31.1 Is Light a Wave or a Particle?FIGURE1.1.Interferenceofwaves emitted fromtwo slits.top right of the figure. Figure 1.2 then plots the intensity (i.e., the square ofthe displacement)as a function of a, with y having the value correspondingto the bottom of Fig.1.1.Despite the convincing nature of Young's experiment, many proponentsof the corpuscular theory of light remained unconvinced. In 1818, theFrench Academy of Sciences set up a competition for papers explainingthe observed properties of light. One of the submissions was a paper byAugustin-Jean Fresnel in which he elaborated on Huygens's wave modelof refraction.A supporter of the corpuscular theory of light, Simeon-DenisPoisson read Fresnel's submission and ridiculed it by pointing out thatif that theory were true, light passing by an opaque disk would diffractaround the edges of the disk to produce a bright spot in the center of theshadow of the disk, a prediction that Poisson considered absurd. Never-theless, the head of the judging committee for the competition, FrancoisArago, decided to put the issuetexperimental test and found thatsuch a spot does in fact occur. Although this spot is often called “Arago'sspot,"or even, ironically,"Poisson's spot,"Arago eventually realized thatthe spot had been observed 100 years earlier in separate experiments byDelisle and Maraldi.Arago's observation of Poisson's spot led to widespread acceptance ofthe wave theory of light. This theory gained even greater acceptance in1865, when James Clerk Maxwell put together what are today known asMaxwell's equations. Maxwell showed that his equations predicted thatelectromagnetic waves would propagate at a certain speed, which agreedwith the observed speed of light. Maxwell thus concluded that light is sim-ply an electromagnetic wave.From 1865 until the end of the nineteenth
1.1 Is Light a Wave or a Particle? 3 FIGURE 1.1. Interference of waves emitted from two slits. top right of the figure. Figure 1.2 then plots the intensity (i.e., the square of the displacement) as a function of x, with y having the value corresponding to the bottom of Fig. 1.1. Despite the convincing nature of Young’s experiment, many proponents of the corpuscular theory of light remained unconvinced. In 1818, the French Academy of Sciences set up a competition for papers explaining the observed properties of light. One of the submissions was a paper by Augustin-Jean Fresnel in which he elaborated on Huygens’s wave model of refraction. A supporter of the corpuscular theory of light, Sim´eon-Denis Poisson read Fresnel’s submission and ridiculed it by pointing out that if that theory were true, light passing by an opaque disk would diffract around the edges of the disk to produce a bright spot in the center of the shadow of the disk, a prediction that Poisson considered absurd. Nevertheless, the head of the judging committee for the competition, Fran¸cois Arago, decided to put the issue to an experimental test and found that such a spot does in fact occur. Although this spot is often called “Arago’s spot,” or even, ironically, “Poisson’s spot,” Arago eventually realized that the spot had been observed 100 years earlier in separate experiments by Delisle and Maraldi. Arago’s observation of Poisson’s spot led to widespread acceptance of the wave theory of light. This theory gained even greater acceptance in 1865, when James Clerk Maxwell put together what are today known as Maxwell’s equations. Maxwell showed that his equations predicted that electromagnetic waves would propagate at a certain speed, which agreed with the observed speed of light. Maxwell thus concluded that light is simply an electromagnetic wave. From 1865 until the end of the nineteenth
41.TheExperimental Origins of QuantumMechanicsFIGURE1.2.Intensityplotforahorizontal lineacrossthebottomof Fig,1.1century, the debate over the wave-versus-particle nature of light was con-sidered to have been conclusively settled in favor of the wave theory1.1.3BlackbodyRadiationIn the earlytwentieth century,thewave theory of light began to experiencenewchallenges.Thefirstchallengecamefromthetheoryofblackbodyradia-tion.In physics, a blackbody isan idealized object that perfectly absorbs allelectromagnetic radiation that hits it.Ablackbody can be approximated inthe real world by an object with a highly absorbent surface such as"lampblack."Theproblem ofblackbodyradiation concerns thedistribution ofelectromagnetic radiation in a cavity within a blackbody. Although thewalls of the blackbody absorb the radiation that hits it, thermal vibrationsof the atoms making up the walls cause the blackbody to emit electromag-netic radiation. (At normal temperatures, most of the radiation emittedwould be in the infrared range.)In the cavity,then,electromagnetic radiation is constantlyabsorbed andre-emitted until thermal equilibrium is reached, at which point the absorp-tion and emission of radiation are perfectly balanced at each frequency.According to the“equipartition theorem"of (classical)statistical mechan-ics, the energy in any given mode of electromagnetic radiation should beexponentially distributed, with an average value equal to kT, whereT isthetemperature andkB isBoltzmann's constant.