81.TheExperimental Origins of QuantumMechanicsthen, is the reverse of that with light. While light was long thought to bea wave(atleastfromthepublication ofMaxwell'sequationsin1865untilPlanck'swork in1900)and was onlylater seen tohave particlelike behavior,electrons were initially thought to be particles and were only later seen tohave wavelike properties.In the end, however, both light and electrons haveboth wavelike and particlelike properties.1.2.1The Spectrum ofHydrogenIf electricity is passed through a tube containing hydrogen gas, the gas willemit light. If that light is separated into different frequencies by meansof a prism, bands will become apparent, indicating that the light is not acontinuous mix of many different frequencies, but rather consists only of adiscrete family of frequencies. In view of the photonic theory of light, theenergyin each photon is proportional to itsfrequency.Thus, each observedfrequency corresponds to a certain amount of energybeing transferred froma hydrogen atom to the electromagnetic field.Now, a hydrogen atom consists of a singleproton surrounded by a singleelectron. Since the proton is much more massive than the electron, onecan picture the proton as being stationary,with the electron orbiting it.The idea, then, is that the current being passed through the gas causes someof the electrons to move to a higher-energy state.Eventually, that electronwill return to a lower-energy state,emitting a photon in the process. In thisway,by observing theenergies(or,equivalently,thefrequencies)oftheemitted photons, one can work backwards to the change in energy of theelectron.The curious thing about the state of affairs in the preceding paragraphis that the energies of the emitted photons-and hence, also, the energiesof the electron-come only in a discretefamily of possible values.Basedon the observed frequencies, Johannes Rydberg concluded in 1888 that thepossible energies of theelectron were of the formR(1.1)En=n2Here, R is the“Rydberg constant,"given (in"Gaussian units")bymeQ4R=2h2，where Q is the charge of the electron and me is the mass of the electron.(Technically,me should be replaced by the reduced mass μ of the proton-electron system; that is, μ = memp/(me + mp), where mp is the massof the proton. However, since the proton mass is much greater than theelectron mass,μ is almost the same as me and we will neglect the differencebetween the two.) The energies in (1.1) agree with experiment, in that all
8 1. The Experimental Origins of Quantum Mechanics then, is the reverse of that with light. While light was long thought to be a wave (at least from the publication of Maxwell’s equations in 1865 until Planck’s work in 1900) and was only later seen to have particlelike behavior, electrons were initially thought to be particles and were only later seen to have wavelike properties. In the end, however, both light and electrons have both wavelike and particlelike properties. 1.2.1 The Spectrum of Hydrogen If electricity is passed through a tube containing hydrogen gas, the gas will emit light. If that light is separated into different frequencies by means of a prism, bands will become apparent, indicating that the light is not a continuous mix of many different frequencies, but rather consists only of a discrete family of frequencies. In view of the photonic theory of light, the energy in each photon is proportional to its frequency. Thus, each observed frequency corresponds to a certain amount of energy being transferred from a hydrogen atom to the electromagnetic field. Now, a hydrogen atom consists of a single proton surrounded by a single electron. Since the proton is much more massive than the electron, one can picture the proton as being stationary, with the electron orbiting it. The idea, then, is that the current being passed through the gas causes some of the electrons to move to a higher-energy state. Eventually, that electron will return to a lower-energy state, emitting a photon in the process. In this way, by observing the energies (or, equivalently, the frequencies) of the emitted photons, one can work backwards to the change in energy of the electron. The curious thing about the state of affairs in the preceding paragraph is that the energies of the emitted photons—and hence, also, the energies of the electron—come only in a discrete family of possible values. Based on the observed frequencies, Johannes Rydberg concluded in 1888 that the possible energies of the electron were of the form En = − R n2 . (1.1) Here, R is the “Rydberg constant,” given (in “Gaussian units”) by R = meQ4 22 , where Q is the charge of the electron and me is the mass of the electron. (Technically, me should be replaced by the reduced mass μ of the proton– electron system; that is, μ = memp/(me + mp), where mp is the mass of the proton. However, since the proton mass is much greater than the electron mass, μ is almost the same as me and we will neglect the difference between the two.) The energies in (1.1) agree with experiment, in that all
