Introduction to the Quantum Theory: Third Edition

By David Park

More than a chance to gain new insights into physics, this book offers students the opportunity to look at what they already know about the subject in an improved way. Geared toward upper-level undergraduates and graduate students, this self-contained first course in quantum mechanics consists of two parts: the first covers basic theory, and the second part presents selected applications. Numerous problems of varying difficulty examine not only the steps of the proofs but also related ideas.
Starting with an introduction that ventures beyond classical physics, the first part examines the physical content of the wave function; general principles; physics in one dimension; hermitian operators, symmetry, and angular momentum; and systems in two and three dimensions. Additional topics include approximate methods of calculation; the theory of scattering; spin and isospin; questions of physical meaning; electromagnetic radiation; systems containing identical particles; and classical dynamics and Feynman's construction.
Focusing on applications, the second part explores the theory of alpha decay; electrons in a periodic lattice; the hydrogen spectrum; the helium atom; interatomic forces; the neutron-proton interaction; and the quark model of baryons.

• Language: English
• Publisher: Dover Publications
• Release date: Nov 20, 2012
• ISBN: 9780486137766

David Park

David Park has written nine previous books including The Big Snow, Swallowing the Sun, The Truth Commissioner, The Light of Amsterdam, which was shortlisted for the 2014 International IMPAC Prize, and, most recently, The Poets' Wives, which was selected as Belfast's Choice for One City One Book 2014. He has won the Authors' Club First Novel Award, the Bass Ireland Arts Award for Literature, the Ewart-Biggs Memorial Prize, the American Ireland Fund Literary Award and the University of Ulster's McCrea Literary Award, three times. He has received a Major Individual Artist Award from the Arts Council of Northern Ireland and been shortlisted for the Irish Novel of the Year Award three times. In 2014 he was longlisted for the Sunday Times EFG Short Story Award. He lives in County Down, Northern Ireland.

Book preview

Park

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TO THE STUDENT

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I should try to make clear at the outset what is entailed in learning this subject. It is not merely learning some new physics. It is much more learning to look at old physics in a new and better way—it sounds new because the ordinary vocabulary of science and everyday life is not well adapted to the uses of quantum theory, and it is better because the new theory is in agreement with experiment over a far wider range of phenomena than the old one. It gets its results by reasoning which sometimes cannot be explained in classical terms, but this does not mean that it renders the classical theory wrong or obsolete. It concerns itself largely with questions to which classical thinking cannot be applied, but the latter has, it seems, a permanent place in physics because it deals in a natural way with phenomena that ordinarily present themselves to the human senses. There exists, therefore, a very special relationship between quantum mechanics and newtonian physics, and it will be noticed that all the final results of calculations in the new theory are expressed in the language of the old. It is thus in the calculation of physical results that one makes the closest contact with what is familiar, and that is why this book offers a variety of worked-out examples of the theory’s application in Part II. Most readers will probably learn more from this part than from the first.

A word of advice. Do not ignore the problems in reading the text; they are in many cases designed to call attention to matters you may not have noticed. Most of the book is designed to be read with a pencil and paper, and you are not supposed to be able to master it otherwise. Steps are omitted from some of the proofs and some of the calculations are curtailed, not because I am lazy, but in order to encourage you to participate. Such gaps are very narrow in the beginning and become wider toward the end. None is very wide; if it appears so, you are probably off the track.

A final word. This is a physical theory, constructed out of concrete physical ideas. It is expressed mathematically because that is the way to express this kind of thinking. Everything possible has been done to make this book easy mathematically; things I have been unable to simplify I have omitted. If, therefore, you have trouble at any point, try to see whether the difficulty is in physics or in mathematics. That usually works.

David Park

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PART

I

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THEORY

The exact sciences also start from the assumption that in the end it will always be possible to understand nature, even in every new field of experience, but that we may make no a priori assumptions about the meaning of the word understand.

Werner Heisenberg

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CHAPTER

1

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BEYOND CLASSICAL PHYSICS

The theory of dynamics that descended from Newton, Lagrange, and their followers is a mathematical procedure for analyzing changes in those observable properties of things that can be described in terms of variables that situate them in newtonian space and time. Such a theory is called classical. Classical dynamicsis conceptually simple, even beautiful; it is self-consistent, and it has been tested thousands of times against experience. But nature, so much more inventive than we are, provides some situations in which it gives wrong answers. An example is the group of phenomena called relativistic, some of which involve objects moving at great speeds with respect to the calculator’s system of reference. It turns out that the classical mode of analysis can take care of relativistic calculations, almost without change. The ancients were wrong in some physical hypotheses but not in their analytical approach. It is enough to touch up the equations and then proceed pretty much as before.

The classical theory is also incomplete in that it does not work for phenomena on the atomic scale. The evidence is all around us. If we believe that an atom consists of a nucleus orbited by electrons, the rigidity of matter is incomprehensible. We cannot understand why our bodies hold together or why we do not fall through the floor. No classical analysis of the inner dynamics of atoms comes out right. There was one decade, beginning in 1913, when people thought that perhaps classical dynamics, with the addition of the quantization rules proposed by Niels Bohr and developed by others, might suffice. Investigations along this line were at first very rewarding. The spectrum of hydrogen was accurately explained and many qualitative features of the periodic table could be understood. Disillusion began in 1923 when Max Born and Werner Heisenberg managed to calculate the spectrum of helium, the next most complicated element after hydrogen, and found that their result was not only very hard to get, it was grossly wrong when they got it. There was the added difficulty that though the quantum rules had a certain limited success, it was impossible to guess why they were true. They were assumed in Bohr’s theory but they looked as if they should have emerged from some more basic line of thought. We will find that they do.

