Ea rly Quantum T heo ry - Michigan State University · Ea rly Quantum T heo ry In sum , one can sa y that there is hardly one among the great pr oblem s in w hi ch m o de rn ph ys

Chapter 1

Early Quantum TheoryIn sum, one can say that there is hardly one among the great problems in which modern physics is sorich to which Einstein has not made a remarkable contribution. That he may sometimes have missedthe target in his speculations, as, for example, in his hypothesis of light-quanta, cannot really be heldtoo much against him, for it is not possible to introduce really new ideas even in the most exact scienceswithout sometimes taking a risk.

Planck and others nominating Einstein for membership in the Prussian Academy of Sciences, 1913.

We have seen just how unusual the ideas of Relativity were and, how slowly they were adopted by thephysics community. In some ways, the insult to common sense that was Relativity is secondary to theinjury that became Quantum Mechanics. By the 1920’s, like a slow-motion, one-two punch, physics realizedthat it had been rocked by the tag-team of Relativity and Quantum Mechanics which called into questionbasic facts about nature, but also the status of Reality and Knowledge. Between Relativity and QuantumMechanics, Ontology and Epistemology—what is and what we can know—took their biggest hits sincePlato.

1

CHAPTER 1. EARLY QUANTUM THEORY 2

Complicated ideas sometimes are unfairly reduced to buzz words and the quantum theory is no di!erent.Its famous buzz words are perhaps: “wave-particle duality” and “uncertainty.” Like all bumper-sticker-sized phrases, they leave a lot unsaid, while they still hint at a part of the truth. Like many revolutionaryideas, the realization that waves and particles somehow share the same reality emerged as the unexpectedand unwelcome outcome of ordinary scientific problem-solving. Remember, this is not a new idea. IsaacNewton had believed that light was made up of “corpuscles,”particles, if you will. He had his reasons, buteven during his time there was evidence to the contrary. But, he was Newton and Newton’s ideas prevailedin England long after his death and long, after they should have yielded to the evidence.

The matter of waves vs. particles as a description light appeared to be settled in the early years of the1800’s when Englishman Thomas Young found and demonstrated interference e!ects of light that couldonly happen if it was a wave1. When Maxwell’s theory of electromagnetism was found to be well-foundedmathematically and experimentally and encompassed Optics as well as electricity and magnetism, theconclusion was clear: By 1900, waves, it was.

I’ll bet you all swallowed that paragraph. Did you notice that contained a gigantic assumption? There wasindeed indisputable evidence for the wave-like nature of light. However, it was always assumed that thereality of waves was simultaneously a disproof of particles. Nobody questioned this either-or presumption,until Einstein.

1.1 Heat Radiation

George Carlin says that you can’t “preheat” an oven (just like you can’t “preboard” an airplane). An obvi-ously trivial observation and yet subtly true in a very deep sense. All objects, above absolute zero of tem-

1The attacks on Young for this a!ront to Newton’s name were so vicious that he dropped out of science and delved intolanguages, solving the riddle of the Rosetta Stone, among other accomplishments.


Few cities in the world were as self-confident as turn-of-the-century Berlin. Since the political unification of Germany in the1870’s, an expectation of and realization of progress in industry, arts, science, and militarism characterized the German outlook.It was at once an optimistic intellectual environment, unusually accepting, especially of intellectual Jews, while simultaneouslya conservative and authoritarian society with open anti-Semetism. It was these latter aspects which troubled the young Einsteinand the former which eventually attracted him to Berlin. Certainly Science had become the German metaphor for progresswith and the similar commitments inherent in Science and Religion became a popular analogy. Unusually connected withprogressive industry, it attracted the "best and brighest" to physiology, chemistry, and physics, with successful members ofone field often trained in another—Helmholtz, for example, trained as a practicing physician and physiologist.

Box (1.1) Berlin, 1900

perature (which cannot be reached...so, I guess it’s all objects, then), emit heat. In the mid-19th century, itwas realized that this heat radiation was likely just a longer-wavelength version of Maxwell’selectromagneticradiation, just a little longer than the color red. So, the idea was broadened: all objects radiate at allwavelengths, including those in the infrared region. “Warm” objects radiate a lot in that region, while“cold” objects radiate much less.