1.2 Is an Electron a Wave or a Particle?the observed frequencies in hydrogen are (at least to the precision availableat the time of Rydberg)of the form1(E, -Em),(1.2)w=for some n > m.It should be noted that Johann Balmer had alreadyobserved in 1885frequencies of the same form, but only in the case m=2,and thatBalmer's work infuenced Rydberg.The frequencies in (1.2)are known as the spectrum of hydrogen.Balmerand Rydberg were merely attempting to find a simple formula that wouldmatch the observed frequencies in hydrogen. Neither of them had a the-oretical explanation for why only these particular frequencies occur.Suchan explanation would have to wait until the beginnings of quantum theoryin the twentieth century.1.2.2TheBohr-deBroglieModel of theHydrogen AtomIn 1913, Niels Bohr introduced a model of the hydrogen atom that at-tempted to explain the observed spectrum of hydrogen.Bohr pictured thehydrogen atom as consisting of an electron orbiting a positively chargednucleus, in much the same way that a planet orbits the sun. Classically,the force exerted on the electron by the proton follows the inverse squarelawoftheformQ?F=(1.3)r2where Qis the charge of the electron, in appropriate units.If the electron is in a circular orbit, its trajectory in the plane of theorbit will take the form(r(t), y(t)) = (rcos(wt),rsin(wt)If we take the second derivative with respect to time to obtain the acceler-ation vectora,weobtaina(t)= (-w*rcos(wt),-wrsin(wt)so that the magnitude of the acceleration vector is w?r.Newton's secondlaw, F= ma, then requires thate2mew?r:r2so thatQ2mer3
1.2 Is an Electron a Wave or a Particle? 9 the observed frequencies in hydrogen are (at least to the precision available at the time of Rydberg) of the form ω = 1 (En − Em), (1.2) for some n > m. It should be noted that Johann Balmer had already observed in 1885 frequencies of the same form, but only in the case m = 2, and that Balmer’s work influenced Rydberg. The frequencies in (1.2) are known as the spectrum of hydrogen. Balmer and Rydberg were merely attempting to find a simple formula that would match the observed frequencies in hydrogen. Neither of them had a theoretical explanation for why only these particular frequencies occur. Such an explanation would have to wait until the beginnings of quantum theory in the twentieth century. 1.2.2 The Bohr–de Broglie Model of the Hydrogen Atom In 1913, Niels Bohr introduced a model of the hydrogen atom that attempted to explain the observed spectrum of hydrogen. Bohr pictured the hydrogen atom as consisting of an electron orbiting a positively charged nucleus, in much the same way that a planet orbits the sun. Classically, the force exerted on the electron by the proton follows the inverse square law of the form F = Q2 r2 , (1.3) where Q is the charge of the electron, in appropriate units. If the electron is in a circular orbit, its trajectory in the plane of the orbit will take the form (x(t), y(t)) = (r cos(ωt), r sin(ωt)). If we take the second derivative with respect to time to obtain the acceleration vector a, we obtain a(t)=(−ω2r cos(ωt), −ω2r sin(ωt)), so that the magnitude of the acceleration vector is ω2r. Newton’s second law, F = ma, then requires that meω2r = e2 r2 , so that ω = Q2 mer3 .
101.TheExperimental Origins of QuantumMechanicsFrom theformulaforthe frequency,we can calculatethat the momentum(mass times velocity)has magnitudemeQ2(1.4)We can also calculate the angular momentum J, which for a circular orbitis just the momentum times the distance from the nucleus, asJ=VmeQ?r.Bohr postulated that the electron obeys classical mechanics, ercept thatits angular momentum is “quantized."Specifically,in Bohr's model, theangular momentum is required to be an integer multiple of h (Planck'sconstant).SettingJequal tonhyieldsn?h2(1.5)rnmeQ2If one calculates the energy of an orbit with radius rn, one finds (Exercise 3)that it agrees precisely with the Rydberg energies in (1.1). Bohr furtherpostulated that an electron could move from one allowed state to another,emitting a packet of light in the process with frequency given by (1.2).Bohr did not explain why the angular momentum of an electron is quan-tized, nor how it moved from one allowed orbit to another. As such, histheory of atomic behavior was clearly not complete; it belongs to the “"oldquantum mechanics" that was superseded by the matrix model of Heisen-berg and the wavemodel of Schrodinger.Nevertheless, Bohr's model was animportant step in the process of understanding the behavior of atoms, andBohr was awarded the 1922 Nobel Prize in physics for his work. Some rem-nant of Bohr's approach survives in modern quantum theory, in the WKBapproximation (Chap.15),where theBohr-Sommerfeld condition givesanapproximation to the energy levels of a one-dimensional quantum system.In 1924,Louis deBroglie reinterpreted Bohr's condition on the angularmomentum as a wave condition. The de Broglie hypothesis is that an elec-tron can be described by a wave, where the spatial frequency k of the waveis related to the momentum of the electron by the relation(1.6)p=hk.Here, "frequency" is defined so that the frequency of the function cos(kr)is k. This is “angular" frequency, which differs by a factor of 2 from thecycles-per-unit-distance frequency. Thus, the period associated with a givenfrequencykis2元/k.In de Broglie's approach, we are supposed to imagine a wave super-imposed on the classical trajectory of the electron, with the quantization