The rest of this chapter will recapitulate some of the ideas that define the limits of classical physics and then come together in the quantum theory. It will mention relevant physical facts, establish some notations, and do a few simple calculations. In the last sections of the chapter the limits will be transcended in a single new idea, that what people had thought to be the particles of matter—electrons, protons, neutrons, and even whole atoms—can sometimes seem to act like waves.

1.1  WAVE NATURE OF LIGHT

Our definite knowledge about the wave nature of light begins with Dr. Thomas Young, who in 1802 published a paper on optical interference (a term he introduced) in which the phenomena are explained in terms that would be perfectly acceptable today. Earlier workers had imagined light as a beam of corpuscles (Newton), a sequence of puffs in a luminiferous medium (Huygens), or a wave in which a definite frequency corresponds to a pure color (Euler), but Young was the first to see that by assuming that light is a wave he could explain interference phenomena. To do this he formulated the principle of superposition: If two light waves arrive at the same point, their effect is exactly equivalent to that of a third wave whose displacement at that point is equal to the (algebraic) sum of the displacements of the component waves. The superposition of two light beams of the same wavelength can be carried out in two ways: coherently, so that there is a definite and constant phase relation between the two beams, or incoherently, so that there is not. The characteristic interference phenomena of light are found only when the superposition is coherent, and this is ordinarily achieved by deriving both beams from a common source. Otherwise, the addition of two beams merely produces an intensity that is the sum of the separate intensities. (Though there is a definite phase relation between the two beams at every moment, it is not constant, and an average washes out the interference patterns.)

The typical arrangement for demonstrating the coherent superposition of two light beams is illustrated in Fig. 1.1, which shows the beams coming from a common source, being divided by a diaphragm, and recombining at a screen, which may be a photographic plate. The intensity pattern at the screen is explained by supposing that the wave amplitudes add. The possibility of performing such an addition is evident if the waves are mechanical in nature (Young thought they probably were) so that the displacement is a mechanical displacement. It is not quite so evident if one does not know what kind of waves they are, but that the coherent superposition of two light beams forms a third is conclusively shown by Young’s experiments.

FIGURE 1.1

Interference produced by a double slit.

Later, in 1817, Young proposed to account for polarization phenomena by supposing the waves to be transverse; he used the now common analogy with waves on a rope. But this does not answer all the questions one can ask about light. First, there is the nature of the medium in which the undulations occur, the so-called ether. Popular writings on relativity often assert that the ether has finally been done away with. This is not the case,¹ but at least we know that there is nothing resembling a fluid or solid medium with mechanical waves running through it. Further, a knowledge of the properties of light waves traveling in free space does not tell us much about light. It does not tell us how light is emitted or absorbed, for example, or what will happen when the waves move from air into glass. (In this case the refraction can be explained à la Huygens but not the partial reflection that occurs.) We need an equation or a set of equations which will describe all these things and of which the monochromatic plane waves of a simple diffraction experiment represent particular solutions.

What is known about these aspects of light can be summed up in the idea of an electromagnetic wave. In 1861, James Clerk Maxwell succeeded in characterizing the electric and magnetic fields, their relation to charges and currents, and the relations between them in eight partial differential equations.² As long as the fields are static or nearly so, the general physical content of these equations can be understood from a knowledge of Coulomb’s law and the right-hand rule for magnetic fields. But when time Variation becomes important, entirely new phenomena begin to emerge, and Maxwell found that it is possible to have wave-like configurations of electric and magnetic fields that are entirely disconnected from their sources and propagate through space at the well-known speed c. Later it was proved that visible light and its prolongations at each end of the spectrum are electromagnetic waves of this kind.

With Maxwell’s equations one can explain phenomena relating to the propagation, dispersion, reflection, refraction, and interference of electromagnetic waves, essentially all of which are found to correspond with known properties of light. Thus they provide the rigorous mathematical foundation of Young’s hypotheses.

1.2  PARTICLE NATURE OF LIGHT

The idea of light as a stream of particles originated with the ancients but in modern times stems from Newton. His espousal of it at first seems paradoxical, especially in view of his own experiments on interference, diffraction, and polarization, but he argued that only the corpuscular theory of light can explain sharp shadows. The wave theory would lead one to expect that light will travel as easily as sound around the corner of a building. (How can Newton’s argument be countered today?) Deeper than this, however, Newton was a thoroughgoing atomist, and it went against his instinct of the uniformity of nature to suppose that matter is one thing and light something entirely different. In this, as we shall see, he was perfectly right.