Suppose you’re on a camping trip with a hot, open fire. If you have a fire pit circled by rocks, even afterthe fire has been put out, without even touching, you can tell which rocks were close to the fire. If onewas close to the coals, such a rock would be still warm, maybe even glowing—it emits electromagneticradiation predominantly in the part of the spectrum that’s in the infrared, (the heat part) and maybe abit in the visible (the glowing part). Your hand detects the heat wavelengths, your eyes, the visible.

In the morning, the rock is cold since during the night it continued to emit radiation into the atmosphere(warming up the air...a little) until it reached the temperature of its surroundings. We say it “cooled,”but a microscopic observer in the rock would say that the agitated molecules of the rock’s atomic latticeconsiderably slowed down their vibratory motions. Because these molecules involve electric charges and,as we have seen, the acceleration of charges produces electromagnetic radiation, they emit waves that youcall predominantly heat and light.


But, by that cold morning had it stopped radiating? No, the molecular motions have not stopped, but theradiation is much di!erent in character, it’s not so concentrated in the infrared portion of the spectrum,and it’s resumed its dark, rock-like appearance. Suppose, once the rock was hot the previous night youhad gingerly picked it up and placed it in an enclosed box, with un-shiny, thermally insulated walls. Now,just like overnight, the rock continues to emit electromagnetic waves of energy...but they don’t disappearinto the atmosphere, they encounter the walls, which warm up and themselves emit back into the enclosureand so on and so on. It would classically appropriate to imagine a linguine mixture of waves inside thebox, at hundreds of di!erent wavelengths. But, this soup, including the wall’s unceasing absorption andemission, has unusual characteristics and Einstein’s eventual model even envisioned it, not as a collectionof waves, but as a radiation gas! Particles.

At some point, the walls and the rock will reach a state in which they each emit as much radiation asthey absorb. If there is no reflection, the absorption becomes total at all wavelengths and its emission isonly due to the thermal motions of the wall’s surface: such completely absorbing (and emitting) objectsare called “blackbodies.” You can make one: take a white box and cut out a hole and look inside: youwould look in and see...nothing. Blackness. There is no radiation coming out that tells you the whitecharacteristics of the wall. Any radiation that escapes through a hole in a blackbody cavity is an unbiasedsample of the radiation that’s filled the box and “bouncing” all around.

Makers of china, blacksmiths, sword-smiths...craftspeople with an ovens and a need to bring products tocertain temperatures have for centuries learned that they can accurately guess at the temperature of aglowing object by looking at its color—sword2, porcelain, or whatever. This trick of relating color totemperature was found to be a universal phenomenon...it doesn’t matter what the material is, radiationpatterns will be the same for objects at the same temperatures. This was even proved theoretically in themid-1800’s by Kirchho! and ovens with holes of the sort described above were crafted in order to preciselymeasure and characterize this radiation.

2The art of forging samurai swords is, by legend, 1300 years old. Among the instructions are to heat the sword “until itturns to the color of the moon about to set out on its journey across the heavens on a June or July evening.”


The results precisely confirmed the idea that the amount of radiation emitted by a blackbody is independentof the material. Understanding such a universal phenomenon must teach something important about itsphysical mechanism and since Maxwell’s electromagnetism had only been shown to be such a good theoryin the 1880’s or so, people at this time were working hard to gain such an understanding. They failed.Miserably.

1.1.1 Early Research

There were essentially two issues that physicists thought should be understandable about radiating objectsusing the new electromagnetic theory: the total amount of energy radiated at a given temperature and thefrequency spectrum of the radiation—how much energy is radiated at each frequency. In terms of our rockthese issues would include things like: how hot does your hand feel; in addition to the rock being warm,does the rock also glow (radiation in the infrared and the visible region?) so how much of the radiationenergy comes from heat and how much from visible light. As new instruments were developed, physicistswere able to measure the total intensity and then the intensity at a given wavelength. It’s like knowingthe final score of a baseball game, and in addition, knowing the inning-by-inning totals.

A number of measurements were done with laboratory-quality blackbody cavities toward the end of the19th century. However, some of the most interesting early precision experiments were done using hot arclamps by Josef Stefan inferred 1879 that the amount of energy radiated per unit area per unit time (calleda “flux”) was proportional to a function of the temperature of the radiator. Today, we call this Stefan’sLaw:

u(T ) = !T 4, (1.1)showing that the functional relation is the fourth power of the temperature. The constant of proportionality,


now called Stefan’s Constant, has the modern value of ! = 5.670400(40) ! 10!8J·s!1·m!2·K!4. (Noticethat the units are energy per unit time per unit area per T!4 as you would expect.) This is a generalrelationship between the energy and the temperature and so ! is independent of the material.