10 1. The Experimental Origins of Quantum Mechanics From the formula for the frequency, we can calculate that the momentum (mass times velocity) has magnitude p = meQ2 r . (1.4) We can also calculate the angular momentum J, which for a circular orbit is just the momentum times the distance from the nucleus, as J = meQ2r. Bohr postulated that the electron obeys classical mechanics, except that its angular momentum is “quantized.” Specifically, in Bohr’s model, the angular momentum is required to be an integer multiple of (Planck’s constant). Setting J equal to n yields rn = n22 meQ2 . (1.5) If one calculates the energy of an orbit with radius rn, one finds (Exercise 3) that it agrees precisely with the Rydberg energies in (1.1). Bohr further postulated that an electron could move from one allowed state to another, emitting a packet of light in the process with frequency given by (1.2). Bohr did not explain why the angular momentum of an electron is quantized, nor how it moved from one allowed orbit to another. As such, his theory of atomic behavior was clearly not complete; it belongs to the “old quantum mechanics” that was superseded by the matrix model of Heisenberg and the wave model of Schr¨odinger. Nevertheless, Bohr’s model was an important step in the process of understanding the behavior of atoms, and Bohr was awarded the 1922 Nobel Prize in physics for his work. Some remnant of Bohr’s approach survives in modern quantum theory, in the WKB approximation (Chap. 15), where the Bohr–Sommerfeld condition gives an approximation to the energy levels of a one-dimensional quantum system. In 1924, Louis de Broglie reinterpreted Bohr’s condition on the angular momentum as a wave condition. The de Broglie hypothesis is that an electron can be described by a wave, where the spatial frequency k of the wave is related to the momentum of the electron by the relation p = k. (1.6) Here, “frequency” is defined so that the frequency of the function cos(kx) is k. This is “angular” frequency, which differs by a factor of 2π from the cycles-per-unit-distance frequency. Thus, the period associated with a given frequency k is 2π/k. In de Broglie’s approach, we are supposed to imagine a wave superimposed on the classical trajectory of the electron, with the quantization
111.2Isan Electron a Wave or aParticle?FIGURE 1.3. The Bohr radii for n =1 to n =10, with de Broglie waves super-imposedforn=8andn=10condition now being that the wave should match up with itself when goingall the way around the orbit. This condition means that the orbit shouldconsist of an integer number of periods of the wave:2元2元T=nkUsing (1.6)along with theexpression (1.4)forp,we obtainh=2元nh2元=n2元meQ?pSolving this equation for r gives precisely the Bohr radi in (1.5)Thus, de Broglie's wave hypothesis gives an alternative to Bohr's quan-tization of angular momentum as an explanation of the allowed energies ofhydrogen. Of course, if one accepts de Broglie's wave hypothesis for elec-trons, one would expect to see wavelike behavior of electrons not just in thehydrogen atom, but in other situations as well,an expectation that wouldsoon be fulfilled.Figure 1.3 shows the first 10o Bohr radi.For the 8th and1Oth radi, the de Broglie wave is shown superimposed onto the orbit.1.2.31ElectronDiffractionIn 1925, Clinton Davisson and Lester Germer were studying properties ofnickel by bombarding a thin film of nickel with low-energy electrons.As aresult of aproblem with their equipment, the nickel was accidentally heatedto a very high temperature. When the nickel cooled, it formed into large
1.2 Is an Electron a Wave or a Particle? 11 FIGURE 1.3. The Bohr radii for n = 1 to n = 10, with de Broglie waves superimposed for n = 8 and n = 10. condition now being that the wave should match up with itself when going all the way around the orbit. This condition means that the orbit should consist of an integer number of periods of the wave: 2πr = n 2π k . Using (1.6) along with the expression (1.4) for p, we obtain 2πr = n2π p = 2πn r meQ2 . Solving this equation for r gives precisely the Bohr radii in (1.5). Thus, de Broglie’s wave hypothesis gives an alternative to Bohr’s quantization of angular momentum as an explanation of the allowed energies of hydrogen. Of course, if one accepts de Broglie’s wave hypothesis for electrons, one would expect to see wavelike behavior of electrons not just in the hydrogen atom, but in other situations as well, an expectation that would soon be fulfilled. Figure 1.3 shows the first 10 Bohr radii. For the 8th and 10th radii, the de Broglie wave is shown superimposed onto the orbit. 1.2.3 Electron Diffraction In 1925, Clinton Davisson and Lester Germer were studying properties of nickel by bombarding a thin film of nickel with low-energy electrons. As a result of a problem with their equipment, the nickel was accidentally heated to a very high temperature. When the nickel cooled, it formed into large