The first contemporary suggestion of the discrete nature of light is found in the work of Max Planck and Albert Einstein around 1900. Although Planck did not really believe that the energy levels of an oscillator are discrete, he found that he could explain blackbody radiation if he counted them as if they were discrete. It was Einstein who expressed this mathematical procedure as a physical hypothesis, showed that it implied the existence of light quanta, and gave some of their physical properties. The most famous of these is his formula for the energy of a light quantum,³

where v is the frequency of the associated light wave and h, Planck’s constant, is

There was, of course, no reason to think that the radiant energy would stay together once it was emitted; rather, one would expect it to spread out like the circular ripple caused by a stone falling into water. Quanta of this kind would carry energy but no momentum. But Einstein (1905), considering the radiation in a cavity as a gas of particles, showed that if each particle has an energy hv and the corresponding momentum hv/c, then it is possible to understand Planck’s formula for the entropy of blackbody radiation as though it were a formula from the kinetic theory of gases.⁴

Einstein was at once able to explain several puzzling facts about radiation, notably that a change in the intensity of the light absorbed in the photoelectric process does not affect the energy of the photoelectrons but produces a proportionate change in their number. Introducing the work function φ, where eφ represents the least energy required to remove an electron from the metal surface, Einstein predicted that the kinetic energy of the fastest electrons emitted would vary with v according to

This relation proved exceptionally difficult to verify, but at length, in 1915, Millikan was able to show that it is accurately fulfilled.⁵ Figure 1.2 shows the agreement between theory and experiment. By this time, Einstein’s theory had received an independent test in its application to Bohr’s theory of atoms, and Einstein received a Nobel prize for it in 1921.

If light is considered to be made up of particles all traveling with the fundamental velocity c, then these particles should possess momentum as well as energy. The relation between energy E and momentum p in relativistic mechanics is

for a free particle whose rest mass is m. [Derived below as Eq. (16.6); we can here set V = 0.] If a photon is a particle whose rest mass is zero, this gives

Unfortunately, this relation cannot be used very simply to test the light-quantum hypothesis, since the densities of energy and momentum in a radiation beam are predicted by Maxwell’s electromagnetic theory to be in the same ratio. But experiments measuring the recoil of a single microscopic particle under the action of a single quantum are decisive. It was shown by Compton in 1922 that the ordinary conservation laws of energy and momentum require that if a photon is scattered through an angle ϑ by collision with a free electron of mass ma relation that has been verified in extensive x-ray experiments by Compton and others. The explicit appearance of h in this formula, unlike (1.4), shows that the phenomenon cannot be explained by Maxwell’s theory.

FIGURE 1.2

Millikan’s data for the potential necessary to stop the fastest photoelectrons produced by monochromatic light plotted against the frequency of the light. [From Millikan (1916). Reproduced by permission.]

Problem 1.1. Use the relativistic formulas for momentum and energy to derive the above formula governing Compton scattering. Why are the experiments performed using x-rays?

Problem 1.2. Show that it is impossible for a photon striking a free electron to be absorbed and not scattered.

Even more direct, though far more difficult, is the verification that when an atom emits a quantum of energy hv in one direction, it recoils with a momentum hv/c in the opposite direction. Though the recoil velocity of the atom is very much less than its thermal velocity, a delicate experiment by R. Frisch (1933) showed that sodium atoms in a narrow beam are deviated from their paths by about the expected amount when they are caused to radiate.

Problem 1.3. What is the expected recoil velocity of a sodium atom which at rest emits a quantum of its D radiation (λ = 5890 Å)? Design a beam experiment in which one might hope to detect this recoil.

Problem 1.4. In the following problems and generally throughout this book, the masses of particles will be given in MeV. Show that

Problem 1.5. If a 40-keV x-ray hits a stationary electron (me = 0.511 MeV), what is the maximum energy it can transfer to it?

Problem 1.6. According to Wien’s displacement law (1894), the wavelength λm at which blackbody radiation at temperature T has its maximum intensity is given roughly by λmT ≈ 3mm⋅K. Assuming that the quantum energy at this temperature is of the order of kT where k is Boltzmann’s constant, estimate the value of Planck’s constant.

Problem 1.7. In an experiment to find the value of h, light at wavelengths 218 and 431 nm was shone on a clean sodium surface. The potentials that stopped the fastest photoelectrons were 5.69 and 0.59 V, respectively. What values of h and φ are deduced?

Maxwell’s theory explains certain experiments well, but the same is true of the photon concept. Do we then have to conclude that light is both a wave and a particle? That would be too bad, since it would not make any sense. But think for a minute: What is a physical experiment and what does it actually tell us? We prepare a piece of apparatus, we turn it on, something happens, a measurement is made. The experiment teils us what happened, and a good theory tells what will happen before any one does the experiment. The crucial point is this: An experiment or a theory tells what happens, not what is.

Experiments in the old days usually involved things one could touch and see, and it is from experience of touching and seeing that we get our firm ideas of what is; therefore, if one deduced what was from what happened, things rarely went wrong. You can see light but you cannot see a light wave; you cannot touch a photon, and it is obvious from the experiments just described that a light wave is not just a scaled-down version of waves in a duck pond, nor is a quantum particle a scaled-down version of a baseball. In fact, the nature of light cannot be represented as a scaled-down version of anything we are familiar with. It has its own nature, whatever that may be, and what we know about it is what happens in different experiments. Luckily for us, the situation is not entirely strange. What happens in some experiments can be modeled by a wave; in others, by a particle. Often, as when we have to deal with the electron cloud surrounding an atom, neither of those models is very helpful in understanding what happens and we trust the model whose root is in mathematics. It may not satisfy all our desires, but at least it works.