(Thetemperature shouldbemeasured on a scalewhere absolute zero corresponds toT=o.)The dif-ficulty with this prediction is that theaverageamountof energy is thesamefor every mode (hence the term “"equipartition"). Thus, once one adds upoverall modes-ofwhichthereareinfinitelymany-thepredicted amountof energy in the cavity is infinite. This strange prediction is referred to astheultraviolet catastrophe,since theinfinitude of the energy comes from theultraviolet (high-frequency)end of the spectrum.This ultraviolet catastro-phe does not seem to make physical sense and certainly does not match upwith the observed energy spectrum within real-world blackbodies
4 1. The Experimental Origins of Quantum Mechanics FIGURE 1.2. Intensity plot for a horizontal line across the bottom of Fig. 1.1 . century, the debate over the wave-versus-particle nature of light was considered to have been conclusively settled in favor of the wave theory. 1.1.3 Blackbody Radiation In the early twentieth century, the wave theory of light began to experience new challenges. The first challenge came from the theory of blackbody radiation. In physics, a blackbody is an idealized object that perfectly absorbs all electromagnetic radiation that hits it. A blackbody can be approximated in the real world by an object with a highly absorbent surface such as “lamp black.” The problem of blackbody radiation concerns the distribution of electromagnetic radiation in a cavity within a blackbody. Although the walls of the blackbody absorb the radiation that hits it, thermal vibrations of the atoms making up the walls cause the blackbody to emit electromagnetic radiation. (At normal temperatures, most of the radiation emitted would be in the infrared range.) In the cavity, then, electromagnetic radiation is constantly absorbed and re-emitted until thermal equilibrium is reached, at which point the absorption and emission of radiation are perfectly balanced at each frequency. According to the “equipartition theorem” of (classical) statistical mechanics, the energy in any given mode of electromagnetic radiation should be exponentially distributed, with an average value equal to kBT, where T is the temperature and kB is Boltzmann’s constant. (The temperature should be measured on a scale where absolute zero corresponds to T = 0.) The dif- ficulty with this prediction is that the average amount of energy is the same for every mode (hence the term “equipartition”). Thus, once one adds up over all modes—of which there are infinitely many—the predicted amount of energy in the cavity is infinite. This strange prediction is referred to as the ultraviolet catastrophe, since the infinitude of the energy comes from the ultraviolet (high-frequency) end of the spectrum. This ultraviolet catastrophe does not seem to make physical sense and certainly does not match up with the observed energy spectrum within real-world blackbodies
51.1 Is Light a Wave or a Particle?An alternativeprediction of the blackbody energy spectrum was offeredby Max Planck in a paper published in 1900. Planck postulated thatthe energy in the electromagnetic field at a given frequency w should be"quantized,"meaning that this energy should come only in integer mul-tiples of a certain basic unit equal to hw, where h is a constant, whichwe now call Planck's constant.Planck postulated that the energy wouldagain be exponentially distributed, but only overinteger multiples of hw.At low frequencies, Planck's theory predicts essentially the same energy asin classical statistical mechanics.At high frequencies, namely at frequen-cies where hw is large compared to kBT, Planck's theory predicts a rapidfall-off of the average energy (see Exercise 2 for details). Indeed, if we mea-sure mass, distance, and time in units of grams, centimeters, and seconds,respectively, and we assign h the numerical valueh=1.054×10-27then Planck's predictions match the experimentally observed blackbodyspectrum.Planck pictured the walls of the blackbody as being made up of inde-pendent oscillators of different frequencies, each of which is restricted tohave energies of hw. Although this picture was clearly not intended as arealisticphysical explanation of thequantization of electromagnetic energyin blackbodies, it does suggest that Planck thought that energy quantiza-tion arose from properties of the walls of the cavity, rather than in intrinsicproperties of the electromagnetic radiation. Einstein, on the other hand, inassessing Planck's model, argued that energy quantization was inherent inthe radiation itself. In Einstein's picture, then, electromagnetic energy ata given frequency—whether in a blackbody