121.TheExperimental Origins of QuantumMechanicscrystalline pieces, rather than the small crystals in the original sample.After thisrecrystallization,Davisson and Germer observed peaksin thepatternof electronsreflecting offofthenickel samplethathad notbeenpresent when using the original sample. They were at a loss to explain thispattern until, in 1926, Davisson learned of the de Broglie hypothesis andsuspected that they were observing the wavelikebehavior of electrons thatdeBrogliehad predicted.After this realization, Davisson and Germer began to look systemati-cally for wavelike peaks in their experiments. Specifically,they attemptedto show that the pattern of angles at which the electrons reflected matchedthe patterns one sees in x-ray diffraction.After numerous additional mea-surements.they wereabletoshow averyclosecorrespondence betweenthe pattern of electrons and the patterns seen in x-ray diffraction. Sincex-rays were by this time known to be waves of electromagnetic radiation,the Davisson-Germer experiment was a strong confirmation of de Broglie'swave picture of electrons.Davisson and Germer published their results intwo papers in 1927,and Davisson shared the 1937Nobel Prize in physicswith George Paget, who had observed electron diffraction shortly afterDavissonand Germer.1.2.4TheDouble-SlitErperimentwithElectronsAlthough quantum theory clearly predicts that electrons passing througha double slit will experience interference similar to that observed in light,it was not until Clauss Jonsson's work in 1961 that this prediction wasconfirmed experimentally. The main difficulty is the much smaller wave-length for electrons of reasonable energy than for visible light.Jonsson'selectrons.forexample.hadadeBrogliewavelengthof5nm,ascomparedtoawavelengthofroughly500nmforvisiblelight(dependingonthecolor)Inresultspublished in 1989,a teamledby AkiraTonomuraatHitachiperformed a double-slit experiment in which they were able to record theresults one electron at a time. (Similar but less definitive experiments werecarried out by Pier Giorgio Merli, GianFranco Missiroli and Giulio Pozziin Bologna in 1974 and publishedin the American Journal of Physics in1976.)In the Hitachi experiment, each electron passes through the slits andthen strikes a screen, causing a small spot of light to appear.Thelocation ofthis spot is then recorded for each electron, one at a time. The key point isthat each individual electron strikes the screen at a single point. That is tosay, individual electrons are not smeared out across the screen in a wavelikepattern, but rather behave like point particles, in that the observed locationof the electron is indeed a point. Each electron, however, strikes the screenat a different point, and once a large number of the electrons have struckand their locations have been recorded, an interference pattern emerges.It is not the variability of the locations of the electrons that is surprising,sincethiscouldbeaccountedforbysmallvariationsinthewaytheelectrons
12 1. The Experimental Origins of Quantum Mechanics crystalline pieces, rather than the small crystals in the original sample. After this recrystallization, Davisson and Germer observed peaks in the pattern of electrons reflecting off of the nickel sample that had not been present when using the original sample. They were at a loss to explain this pattern until, in 1926, Davisson learned of the de Broglie hypothesis and suspected that they were observing the wavelike behavior of electrons that de Broglie had predicted. After this realization, Davisson and Germer began to look systematically for wavelike peaks in their experiments. Specifically, they attempted to show that the pattern of angles at which the electrons reflected matched the patterns one sees in x-ray diffraction. After numerous additional measurements, they were able to show a very close correspondence between the pattern of electrons and the patterns seen in x-ray diffraction. Since x-rays were by this time known to be waves of electromagnetic radiation, the Davisson–Germer experiment was a strong confirmation of de Broglie’s wave picture of electrons. Davisson and Germer published their results in two papers in 1927, and Davisson shared the 1937 Nobel Prize in physics with George Paget, who had observed electron diffraction shortly after Davisson and Germer. 1.2.4 The Double-Slit Experiment with Electrons Although quantum theory clearly predicts that electrons passing through a double slit will experience interference similar to that observed in light, it was not until Clauss J¨onsson’s work in 1961 that this prediction was confirmed experimentally. The main difficulty is the much smaller wavelength for electrons of reasonable energy than for visible light. J¨onsson’s electrons, for example, had a de Broglie wavelength of 5 nm, as compared to a wavelength of roughly 500 nm for visible light (depending on the color). In results published in 1989, a team led by Akira Tonomura at Hitachi performed a double-slit experiment in which they were able to record the results one electron at a time. (Similar but less definitive experiments were carried out by Pier Giorgio Merli, GianFranco Missiroli and Giulio Pozzi in Bologna in 1974 and published in the American Journal of Physics in 1976.) In the Hitachi experiment, each electron passes through the slits and then strikes a screen, causing a small spot of light to appear. The location of this spot is then recorded for each electron, one at a time. The key point is that each individual electron strikes the screen at a single point. That is to say, individual electrons are not smeared out across the screen in a wavelike pattern, but rather behave like point particles, in that the observed location of the electron is indeed a point. Each electron, however, strikes the screen at a different point, and once a large number of the electrons have struck and their locations have been recorded, an interference pattern emerges. It is not the variability of the locations of the electrons that is surprising, since this could be accounted for by small variations in the way the electrons