The reason it’s an interesting story is that: after understanding laboratory-based arcs, by comparison withlight from a telescope, he then took the temperature of the sun and found that its surface temperaturemust be approximately 5430C. Further, in 1885 his empirical relationship was derived theoretically by hisstudent Boltzmann in 1884 who derived it from thermodynamic and electromagnetic theory.

At this point, the story is really interesting: the total amount of radiation is precisely measurable inthe laboratory and those results are deducible from accepted theory. So. Any more detailed account ofblackbody radiation should include the Stefan-Boltzmann Law logically within it. This satisfies the firstmatter of interest—how much radiation is emitted at a given temperature. Now, the interesting questionis how much radiation is emitted at each wavelength. ?

Figure 1.1 shows schematically how this was done. Precision blackbody cavities were machined withuniform characteristics and heated by a furnace. Radiation was collected from a hole and passed through aslit to define its shape into a ribbon. It was then passed through a pure glass prism to refract the di!erentwavelengths into di!erent angles. Then a device called a bolometer sampled the radiation and determinedits strength at each angle. That strength was then plotted at each angle (converted to wavelength orfrequency) in order to produce its distribution.

1.1.2 The Blackbody Spectrum

Let’s skip ahead a little and see what the actual energy distribution from radiating substances looks like.Figure 1.2 is particularly instructive for a number of reasons. It shows the energy density (an energy per


Figure 1.1:

unit volume within the blackbody cavity) at di!erent wavelengths for blackbodies at two temperatureswhich are a factor of 2 apart: 5780K and 2890K. (Wait a second for why such odd choices.) What thisflux distribution means is the following: Look at the yellow curve which peaks at a wavelength just under" "= 0.5!10!6 m. (This is a half of a micron, µm, or it’s also 500 nm, or 5000Å. Electromagnetic radiationof this wavelength is in the visible wavelength band and about corresponds to the color cyan.) Follow theyellow curve to the right, from the maximum value, until a point halfway down. This point correspondsto about " "= 0.9µm so about half as much energy would be radiated by such an object to a given area ina given time at a " "= 0.9µm which is now outside of the visible range, into the near-infrared. A reflectionof the fact that when an object that’s glowing-hot cools, it ceases to glow, but still remains warm.


Figure 1.2: This shows the energy emitted by a blackbody at di!erent wavelengths for objects at twotemperatures, di!ering by a factor of 2: 5780K (yellow) and 2890K (red). The inset shows the hotter curveon an expanded horizontal scale with the visible light wavelength range indicated.

This figure demonstrates a couple of interesting things: First, notice how tiny the red curve is relativeto the yellow—that factor of only two in temperature changes considerably the radiative flux, which iswhat Stefan suggested: The total amount of energy received by a given area during a given time is the


area under these curves and if the temperature changes by a factor of two, then the areas will di!er by afactor of 24 or 8. You instinctively know this already. Think about how di!erent your skin feels if exposedfor an hour on a 90 degree day in August as compared with one in October at 45 degrees. Second, lookat the inset figure, which is just a replicate of the yellow curve on an expanded horizontal scale. Thewavelengths of light visible to humans is bounded by the two vertical arrows at about 400nm (aroundviolet) and 750nm (red). On the high side of this boundary is the infrared wavelength band, which is quitewide and extends way o! the scale to the right. On the low side of the band is the ultraviolet region,bounded below at about 10nm3. That this particular temperature includes mostly the visible wavelengthsis probably not an accident: the yellow curve corresponds to the temperature of the surface of the sun. Ofcourse, the name “visible” wavelengths is only a human-physiological distinction. That we have evolved avision sensitivity in the wavelength range dominated by the sun’s emission spectrum seems pretty natural.Further, some animals such as snakes and bats have sensitivities further to the infrared than we, indicatingan evolutionary preference for detection of heat (the infrared region), while others, like penguins, can senseinto the ultraviolet region.