cavity or not-comes in pack-ets or quanta having energy proportional to the frequency.Each quantumof electromagnetic energy constitutes what we now call a photon, whichwe may think of as a particle of light. Thus, Planck's model of blackbodyradiation began a rebirth of the particle theory of light.It is worth mentioning, in passing, that in 1900, the same year in whichPlanck's paper on blackbody radiation appeared, Lord Kelvin gave a lec-ture that drew attention to another difficulty with the classical theoryof statistical mechanics.Kelvin described two "clouds"over nineteenth-century physics at the dawn of the twentieth century.The first of theseclouds concerned aether-a hypothetical medium through which electro-magneticradiationpropagates--andthefailureofMichelsonandMorleytoobservethemotion of earthrelativetotheaether.Under thiscloud lurkedthe theory of special relativity.The second of Kelvin's clouds concernedheat capacities in gases. The equipartition theorem of classical statisti-cal mechanics made predictions for the ratio of heat capacity at constantpressure (cp) and the heat capacity at constant volume (cu). These pre-dictions deviated substantially from the experimentally measured ratios.Under the second cloud lurked the theory of quantum mechanics, because
1.1 Is Light a Wave or a Particle? 5 An alternative prediction of the blackbody energy spectrum was offered by Max Planck in a paper published in 1900. Planck postulated that the energy in the electromagnetic field at a given frequency ω should be “quantized,” meaning that this energy should come only in integer multiples of a certain basic unit equal to ω, where is a constant, which we now call Planck’s constant. Planck postulated that the energy would again be exponentially distributed, but only over integer multiples of ω. At low frequencies, Planck’s theory predicts essentially the same energy as in classical statistical mechanics. At high frequencies, namely at frequencies where ω is large compared to kBT, Planck’s theory predicts a rapid fall-off of the average energy (see Exercise 2 for details). Indeed, if we measure mass, distance, and time in units of grams, centimeters, and seconds, respectively, and we assign the numerical value = 1.054 × 10−27, then Planck’s predictions match the experimentally observed blackbody spectrum. Planck pictured the walls of the blackbody as being made up of independent oscillators of different frequencies, each of which is restricted to have energies of ω. Although this picture was clearly not intended as a realistic physical explanation of the quantization of electromagnetic energy in blackbodies, it does suggest that Planck thought that energy quantization arose from properties of the walls of the cavity, rather than in intrinsic properties of the electromagnetic radiation. Einstein, on the other hand, in assessing Planck’s model, argued that energy quantization was inherent in the radiation itself. In Einstein’s picture, then, electromagnetic energy at a given frequency—whether in a blackbody cavity or not—comes in packets or quanta having energy proportional to the frequency. Each quantum of electromagnetic energy constitutes what we now call a photon, which we may think of as a particle of light. Thus, Planck’s model of blackbody radiation began a rebirth of the particle theory of light. It is worth mentioning, in passing, that in 1900, the same year in which Planck’s paper on blackbody radiation appeared, Lord Kelvin gave a lecture that drew attention to another difficulty with the classical theory of statistical mechanics. Kelvin described two “clouds” over nineteenthcentury physics at the dawn of the twentieth century. The first of these clouds concerned aether—a hypothetical medium through which electromagnetic radiation propagates—and the failure of Michelson and Morley to observe the motion of earth relative to the aether. Under this cloud lurked the theory of special relativity. The second of Kelvin’s clouds concerned heat capacities in gases. The equipartition theorem of classical statistical mechanics made predictions for the ratio of heat capacity at constant pressure (cp) and the heat capacity at constant volume (cv). These predictions deviated substantially from the experimentally measured ratios. Under the second cloud lurked the theory of quantum mechanics, because
