigenfunctions of the operator (id/dx)² if the eigenfunctions are required to be zero when x = 0 and 2?

      BIBLIOGRAPHY

      Suggested books for further reading:

      Bergmann, P. G. Basic Theories of Physics, Vol. II. Englewood Cliffs, N. J., Prentice-Hall Inc., 1951. This book contains an excellent and detailed treatment of black-body radiation and the early quantum theory, on which much of this chapter is based. The author is particularly gifted in presenting apt examples and models in his lucid presentation.

      Slater, J. C, Modern Physics. New York: McGraw-Hill Book Co., 1955. This book is notable for its exceptionally clear and interesting exposition.

      II

      FOURIER ANALYSIS

      In describing the electric field vector everywhere within the box (the black-body cavity) at a given instant of time through Eq. (1.10), we obtained an alternative way of specifying the condition of the radiation field within the box. It is equally possible to specify the electric field vector as an explicit function of the spatial coordinates x, y, and z, or alternatively to specify the amplitude of each standing wave, as is suggested by Eq. (1.10). Either description serves to represent completely the physical state of the interior of the box. The method of representing a state of radiation in terms of its various modes of oscillation is developed more generally in this chapter.

      The customary representation in terms of spatial coordinates is replaced by alternative representations in terms of modes, which are of great utility in computing and understanding quantum mechanical behavior. It will be shown, by representing a wave in terms of its frequency dependence as well as in terms of its time dependence, that one cannot simultaneously measure the frequency of a wave and the time at which the wave is absorbed in a detector with unrestricted precision. This imprecision or uncertainty is a necessary consequence of all wave motion and is of central importance to the quantum theory. Indeed, Heisenberg originally derived quantum mechanics by postulating this uncertainty as fundamental to all physical measurement. In later chapters it will transpire that a description of any physical system can be made only in terms of pairs of complementary physical variables, both of which cannot be measured simultaneously with infinite precision. In this chapter it will be shown that it is impossible to limit the range of frequency or wave number and the temporal duration or spatial position of a wave; in the following chapter, in addition to discussing the complementary variables of time and frequency or energy, we will take up the physical impossibility of simultaneously measuring position and momentum with infinite precision.

      Sines and cosines are good examples of functions whose product with any other function of the same class gives zero when integrated over all ranges of the variable, unless the two multiplied functions are identical. Such sets of functions are said to be orthogonal. For example, the integrals over a period of the fundamental frequency ω0 of the products sin mω0t sin nω0t, sin mω0t cos nω0t, and cos mω0t cos nω0t, where m and n are integers, are

      and similarly

      where δmn, called a Kronecker delta, is zero for m ≠ n and unity for m = n. Besides sines and cosines, other examples of orthogonal functions that we shall encounter are Bessel functions and Legendre polynomials. The coefficients of such functions may frequently be used to describe a given function of coordinates or some other variable, so that such functions furnish a new and frequently more convenient kind of space in which to describe the given function.

      2.1 FOURIER SERIES

      Suppose that a given oscillation or periodic function of time f(t), with period 2π/ω0, is to be described by a sum or superposition of many waves, specifically an infinite sine and cosine series, called a Fourier series:

      where the n’s are integers, ω0 is a given fundamental frequency, and the A’s and B’s are the amplitudes of each component frequency. Multiplying both sides of Eq. (2.1) by sin mω0t and integrating over the fundamental period, we find that

      The sines and cosines are orthogonal functions, and all the terms of the infinite series in n on the right-hand side of Eq. (2.2) give zero except for the one term in the sine series for which n = m, which gives

      Similarly by multiplying (2.1) by cos mω0t,

      As in the preceding, our function may be given as an explicit function of time, f(t); alternatively, however, we may compute the A’s and B’s from a knowledge of f(t) and use them to describe our function :

      where the quantities in brackets are functions only of frequency nω0, since the time will have been integrated out. In this way f(t) may be described, for example, as being made up of oscillations having frequencies nω0, each of whose relative amplitudes An and Bn represent the quantity just as completely as f(t) does. There is one essential difference, however, in that while all values of t are possible in the time representation, only certain values of frequency, integral multiples of ω0, are possible in the frequency representation. If f(t) is not periodic this difference disappears, as will be shown in the following section.

      As an example, let us consider a 60-cycle alternating current of unit amplitude f(t) = sin 2π60t. If we attempt to find a series to describe it using the basic frequency ω0 = 2π60, we immediately find the trivial result A1 = 1. All other coefficients are zero. For graphs of alternative representations of 60-cycle ac current see Figure 2.1. Note that in this example the Fourier series consists of only one term, because we implicitly assumed that the wave started at t = − ∞ and continued to t = + ∞. Actually, every time an electric switch is turned on or off, higher frequencies called transients are introduced (see Problem 2.2). These are frequently picked up by household radios as sharp audible clicks.

      For a second example, suppose that at

      FIGURE 2.1 Alternative representations of 60-cycle ac; the second graph illustrates the fact that only one frequency is represented.

      and then the function is repeated for larger and smaller integral multiples of t0. Obviously, a convenient fundamental frequency will be π/t0:

      Similarly,

      and hence

      Alternate representations of this system are presented in Figure 2.2.

      Problem 2.1 What are the amplitudes of the various wavelengths present in a vibrating violin string of length l plucked initially to form an isoceles triangle?

      FIGURE 2.2 Alternative representations of a periodic step function pulse.

      Problem 2.2 What are the amplitudes of the various frequencies present in half-wave rectified 60-cycle ac, that is, 0 < t < 1/120, f(t) = sin 2π60t, and 1/120 < t < 1/60, f(t) = 0, and then the function repeats?

      Problem 2.3 What are the amplitudes of the various frequencies present when 60-cycle ac is repeatedly turned on for 1/10 second and then turned off for 4/10 second? Do not bother to carry out the integration.

      Problem 2.4 Find the two Fourier series for f(x) = x and |x| for − l < x < l (f(x) repeats for other values of x); then compare the series for 0 < x < l.

      Problem 2.5 Find the Fourier series for f(x) = cos µx for 0 < µ < 1, − π < x < π (f(x) repeats for other values of x).

      Problem 2.6 Find the two Fourier series for f(x) = x² and x³, over − l < x < l (f(x) repeats for other values of x).

      2.2 FOURIER TRANSFORMS

      The preceding discussion was devoted to alternative representations of functions varying periodically with time. The periodicity resulted in a discrete frequency representation, called a Fourier

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