There is another temperature scale of interest to us, namely “room temperature,” where we are mostcomfortable, at about 300K. Figure 1.3 shows the same solar spectrum with that of a blackbody at aroundroom temperature. (The latter is multiplied by a factor of 2,000,000 in order to show its wavelengthdistribution more prominently.) Notice that it peaks at about " "= 10µm, which is just about at thewavelength at which humans radiate most strongly. We are all pretty good blackbody radiators whichhelps to explain why a lecture hall can become warm when full of 200 students, each radiating like a 300Watt light bulb.

3Notice that there is a little ultraviolet light (to the left of the purple arrow) emitted by the sun’s surface, but even thatamount would be deadly to animal and plant life. The actual distribution of radiation on the Earth’s surface is di!erent fromthat shown in 1.3 as the atmosphere absorbs and emits in a non-uniform way at di!erent wavelengths. In particular, theatmosphere absorbs in the ultraviolet wavelengths due to molecular O3- Ozone. The infrared band to the right of the redarrow is also depleted by water vapor. So, the visible wavelengths are even relatively more pronounced on the surface of theEarth than the blackbody spectrum suggests.


Figure 1.3: On the same horizontal scale as Figure 1.2 is plotted the Sun’s blackbody spectrum as well asthat of a blackbody at a temperature of 300K...times a factor of 2 million. The inset shows an infraredcamera image with a Fahrenheit temperature scale calibrated to the colors. This lady has a cold nose.

1.1.3 Classical Physics Attack on Blackbody Radiation

So, understanding this universal phenomenon became a research problem of considerable interest and highstakes. Figure 1.4 shows some turn-of-the-century experimental results with measurements noted as thecircles. These are very precise measurements, their experimental error bars are inside of the little circles.The dashed curves show the best early e!ort at modeling these results by Wilhelm Wien in 1896. These are


good fits to a particular function of temperature and frequency (or wavelength) in terms of two parameterswhich he had to extract from the shapes. Initially, it looked like a part of the truth, as it fit all of the data.However, this success had two problems associated with it: the function that he chose had no real physicalmodel to motivate it or that could be suggested by it—it was just a fit of a trial function to arbitraryparameters. Such a deficiency could just be a lack of imagination to that point, but second problem was itsdownfall: Prior to about 1900, experiments had been done on solar or hot, electrical arcs or heated wireswhich are very hot and similar temperature ranges. As experiments became more and more sophisticated,physicists were able to measure increasingly longer wavelengths and it became apparent by 1899 thatWien’s fit failed completely in the infrared region as can be seen by the dashed curves. Something elseneeded to be done and some of the giants of theoretical physics attacked the problem, among them one ofthe most noble...literally, noble of them all. The great John William Strutt (the future

Lord Rayleigh) had a problem when he wasa student—he was brilliant and at the top ofhis class at Cambridge. But, his problemsare not like our problems: as a future BaronRayleigh, a scientific career was considered asignificantly lower-class occupation and was notpopular with his family. Nonetheless, he pur-sued his calling and, as a wealthy man, didn’tneed an academic livelihood and could devotehimself to independent experimental and math-ematical research, which he did at his family es-tate. See, not our kind of problems. He was thefirst to explain, among many other things, whythe sky is blue and he shared the 1904 NobelPrize for his discovery of the element Argon.

In 1900 Lord Rayleigh attacked the blackbody radiation problem in a program of research which lasteduntil after 1905. With James Jeans, they applied the most sophisticated application of classical electromag-netism, mechanics, and thermodynamics attempted. They using the model that vibrations of the atomsin a radiator’s walls radiated and set up electromagnetic standing waves of many frequencies. By usingthe accepted ideas of how energy is distributed among components of a system that the energy densitydistribution of these waves came out to be:

UR(#, T ) =8$#2

c3kBT (1.2)

where kB is Boltzmann’s Constant and # refers to the frequency of the radiation. This formula fit the datain the low frequency (large wavelength) region—where Wein’s formula failed. But, it is obviously nonsense!The energy increases according to the square of the frequency—without bound! There is no upper limit andthe energy could approach infinite quantities, an embarrassment that was given the nickname “ultravioletcatastrophe.” Within the rules of Maxwell’s electromagnetism and the energetics of thermodynamics andmechanics, this calculation is unassailable. It has to be right, or the rules are wrong. The Rayleigh-JeansFormula became the poster child for the failure of classical physics.