1.3  BEHAVIOR OF ELECTRONS

That electric charge comes in discrete quanta was first suggested⁶ by experiments on electrolysis and gaseous ions, and it was presently shown, by J. J. Thomson and others, that if one assumes that the quanta of charge are particles with a definite mass obeying Newton’s laws of motion, several phenomena of cathode rays can be explained.

Problem 1.8. Electrons are accelerated by falling through a potential V and then deflected by passing between a pair of parallel plates of given size across which there is a transverse electric field of E V/m. Through what angle is the beam deflected? (Neglect edge effects in comparison with the dimensions of the plates.)

Problem 1.9. Electrons are shot into a uniform magnetic field of 10–3 T, entering with a velocity perpendicular to the lines of force. What must be their energy in eV if they are to travel in a circle of radius 10 cm? What must it be if the electrons are to spiral upward at an angle of π/4 to the lines of force?

For electrons confined to a small region of space, as in an atom, the situation was more obscure, since Newton’s laws do not explain the stability of an atomic system. That this is so is seen in Rutherford’s planetary model of hydrogen, since an electron may revolve around a proton in an elliptical orbit that has any size and binding energy whatever, depending only upon the initial conditions. This fact can be seen in a formal way if we note that the only constant parameters which enter the theory are the masses and charges of the component parts and that no constant of the dimension of a length can be formed from these numbers. (Try it.) Thus newtonian mechanics unaided cannot account for the well-known st able structure, fairly rigid and a few times 10–8 cm in size, that all atoms exhibit in the structure of liquids and crystals and in various dynamical processes. The remark that with the introduction of h cm using the m and e of the electron led Niels Bohr (1913) to the hypothesis that h enters the theory as a mechanical quantity.⁷

Bohr’s famous assumptions which led him to explain the spectrum of atomic hydrogen are, first, that the atom obeys the laws of newtonian mechanics except that

The electron’s orbital angular momentum is restricted to be an integral multiple of h/2π.

But the electromagnetic radiation from the atom does not follow Maxwell’s theory; rather, it takes place one photon at a time, so that

A quantum of radiation is emitted when the electron changes from one orbit to another of lower energy, the quantum’s energy hv being equal to the energy released.

Further

An electron in its orbit of lowest energy does not radiate.

(The fact that most of the ideas expressed and implied here had been put forth by others at various earlier dates does not in the least detract from Bohr’s merit in being the first to combine them into a definite theory.) We shall speak here only of circular orbits; elliptical orbits require a further hypothesis.

Let a be the radius of the orbit and Ω the electron’s angular velocity. Then Bohr supposes that all possible orbits must satisfy

and

where

and we shall rarely refer again to plain h. From these relations one finds that the radii of the possible orbits are⁸

where the quantity

is known as the radius of the first Bohr orbit. This accounts for the size of hydrogen atoms. The electron’s total energy in its nth orbit is

where Ry, standing for rydbergs, is given by

In terms of this quantity, the angular velocity of the electron in its orbit is

Problem 1.10. Derive Eqs. (1.8), (1.10), and (1.12).

The possible spectral frequencies will be given by

where ω is 2π times the usual frequency. This coincides with Balmer’s famous empirical formula (1885). As already mentioned, there will, of course, be elliptical orbits as well; but these are found to introduce no new energies, and so they introduce no further spectral lines.

Correspondence Principle

in the energy formula arising from both the angular-momentum hypothesis and the light-quantum hypothesis, and there is also the basic fact that the emitted frequencies are associated with differences in the states of the atom rather than with the states themselves. Classical physics predicts that the fundamental frequency of the radiation emitted by an electron in a certain orbit will be that of the electron in that orbit and that there will be higher harmonics as well,⁹

A look at (1.8) now shows that a remarkable situation exists. If n is chosen large enough, the radius of the atom will reach a size (say, 1 m) where one would expect the classical theory to be valid. We therefore have in this ränge two different theories applying to the same phenomenon, and we must see whether their results agree. The lowest frequencies emitted will correspond to transitions between states for which nf is not much less than ni. If ni – nf, which we shall call Δn, is much less than ni, then the parentheses in (1.13) contain

Expanding the fraction in powers of Δn/n gives

where we have neglected higher-order terms, unimportant when ni→ ∞. This gives

by (1.12). Comparing this with (1.14), we see that there is an exact correspondence between the frequencies predicted by the classical and the quantum theories in the limiting case in which the two theories overlap. Bohr pointed out how important it is that classical and quantum theories should agree in the domain of high quantum numbers; he called this the correspondence principle, and it is one of the ways we know that quantum mechanics is a good theory. Of special interest now are situations in which they do not agree; these can occur when the classical equations have chaotic solutions, solutions in which a very small difference in initial conditions makes a large difference in the resulting orbit. A hydrogen atom in a very high quantum state (called a Rydberg state) and immersed in a very strong magnetic field is an example of such a system; experiment agrees with a quantum-mechanical calculation and not with one based on the correspondence principle.

The correspondence principle governs the intensities as well as the frequencies of spectral lines. If the classical theory does not yield a certain frequency of radiation at all, in quantum theory the corresponding transition should not take place. The selection rules to which this argument gives rise are found to hold for small quantum numbers also and will be discussed in Sec. 11.3.