61.The Experimental Origins of Quantum Mechanicsthe resolution of this discrepancy is similar to Planck's resolution of theblackbodyproblem.As in the caseof blackbodyradiation,quantumme-chanics gives rise to a correction to the equipartition theorem, thus result-ing in different predictions for the ratio of cp to Cu, predictions that can bereconciled with the observed ratios.1.1.4ThePhotoelectricEffectThe year 1905 was Einstein's annus mirabilis (miraculous year), in whichEinsteinpublishedfourground-breakingpapers,two onthespecial theoryof relativityand oneeach onBrownianmotion andthephotoelectriceffect.It wasforthe photoelectric effectthatEinstein won theNobel Prize inphysics in 1921.In the photoelectric effect,electromagnetic radiation strik-ing a metal causes electrons to be emitted from the metal.Einstein foundthat as one increases the intensity of the incident light, the number of emit-ted electrons increases, but the energy of each electron does not change.This result is difficult to explain from theperspective of the wavetheory oflight. After all, if light is simply an electromagnetic wave, then increasingthe intensity of the light amounts to increasing the strength of the electricand magnetic fields involved. Increasing the strength of the fields, in turn,oughttoincreasetheamount of energytransferred to theelectrons.Einstein'sresults,ontheotherhand,arereadilyexplained fromaparticletheory of light.Suppose light is actually a stream of particles (photons)withthe energy of each particle determined by its frequency. Then increasingthe intensity of light at a given frequency simply increases the number ofphotons and does not affect the energy of each photon.If each photon hasa certain likelihood of hitting an electron and causing it to escape fromthe metal, then the energy of the escaping electron will be determinedby the frequency of the incident light and not by the intensity of thatlight.The photoelectric effect,then,provided another compelling reasonfor believing that light can behave in a particlelike manner.1.1.5TheDouble-SlitErperiment,RevisitedAlthough the work of Planck and Einstein suggests that there is a par-ticlelike aspect to light, there is certainly also a wavelike aspect to light,as shown by Young, Arago, and Maxwell, among others.Thus, somehow,light must in some situations behave like a wave and in some situationslike a particle,a phenomenon known as"wave-particle duality."WilliamLawrence Bragg described the situation thus:God runs electromagnetics on Monday, Wednesday, and Fridayby the wave theory, and the devil runs them by quantum theoryon Tuesday,Thursday, and Saturday.(Apparently Sunday, being a day of rest, did not need to be accounted for.)
6 1. The Experimental Origins of Quantum Mechanics the resolution of this discrepancy is similar to Planck’s resolution of the blackbody problem. As in the case of blackbody radiation, quantum mechanics gives rise to a correction to the equipartition theorem, thus resulting in different predictions for the ratio of cp to cv, predictions that can be reconciled with the observed ratios. 1.1.4 The Photoelectric Effect The year 1905 was Einstein’s annus mirabilis (miraculous year), in which Einstein published four ground-breaking papers, two on the special theory of relativity and one each on Brownian motion and the photoelectric effect. It was for the photoelectric effect that Einstein won the Nobel Prize in physics in 1921. In the photoelectric effect, electromagnetic radiation striking a metal causes electrons to be emitted from the metal. Einstein found that as one increases the intensity of the incident light, the number of emitted electrons increases, but the energy of each electron does not change. This result is difficult to explain from the perspective of the wave theory of light. After all, if light is simply an electromagnetic wave, then increasing the intensity of the light amounts to increasing the strength of the electric and magnetic fields involved. Increasing the strength of the fields, in turn, ought to increase the amount of energy transferred to the electrons. Einstein’s results, on the other hand, are readily explained from a particle theory of light. Suppose light is actually a stream of particles (photons) with the energy of each particle determined by its frequency. Then increasing the intensity of light at a given frequency simply increases the number of photons and does not affect the energy of each photon. If each photon has a certain likelihood of hitting an electron and causing it to escape from the metal, then the energy of the escaping electron will be determined by the frequency of the incident light and not by the intensity of that light. The photoelectric effect, then, provided another compelling reason for believing that light can behave in a particlelike manner. 1.1.5 The Double-Slit Experiment, Revisited Although the work of Planck and Einstein suggests that there is a particlelike aspect to light, there is certainly also a wavelike aspect to light, as shown by Young, Arago, and Maxwell, among others. Thus, somehow, light must in some situations behave like a wave and in some situations like a particle, a phenomenon known as “wave–particle duality.” William Lawrence Bragg described the situation thus: God runs electromagnetics on Monday, Wednesday, and Friday by the wave theory, and the devil runs them by quantum theory on Tuesday, Thursday, and Saturday. (Apparently Sunday, being a day of rest, did not need to be accounted for.)