Figure 1.4: Experimental (the circles) results for blackbody radiation of the noted temperatures. Thedashed curve is Wien’s formula and the solid curves are from Planck’s relation.

The story of quantum theory is often told as if the eventual solution was pursued as a reaction to thefailure of the Raleigh-Jeans formula. In fact, their work was published after Max Planck had solved theproblem. Yet, nobody doubted that some missing ingredient to the Rayleigh Jeans model was all that wasnecessary in order to rid it of its embarrassment and nobody accepted the bizarre, real solution, includingits father.


1.2 The Quantum Is BornMax Karl Ernst Ludwig Planck received the No-bel Prize in Physics in 1918 "in recognition ofthe services he rendered to the advancementof Physics by his discovery of energy quanta."From his Nobel address: "But numbers decide,and the result is that the roles, compared withearlier times, have gradually changed. Whatinitially was a problem of fitting a new andstrange element, with more or less gentle pres-sure, into what was generally regarded as a fixedframe has become a question of coping withan intruder who, after appropriating an assuredplace, has gone over to the o!ensive; and to-day it has become obvious that the old frame-work must somehow or other be burst asunder.It is merely a question of where and to whatdegree. If one may make a conjecture aboutthe expected escape from this tight comer, thenone could remark that all the signs suggest thatthe main principles of thermodynamics from theclassical theory will not only rule unchallengedbut will more probably become correspondinglyextended."

It’s not unusual to find that physicists, theoretical and experimental, often work on problems that fit theirown specialties. Like most people, they are comfortable in some areas, and less so in others. Max Planck,while at the top of his career in around 1900, was an expert in thermodynamics and less enamored of theuse of statistical methods in physics championed by Boltzmann and others. As a result, he came to theproblem of blackbody radiation with di!erent tastes and approaches than those employed by Raleigh andothers. In December of 1900 after a series of fits and starts, he managed to account for the full curvethat fit the intensity of radiation as a function of wavelength. He announced it in a meeting in his homeinstitution in Berlin and then one of his experimental colleagues spent the whole night comparing it withthe most recent infrared data that were fresh from experiments in that institution: it was bang-on.

So, great. A curve that fit the data. At first, this success was, in a way, mechanical: he introduced anoscillator model like Rayleigh and added some mathematical steps (following Boltzmann) which were verydi"cult to interpret, but which seemed necessary. He was a fine physicist and was devoted to going beyondjust producing a formula that fit data—he needed an actual physical interpretation to go along with it andwas determined to find one.

But, even if the radiation formula proved to be perfectly correct, it would after all have been only an interpolationformula found by lucky guess-work and thus would have left us rather unsatisfied. I therefore strived from the day ofits discovery to give it real physical interpretation...After some weeks of the most intense work of my life, light beganto appear to me and unexpected views revealed themselves in the distance. (Max Planck, 1920 Nobel Address)It was an act of desperation. For six years I had struggled with the blackbody theory. I knew the problem wasfundamental and I knew the answer. I had to find a theoretical explanation at any price...(Max Planck, 1931 corre-spondence)

Planck’s formula had a number of serious things going for it: it fit the data over all of the wavelengthsthat were measurable; it reduced to the classically-acceptable Raleigh-Jeans formula at small frequencies,and it contained within it the Stefan-Boltzmann result when all of the intensity was added up at each


Figure 1.5: Max Planck (1858-1947), German theoretical physicist and 1918 Physics Nobel Laureate.

wavelength. Further, he could predict Avogadro’s Number, Boltzmann’s Constant, and the electric charge.The exchange for all of this success was the introduction of a bizarre idea and the tiniest fundamentalconstant into physics yet devised.

His was a model, not so much of the radiation, but of the walls of the box. After all, that’s what wasresponsible for the radiation and that’s what was thought to be the heart of the problem. Further, thewalls of his radiator were presumed to be oscillators, much like those of Raleigh, but how their motionscame to equilibrium with the radiation that it produced/absorbed was totally di!erent. For any givenfrequency, he found that he had to limit the energies that would be radiated into what he called “packets”or “quanta4.” The energies at a given wavelength could not become arbitrarily small, but they were limitedto a finite, but tiny value. In fact, he worked in frequency, rather than wavelength, and that’s a part of

4Here is the origin of the word “quanta” or “quantum.” It is Latin in origin for “quantus,” meaning “how much.”


the clue of what becomes the eventual weird, but correct description. So, we’ll start speaking of frequencyfrom this point, rather than wavelength.