Problem 1.11. Discuss the correspondence limit of Eq. (1.4).

Problem 1.12. Show, using Taylor’s series, that the frequency of light emitted in a transition out of a state whose energy is Wn is given by

where Δn is the change in the quantum number n. Compare this with (1.15) and state the conditions under which the second term can be neglected in discussing Bohr’s theory.

Problem 1.13. In the Bohr model, how fast is an electron in the nth circular orbit of hydrogen actually moving?

Problem 1.14. In an atom, the innermost electrons are not much affected by the outer ones. In the Bohr theory, how fast do the inner electrons of ⁹²U travel? Comment on the limitations of the theory as given in the text.

Problem 1.15. Starting from Hamilton’s equations, make a relativistic version of Bohr’s theory for circular orbits and apply it to ⁹²U. The appropriate hamiltonian is

Calculate the energy difference in eV between the states n = 2 and n and assuming that force = dp/dt.]

Problem 1.16. Compare the frequency of light emitted from hydrogen in the transition from n = 10 to n = 9 with the electron’s orbital frequency in the state n = 10.

Spatial Quantization

There is another important aspect of Bohr’s theory to compare later with the quantum-mechanical result. Suppose that an atom is put into a weak, uniform magnetic field. It is shown in classical electrodynamics (Larmor’s theorem) that the atom will continue to function undisturbed, except that it will process as a whole around the direction of the field, , which according to Bohr’s postulates will be equal to l , with l a positive integer. The extension of Bohr’s original theory to other types of periodic motion such as this precession led to the conclusion that not only the magnitude of the angular-momentum vector but also the value of its component in the direction of the field (call it z) take on discrete values:

where m is an integer. Figure 1.4 shows the evenly spaced z components corresponding to l=3; clearly, m can go in integer steps from − l to + l, a total of 2l + 1 different values. This phenomenon is known as spatial quantization.

FIGURE 1.3

Larmor Precession of Bohr’s hydrogen atom in a magnetic field.

FIGURE 1.4

Spatial quantization of the direction of an angular-momentum vector l = 3 in Bohr’s theory. (The arrows were imagined as precessing about the vertical axis.)

will exhibit a magnetic moment proportional to it,

with γ, the gyromagnetic ratio, given by

where me is the electron’s mass. (The same is true of the spin motion also, except that its gyromagnetic ratio is twice as great. This leads to complications we can avoid here.) Since the energy of a magnetic dipole in a magnetic field of induction B is − B ⋅ M, or in this case

we see that an atom in this situation has a potential energy that changes in steps as its orientation in space changes. This produces the Splitting of spectral lines known as the Zeeman effect. It is not necessary here to go into the proofs of formulas such as (1.17) and (1.18), since we shall derive them from quantum mechanics as we go along.

z. One would expect to find 2l + 1 such components.

In both the experiments just described, there is poor agreement between Bohr’s theory and the empirical facts. Generally, there are more lines than the theory predicts; this is due to the effect of electron spin, and it can be taken into account if one also assumes that the quantum number l can go down to zero. With atomic hydrogen, for example, the Stern-Gerlach experiment reveals only two traces, and this is explainable if one assumes that the electron has a spin angular momentum whose z and which can therefore take on two different orientations, while it has no orbital moment at all. Setting l equal to zero is difficult to explain intuitively (what is the nature of an orbit with no angular momentum?); this is a paradox that quantum mechanics will resolve.

The most peculiar thing about space quantization is this: It is established at once, and it is independent of the strength of the field. In fact, it exists in arbitrarily weak fields and so presumably in the limit of no field. But in the limit, where is the z is every . The paradox here is conceptual rather than experimental, since one cannot detect the space quantization without applying a field, but it is perhaps the most extreme Illustration of how far one can be led by Bohr’s postulate of the quantization of periodic motions, a postulate which is purely ad hoc, with nothing in ordinary experience or intuition to justify it.

As to the ultimate results of Bohr’s theory, its failures are as striking as its successes. It ended up by explaining very well the spectrum of hydrogen (and of alkali elements like sodium and potassium which have hydrogen-like spectra), but it provided no understanding of the structure or spectrum of even the next most complicated element, helium. It established the nomenclature and basic concepts for explaining atomic spectra as well as the whole periodic system of elements; yet, except for a few arguments based solidly on the correspondence principle, it was unable to derive more than the simplest quantitative results in those fields. Further, an essential part of Bohr’s theory is a set of selection rules designed to state which of the dynamically possible transitions will take place and which will not. In atomic spectra, for example, they take the form

We shall see in Sec. 11.3 an example of how the correspondence principle can be used, in the limit of large quantum numbers, to derive selection rules, but the inability of Bohr’s theory to give any clear dynamical argument to explain such rules was a serious gap.

Problem 1.17. Verify (1.18) and (1.19) for an electron moving in a circular orbit.

Problem 1.18. To illustrate Larmor’s theorem, imagine an electron in a circular orbit under the influence of some force attracting it toward the center. Now slowly establish a magnetic field B, perpendicular to the plane of the orbit. Verify that the electron’s angular velocity changes by ±eB/2m, regardless of the nature of the attracting force or the size of the orbit.