1.2IsanElectronaWaveoraParticle?In particular, we have already seen that Young's double-slit experimentin the early nineteenth century was one important piece of evidence in fa-vor of the wave theory of light. If light is really made up of particles, asblackbody radiation and the photoelectric effect suggest, one must give aparticle-based explanation of the double-slit experiment.J.J.Thomson sug-gested in 1907 that the patterns of light seen in the double-slit experimentcould be the result of different photons somehow interfering with one an-other.Thomson thus suggested that if the intensity of light were sufficientlyreduced, the photons in the light would become widely separated and theinterference pattern might disappear.In 1909,Geoffrey Ingram Taylor setout to test this suggestion and found that even when the intensity of lightwasdrasticallyreduced(tothepointthatittookthreemonthsforoneoftheimagestoform),the interferencepattern remainedthe same.Since Taylor's results suggest that interference remains even when thephotons are widely separated, the photons are not interfering with one an-other.Rather, as Paul Dirac put it in Chap.1 of [6],"Each photon theninterferes only with itself."To state this in a different way, since there is nointerference when there is only one slit, Taylor's results suggest that eachindividualphotonpassesthroughbothslits.Bytheearly1960s,itbecamepossibletoperformdouble-slit experiments withelectrons instead of pho-tons, yielding even more dramatic confirmations of the strange behavior ofmatter in thequantum realm.(See Sect.1.2.4.)1.2Is an Electron a Wave or a Particle?In the early part of the twentieth century, the atomic theory of matterbecame firmly established. (Einstein's 1905 paper on Brownian motion wasanimportant confirmationofthetheoryandprovidedthefirst calculationof atomic masses in everyday units.)Experiments performed in 1909 byHans Geiger and Ernest Marsden, under the direction of Ernest Rutherford,led Rutherford to put forward in 1911 a picture of atoms in which a smallnucleus contains most of the mass of the atom. In Rutherford's model,each atom has a positively charged nucleus with charge ng, where n isa positive integer (the atomic number) and q is the basic unit of chargefirst observed in Millikan's famous oil-drop experiment.Surrounding thenucleus is a cloud of n electrons, each having negative charge -q. Whenatoms bind into molecules, some of the electrons of one atom may be sharedwithanotheratomtoformabond betweentheatoms.Thispictureofatomsand their binding led to the modern theory of chemistry.Basic to the atomic theory is that electrons are particles; indeed, thenumber of electrons per atom is supposed to be the atomic number. Never-theless, it did not takelong after the atomic theory of matter was confirmedbefore wavelike properties of electrons began to be observed.The situation
1.2 Is an Electron a Wave or a Particle? 7 In particular, we have already seen that Young’s double-slit experiment in the early nineteenth century was one important piece of evidence in favor of the wave theory of light. If light is really made up of particles, as blackbody radiation and the photoelectric effect suggest, one must give a particle-based explanation of the double-slit experiment. J.J. Thomson suggested in 1907 that the patterns of light seen in the double-slit experiment could be the result of different photons somehow interfering with one another. Thomson thus suggested that if the intensity of light were sufficiently reduced, the photons in the light would become widely separated and the interference pattern might disappear. In 1909, Geoffrey Ingram Taylor set out to test this suggestion and found that even when the intensity of light was drastically reduced (to the point that it took three months for one of the images to form), the interference pattern remained the same. Since Taylor’s results suggest that interference remains even when the photons are widely separated, the photons are not interfering with one another. Rather, as Paul Dirac put it in Chap. 1 of [6], “Each photon then interferes only with itself.” To state this in a different way, since there is no interference when there is only one slit, Taylor’s results suggest that each individual photon passes through both slits. By the early 1960s, it became possible to perform double-slit experiments with electrons instead of photons, yielding even more dramatic confirmations of the strange behavior of matter in the quantum realm. (See Sect. 1.2.4.) 1.2 Is an Electron a Wave or a Particle? In the early part of the twentieth century, the atomic theory of matter became firmly established. (Einstein’s 1905 paper on Brownian motion was an important confirmation of the theory and provided the first calculation of atomic masses in everyday units.) Experiments performed in 1909 by Hans Geiger and Ernest Marsden, under the direction of Ernest Rutherford, led Rutherford to put forward in 1911 a picture of atoms in which a small nucleus contains most of the mass of the atom. In Rutherford’s model, each atom has a positively charged nucleus with charge nq, where n is a positive integer (the atomic number ) and q is the basic unit of charge first observed in Millikan’s famous oil-drop experiment. Surrounding the nucleus is a cloud of n electrons, each having negative charge −q. When atoms bind into molecules, some of the electrons of one atom may be shared with another atom to form a bond between the atoms. This picture of atoms and their binding led to the modern theory of chemistry. Basic to the atomic theory is that electrons are particles; indeed, the number of electrons per atom is supposed to be the atomic number. Nevertheless, it did not take long after the atomic theory of matter was confirmed before wavelike properties of electrons began to be observed. The situation