Suppose an oscillator in Planck’s imagined wall vibrates at given frequency, f5. Planck’s Formula is

UP (f, T ) =8$f3

c3

h

ehf/kBT # 1(1.3)

and it forced on him was the interpretation that the energy emitted could not be any lower than % = nhfwhere n is an integer, 0,1,2... and h is a new constant of nature, now called Planck’s Constant with amodern value of 6.6260755(40)!10!34J·s. So, for wavelengths corresponding to visible light, say few !1014

cycles per second, the smallest energy that can be radiated is h times that of order few !10!19 Joules...notzero. The energy can go to zero, but only if the frequency of the radiator goes to zero—that is, if it does notvibrate. For any finite frequency, the radiated energy is finite and equal to h#, 2h#, 3h#, ...and so on. Theidea is precisely as if on a child’s swing, the frequency of the swing is only adjustable in discrete amounts.No matter how precisely a steadily increasing pushing-force is calibrated, no change is observed in the rateof swinging back-and-forth until a “quantum” of energy is delivered and then the swing would jump to thenew frequency all at once, corresponding to the energy going from nhf to (n + 1)hf . This weird hf is atiny, tiny energy and could indeed be mistaken for zero in anything but an atomic environment. But...theidea that it could not be zero in principle, was fundamentally disturbing and Planck fought against hisown interpretation6. In that fight he was in good company, since nobody liked his idea. Nobody, exceptEinstein.

Just how Planck dealt with his forced-hand is controversial. It was clear that there was a huge paradox in5I’m intentionally using ! to represent the frequency of the radiation and f to represent the frequency of the walls’

molecular vibrations. You’ll see.6Although he suspected that he had stumbled onto something significant and on a stroll with one of his young sons

indicated that he had had “a conception today as revolutionary and as great as the kind of thought that Newton had.”


his logic and his derivation ignores it: he equated the radiators (wall oscillators) and the radiated (elec-tromagnetism) by literally allowing the former to be discrete and the latter to be explicitly the continuouselectromagnetism of Maxwell and Lorentz. This is a flawed logic and probably was why Planck himself wasso equivocal. His actions in the years that followed his 1900 announcement were not always on behalf ofthe very ideas that he originated. This is not the mark of someone behaving unscientifically...it’s the markof someone facing nearly insurmountable conceptual di"culties. His scientific stature surely even raisedthe stakes for him personally, as when one is as distinguished as he was, a mistake could carry a significantembarrassment. But, in spite of that risk, Planck followed the physics as far as he could, publicly, andforthrightly.

If one compares Eq.1.2 with Eq. 1.3, one can see the di!erences and the similarities. The former isproportional to T, the latter’s temperature dependence is buried. The classical, characteristic unit ofenergy of interest is kBT but in Planck’s formula, it’s h#. One can see that in each formula, that they areseparately proportional to their respective characteristic energies (leaving a #2 dependence in the Planckformula). But, that the new description carries the now important exponential term. With respect to thatterm, notice how Eq. 1.3 behaves well: for energies of the oscillator (hf) which are very much less thanthe characteristic thermodynamic energy of the system (kBT ) that the exponential can be expanded usingthe Binomial Theorem and the Planck Formula reduces to the classically-correct Rayleigh-Jeans formula.

This is characteristic of quantum phenomena: classical results can be extracted as a limit in which thequantum energy (characterized in magnitude by Planck’s Constant) is tiny relative to some other energy. Inthis sense, quantum ideas had been hidden from view while typical energies were large, just like relativistice!ects were unnoticed as long as characteristic speeds were much smaller than those of the speed of light.Likewise, Wein’s Law is recovered from Eq. 1.3 by allowing the frequency to become very large. Finally,by integrating Eq.1.3 over all wavelengths and correcting from energy density to power density (energy perunit area, per unit time) gives Stefan’s Constant in terms of other fundamental constants:

u(T ) =2$k4

15c2h3T 4.


Planck was a leader of European physics for many years. His position at the most prestigious researchinstitute was well-earned and his perspective was extremely influential. As I noted earlier, it was Planckwho first took Einstein’s relativity ideas seriously, which must have had a sobering e!ect on the youngclerk. It was Planck who eventually attracted Einstein to join him in Berlin and it was Planck who wasforthright and honest during the Nazi takeover of Germany. He remained, but tried unsuccessfully to usehis influence directly with Hitler on behalf of individuals and the political climate generally.