Solution. , and the induced electric field E acts tangentially to exert on the electron a force eE = mr dω/dt, we have

or

Integrating from b = 0 gives a change in ω

If B is very strong, there will also be a stretching of r due to the changed centrifugal force. The above formula is valid only to first order in B.

1.4  MATTER WAVES

Finally, in the latter part of 1923, a new idea led to an explanation of the mysteries and inaccuracies of Bohr’s theory. As things stood in 1923, one could draw up the scheme shown in Table 1.1. On the first line we have the well-established phenomena governed by detailed mathematical equations: Maxwell’s equations on the one hand and Newton’s laws, supplemented by Einstein’s relativistic modifications, on the other. On the second line the phenomena are again well established, but instead of a system of equations we have only a few empirical and not wholly successful rules for calculation. The principal problem facing physics at that moment was the completion of this table.

The necessary new idea was furnished by Louis de Broglie (1923),¹⁰ then a graduate Student in Paris, who noted that one could attain a sort of symmetry in the foregoing table by saying that matter resembles light in having both wave and corpuscular aspects. His arguments¹¹ were rather impressionistic and based on relativistic properties, and so I shall not reproduce them here; a somewhat plausible account can be given as follows. The equations E ω, p = E/c were successful in describing the particle properties of light. Let us turn them around and use them to describe wave properties of matter. Yet there is a difficulty posed by the presence of c in the equations. A nonrelativistic theory should not contain c, and the obvious correctness of Bohr’s results, in which c is not involved, suggests that (at low energies) it should not enter the wave theory. The difficulty is removed by rewriting the second equation in terms of the light’s wavelength λ:

De Broglie suggested that a beam of particles of energy E and momentum p and a frequency ω/2π, and in his first paper he predicted that suitable experiments on a beam of electrons would reveal interference phenomena analogous to those of light.

TABLE 1.1

Classical and quantum physics in 1923

Problem 1.19. What would be the approximate wavelength and frequency associated with an electron traveling at 1 cm/s? A 1-eV electron?

It is now easy to give a physical picture of the origin of Bohr’s angular-momentum rule. We have noted that quantization occurs only in systems that are restricted in space. This suggests resonances of some kind, characterized by Standing waves. Consider, for example, a hydrogen orbit of radius a. If the electron forms a Standing wave in this orbit, then, as shown in Fig. 1.5, the circumference will be an integral number of wavelengths, or

From this and (1.23) we have for the angular momentum

which is Bohr’s rule for angular momenta. The important idea here is that the discrete atomic energy states are analogous to the discrete overtones of a vibrating system such as a violin string or a column of air. This suggestion is the basis of our present understanding of stationary atomic states.

De Broglie’s first paper closes with the words, We are now in a position to explain the phenomena of diffraction and interference in terms that take account of the existence of light quanta. In fact, it was reserved for others to do this, since de Broglie had not fully realized how the principle of superposition applies to matter waves. But he had discovered a new manifestation of the uniformity of nature, the universality of the relation between waves and particles, calling for a new form of dynamics, which should be to newtonian dynamics as wave optics is to geometrical optics.

FIGURE 1.5

Orbital standing waves according to de Broglie’s hypothesis.

The wave nature of electrons was exhibited four years later in the beautiful experiments of Davisson and Germer, of the Bell Laboratories,¹² who directed a beam of electrons onto a suitably oriented nickel crystal. The reflected beams showed a lobed pattern which could be analyzed exactly analogously to the Laue patterns of x-ray diffraction and was in excellent agreement with de Broglie’s formula. The diffraction of neutrons by crystals is now an everyday matter. But one may still ask whether the wave property is a general one—does it apply to composite objects as well as electrons, neutrons, and photons? The answer was provided in 1931 by a remarkable experiment of Stern and his collaborators, who produced a beam of slow helium atoms by means of a mechanical velocity selector and diffracted it from a lithium fluoride crystal. The measured wavelength corresponded exactly with (1.23) if p was taken to be the momentum of the atom as a whole.

More than 50 years later it was possible for physicists at MIT [Keith et al. (1988)] to obtain a diffraction pattern from a beam of sodium atoms diffracted by slits in a gold membrane. Such an atom, considered as formed from neutrons, protons, and electrons, is a 34-particle system.

Problem 1.20. The wavelength λ as measured in this experiment was about 17 pm. What was the velocity of the incident Na atoms? To what temperature would this velocity correspond?

It is natural to ask how fast a matter wave travels through space. Suppose first that we have a uniform monochromatic beam containing only a single frequency and wavelength:

or, introducing the abbreviation k 2π/λ ,

The velocity of a point in this wave which has a certain fixed phase, say a particular crest or trough, is clearly ω/k, corresponding to the usual expression λv. If we consider a free particle with velocity υ, for which

this phase velocity υϕ is

At first sight this seems to be in error, and it looks worse when we remember that energy has no natural zero point, so that E is really arbitrary. We must therefore be careful. The reason why there is no conflict with experiment here is that υϕ is not an observable quantity. A uniform beam is essentially a stationary object, for it has no distinguishing mark by which its velocity could be measured. The beam needs to be modulated in some way; and we shall see that the speed of the modulated pattern is in fact υ, the corresponding particle velocity. Of this I shall give three proofs, general ones in Secs. 2.5 and 3.3, and a more specific one now.