Planck’s personal life was di"cult and in many ways, unbearably sad. He lost a son in World War I, twoof his three daughters (twins) each giving birth as young women, and his wife in 1909. His oldest son wasexecuted by the Nazi’s during World War II as he was captured as a part of a failed assassination attempton Hitler’s life. At the end of the war, he was barely rescued by Allied forces in Berlin as the Russian armyapproached. He was by then a frail, homeless elderly man.

1.3 The Quantum Grows Up

We’ve already met Einstein’s singularly unusual way of looking at the world through relativity. In thatsame year that he was not working very hard at the Patent O"ce, he was literally changing the world inone time-gulp: remember, in 1905, now called his “miracle year,” he published his relativity theory, provedthe existence of atoms by explaining Brownian Motion, and correctly reinterpreted Planck’s formula as adescription of light. As I write this, the year is 2005. While the physics profession celebrated the centenaryof the “birth of the quantum” in the year 2000, as we’ll see there is an argument that that party was fiveyears too early and that Einstein and Neils Bohr are the first Quantum Mechanics.


1.3.1 Photoelectricity, Revisited

Recall that one of the results of Hertz’s experiments was the inadvertent discovery that illuminating somemetals with ultraviolet (UV) light prompted a current to flow from the surface of the metal. This currentcould be accelerated in an electric field and was determined to be negative and likely to be the electrondiscovered by J. J. Thomson. But, the characteristics of this photocurrent were inconsistent with whatone would expect employing a Maxwellian electromagnetism description of the illuminating UV light:

• If one changed the intensity of the illuminating light, the kinetic energies of the emitted electronsdid not change.

• If one changed the intensity of the illuminating light, the current did increase, as if there were moreelectrons.

• If one increased the frequency, #, of the light, the kinetic energies increased.

• If one switched the light source on and o!, the photocurrent started/stopped immediately.

All four of these results were counter to what would be expected if UV light acted as waves and shook theatoms of the material until they released photoelectrons. While the early observations were qualitative,by the early 1900’s physicists were beginning to make very precise measurements of the characteristics ofphotoelectricity and the precision continued as surface preparation and vacuums improved. By the late1800’s it was clear that the following was the sequence of events in establishing a photocurrent: one shinesUV light on a highly polished surface of (predominantly, an alkaline) metal in a good vacuum. Beginningwith a very low frequency, nothing is observed until at some particular frequency, #0, a small photocurrentbegins to flow from the surface. As the frequency is increased beyond that minimum, the kinetic energiesof the electrons grows: E(photoelectrons) " f(#).

Not only did the observations appear to contradict a wave-like behavior for the UV light, it also ran counterto the notions of what held electrons in the metal. It was believed that either the electrons were freely


moving in the metal or that they were bound in oscillators. In either case, the application of a wave oflight energy should cause them to absorb that energy and eventually break free, either of the surface or theoscillator. Calculations suggested that the delays that would be expected would be of the order of secondsor even a minute or so. And yet, the near-instantaneous release of photoelectrons was measured to be onthe order of nanoseconds. So, the problems with photoelectricity caused problems for both Maxwell andLorentz’ electromagnetism as well as the most promising notions of the electronic structure of matter. Itshould be noted that there was known another way to get electrons to be ejected from metallic surfaces...byheating them. This “thermionic” emission mechanism had a similar tendency for the electrons to not beemitted until a minimum amount of energy in the form of heat was applied. This minimum energy iscalled the work function and was the same for similar materials as the low-frequency threshold for photo-emission. This led to the conclusion that the photoelectrons and thermionic electrons originate from thesame material structure.

1.3.2 Einstein, #1

It was into this fray that Einstein leaped with his first crucial paper of June 9, 1905, “On a Heuristic Pointof View Concerning the Generation and Transformation of Light.” (His first relativity paper was secondthat year, the Brownian motion paper, third.) He had been a “closet” devotee of Maxwell’s theory, havinglearned it on his own, given the reluctance in the German and Swiss-German university system to sacrificeGerman theories of light in favor of the British Maxwell theory. So, it was not taught. To its successin explaining the propagation of free electromagnetic waves in optics and other frequencies, Einstein wasunstinting in his praise. It works.