The simplest way to represent a modulated beam is by the coherent superposition of two beams having slightly different frequencies and wavelengths; for example,

This sum is equal to

where k‵ and ω‵ are the averages of the two constituent wave numbers and frequencies. The first cosine represents the modulation; and its phase velocity, which is termed the group velocity υg of the combination, is Δω/Δk. For a practical experiment, Δω and Δk would have to be rather small compared with ω and k (how small?); it is convenient to take the limit and write

with ω expressed as a function of k. With (1.24) this is dE/dp, and with E = p²/2m for a free particle we have

so that the apparent contradiction is reconciled. This method of resolving the difficulty should be taken as warning that quantum mechanics, like electromagnetism, contains unphysical quantities, and one must therefore always think about it in experimental terms. This is a hard lesson to learn, and it is here, and not in the mathematics, that most of the difficulty of the subject lies.

If there is a wave, what is waving? To nineteenth-century physicists, trying to understand Maxwell’s theory, this was a reasonable question, and they invented theories of the ether. Now we know how to say that the waves of Maxwell and de Broglie are waves in a field. But even this can be understood too naively, if one thinks of the field as some sort of continuous physical system with waves in it. Ocean waves have a wavelength that can be objectively measured, but what is the wavelength of an electron? If the electron moves and you run beside it but I do not, you give it p = 0 and I do not. What is the wavelength? The formula says λ = h/p. Evidently the length of the wave, like the momentum of the particle, is a relational quantity and not an absolute one, so you must not think that the wave is there, in space. The wave must be understood as the theory requires it to be understood, and not by analogy with something else.

Problem 1.21. Is it possible for vϕ and vg to be directed oppositely?

Problem 1.22. Low-energy neutrons are produced by letting them wander around in a moderator, a substance that does not absorb them, until they reach thermal equilibrium. The average kinetic energy of a particle at temperature T . What must be the temperature of the moderator to produce neutrons of wavelength about 10 pm?

1.5  PRINCIPLE OF SUPERPOSITION

The foregoing explanation of the velocity paradox involves no new assumptions; the basic trick, the representation of a modulated wave as the superposition of two (or more) unmodulated ones, has already been used to explain interference phenomena. One must be able to add amplitudes. But it is precisely this coherent superposition of amplitudes that is unknown in the classical theory of particles, and this is why the particle theory can never in any way explain interference phenomena. The idea that the amplitude corresponding to a physical object may be written as the sum of other amplitudes is as fundamental to quantum mechanics as it is to optics and acoustics. And one is quite accustomed to treating electrical problems in the same way. To analyze the action of an LC filter in smoothing a rectified voltage, one performs a Fourier analysis of the latter, considers it as the sum of the sinusoidal voltages, and calculates the response of the filter to each. Whether the applied signal actually is an irregular waveform or the sum of sinusoidal ones is a meaningless question. It is both, or it may be represented in other ways—for example, as the sum of a series of brief pulses. The superposition may originate as an actual physical superposition (coherent!) of two systems, or it may be merely a mathematical device to simplify calculation. In many cases it can be regarded both ways.

In order to express this principle mathematically, we must have something to superpose, something which will be analogous to the potentials superposed in network analysis or the vectors E and B that are superposed in Maxwells theory. This quantity has the neutral name wave function and is denoted by ψ(x, y, z, t). It is a scalar quantity, and, as in the other two cases just mentioned, we shall suppose that it specifies all that can be known about the system it represents. A major problem is to decide what is the right physical Interpretation of ψ. Also, the object of any theory is to enable one to understand specific experimental results from a general Standpoint, so that we must have a set of rules that will enable us, knowing ψ, to predict what results a certain measurement will give.

Another major problem can be spotted in Table 1.1. The reason why the classical theories work so well is that they are based on equations of motion. As set forth in the table, the quantum aspect is not so fortunate, since it is expressed only in a set of rules due to Einstein, Bohr, and now de Broglie. The quantum theory cannot hope to have anything like the power and generality of the classical theory until it too is expressed in terms of an equation of motion.

The analogy of Maxwell’s theory suggests that ψ will satisfy a partial differential equation. Such an equation cannot be derived a priori any more than Newton’s second law can be; but one can at least suggest how such an equation can be arrived at and learn something about ψ in the process. Consider a wave periodic in Space and time corresponding to a beam of free particles moving along the x axis. Its ψ will be a sinusoidal function of kx − ωt, where k and ω satisfy (1.24) and are therefore related by

The differential equation must contain this relation. By differentiating ψ with respect to x or t we obtain a multiplicative factor of k or ω. It must therefore be differentiated once to get ω and twice to get k². The only function that retains its form whether differentiated once or twice is the exponential, so that writing

where A is a constant amplitude, we find that

This equation

is satisfied by the proposed wave function (1.30) though it is not the only such equation, nor is (1.30) the only wave function that fulfills the requirements from which it was derived, but it is the simplest possibility. It is derived above only for the simple exponential wave function (1.30), but it is valid for any k and the corresponding ω. Now, since the sum of any set of solutions of (1.31) is itself a Solution, we can use the principle of superposition. If we assume (see Sec. 2.4) that all possible wave functions can be represented by Fourier superposition of such exponentials, we conclude that all possible wave functions for a free particle will also satisfy (1.31). But a plausible wave equation for a beam of free particles is not the basis for a good theory of matter waves. Such a theory must be able to describe electrons interacting with electric and magnetic fields of force, and the preceding argument does not suggest how to introduce them. Two paths have been followed. The first, by Erwin Schrödinger in 1926, was to guess, and that is what will be done at the beginning of Chap. 3. For practical purposes what is important is to have the equation, to know what the variables mean, and to solve it and interpret the answer. For more profound explorations one needs to know more about where the equation comes from so as to understand more precisely what it means. We can study an argument presented by Richard Feynman in 1942: Simple assumptions plus some mathematical wizardry lead once more to what Schrödinger guessed. Feynman’s approach depends on knowing some hamiltonian dynamics and is best appreciated if one already knows something about quantum mechanics; accordingly it is put into Chap 13. For the present we shall think at some length about what it means to represent matter as a wave. Interactions will come later.

Problem 1.23. [and (1.31) is such an equation] has solutions f1 and f2, then f1 +f2 is also a Solution, but that this is not true for a nonlinear equation.

. One is used to encountering this quantity in alternating-current theory, but only because it is put there as a computational device. The basic equations, like the currents and voltages, are real. But here i enters into both the equation (1.31) and its simplest form of Solution, (1.30). The actual results of quantum mechanics, expressed as observable quantities, will be real numbers, and much will be made of this fact later; but an equation is not an observable thing, and there is no reason why it should not involve i. Further, we shall see that ψ is not itself an observable, though its wavelength, for example, is. This is less surprising if one remembers that E and B are not observables either, though they have a simple connection with observable quantities in terms of forces they exert. The ψ function has no simple connection of this kind.

Problem 1.24. Replace Schrödinger’s complex equation with a pair of real equations by (a) writing ψ(x, t) = ψ1(x, t) + iψ2(x, t) with and ψ1 and ψ2 real and separating real and imaginary parts, (b) writing ψ(x, t) = A(x, t)eiφ(x, t) with A and φ real and finding the real equations satisfied by the two real functions A and ψ. Is any simplicity gained by this?

To work in three dimensions we shall need the three-dimensional analog of (1.30), describing a plane wave that moves through space. Figure 1.6 shows the wavefronts of such a wave as a series of parallel planes separated by a distance L Let n be the unit vector normal to the wavefronts. Then if r gives the position of any point P on a certain wavefront A, it satisfies the relation n ⋅ r = d, where d is the normal distance from the origin to the wavefront A. The spatial part of the phase of this wavefront with respect to a plane passing through the origin is 2πd/λ = d= (n . r. Analogous to the number k introduced above, define

FIGURE 1.6

Side view of plane waves in three dimensions. The wavelength is λ, and n is a unit vector normal to the plane of the waves.

the propagation vector of the wave. Then, including the time factor, the wave function for a plane wave is

k and the direction of n; it is given by

analogously to (1.24).

Problem 1.25. Show that becomes ∇².

Energy Levels

The occurrence of stationary energy levels in nature can be qualitatively understood from (1.31); its spatial part [see (1.35) below] has exactly the same form as the equation which arises in the study of a vibrating string (see any text on mechanics or general theoretical physics). A treatment in parallel columns, starting with (1.31), will exhibit the similarities and differences.

where v is the velocity of waves in the string. We now consider the normal modes, that is, the modes of motion that are simply periodic in time.

The string’s energy depends on the amplitude A of its Vibration as well as its density ρ, and it is calculated [Coulson (1944)] as

We see how stationary energy levels are going to be explained as standing-wave phenomena, but we note the characteristic difference that the energy of the matter wave does not depend on its amplitude. This is, of course, already reminiscent of the photoelectric effect, in which the energy of the ejected electrons is independent of the amplitude of the incident light waves; but in order to understand it, we shall have to have a clear idea of what the wave function ψ actually represents. Subsidiary questions are: What is the special importance of normal modes? Why does one impose the boundary condition that ψ(0) = ψ(a) = 0? Such questions can be answered only if we have a physical Interpretation for ψ; the next chapter is devoted to this. Chapter 3 studies the wave equation for a particle acted upon by external forces and discusses some general quantum-mechanical properties of physical systems. Then Chap. 4 will return to a more detailed study of phenomena of free and bound electrons, considered in one dimension for simplicity, and take a closer look at discrete energy levels.

Problem 1.26. To get an idea of the magnitudes involved in (1.36), let the particle be an electron and let the size of the region be atomic, say a = 0.1 nm. In electron volts,

what are the first three energy levels? (Note for future reference that zero is not a possible energy.) If the electron makes a transition from level 2 to level 1, what is the wavelength of the emitted radiation? How does this compare with those of typical atomic spectral lines? (The yellow sodium D lines, for example, have λ ≈ 189 nm.)

Problem 1.27. If a wall is pushed in so that a in (1.36) is made a little smaller, the particle’s energy rises. This means that the electron is exerting a force on the wall. Calculate this force in the rcth State, noting that it is present even in the lowest possible State. If the particle is an electron, how much work is done (in eV) in pushing the wall from a = 0.1 nm to 0.05 nm?

REFERENCES

Balmer, J. (1885): Ann. Phys. 25, 80.

Bergmann, P. G. (1942): Introduction