However, in description of interactions between light and matter there were unexplained di"culties: namely,blackbody radiation (the emission and absorption by material oscillators) and the photoelectric e!ect. Heemphasized the hole in Planck’s argument that had linked together quantized and continuous phenomenaand drilled right into it proposing that not only would the radiators oscillate according to quantized


frequencies, but that that the quantization would remain...a kind of lumpy, resultant wavefront—not themore familiar continuous wave:

“monochromatic [constant frequency] radiation of a small density...behaves as if it consisted of independent energyquanta of magnitude [h!].”

This lumpiness persists after emission, and as the light spread from the point of emission, he acknowledgedthat it would become less intense, but not as the reduction of the amplitude of a wave, but as the result ofthe lumps getting further away from one another at the “wavefront.” The lumps eventually got a name,although not from Einstein and not immediately—they were eventually named “photons” by G. N. Lewisin 1926.

So, this quantum picture is not one that springs easily to the mind! How do all of the interferencephenomena that could only occur with wave-like properties be explained in a lumpy way? Previously“obvious” wave-like observations would have been done with bright light, which in Einstein’s picture, islight containing enormous numbers of photons. Then, he argued, the wavelike properties are more apparentas a kind of cooperative relationship among the photons. But, when one is dealing with the emission ofsingle or few oscillators, then the lumpiness becomes apparent when the large numbers cannot hide light’sparticulate nature.

If we take # to be the frequency of the light, Einstein was proposing that the energy of the particles oflight was identical to Planck’s formula for the oscillators. For the energy of a single quantum of light—aphoton:

E = h#, (1.4)while the energy of an intense beam of light of frequency # the energy would be

E = nh# (1.5)


where now that intensity is the sum of n individual photons.He considered current and future experiments and made precise predictions. For example, his explanationof photoelectricity was natural in this “billiard ball” picture. When a photon enters a material, it carrieswith it energy according to Eq.1.4. If it has enough energy (a high enough “frequency”), it can kick abound electron free of the material. If it does not have enough energy to liberate an electron from itsatom, it doesn’t. As the frequency goes up, the amount of kinetic energy that can be delivered to anelectron goes up, with the maximum amount corresponding to the whole energy of the photon. Then,energy conservation would demand that

energy of photon = energy to free electron + KEmax (1.6)

h# = & +12me(vmax)2 (1.7)

where & is the work function. When turned around to be the maximum kinetic energy of an ejectedphotoelectron,

max(KEe) = h# # & (1.8)the linear dependence on frequency in Equation 1.8 functioned as a prediction.But, while there were many attempts (and many di!erent results) in later years at trying to figure outthe correct functional dependence of the kinetic energy and frequency, it was not until about 1914 thatthe linear relationship predicted by Eq.1.8 gained ground and, in a series of famously precise and carefulexperiments in 1916, (reluctantly) determined to be correct. I say reluctantly, as it was Robert Millikan atthe University of Chicago who performed them, first calling Einstein’s idea “bold, and not to say, reckless”and then later lamenting:

I spent ten years of my life testing that 1905 equation of Einstein’s and contrary to all my expectations was compelledin 1915 to assert its unambiguous verification in spite of its unreasonableness, since it seemed to violate everythingwe knew about the interference of light.

He learned to console himself with his Nobel Prize.


1.3.3 The Compton E!ect

Einstein has not ever been called a reluctant revolutionary, as certainly was Planck. Rather, he was wellaware of the revolutionary aspects of all of his 1905 work, but especially this idea. That is not to say thathe did not waiver, sometimes he insisted that it was only a provisional theory7. But, he was persistent andfound new ways to approach the radiation problem as he mulled the consequences of his original hypothesis:

I have already attempted earlier to show that our current foundations of the radiation theory have to be abandoned...Itis my opinion that the next phase in the development of theoretical physics will bring us a theory of light that can beinterpreted as a kind of fusion of the wave and the emission theory. (1909)

Finally, far from giving up, in 1916 he kicked it up a notch to conclude that the light quantum would notonly kick out electrons, but if it hit a molecule, then that molecule would recoil mechanically and that thephoton that rebounded o! would itself have to change its frequency in order to conserve momentum.

Remember the relativistic formula for the total energy of an object having momentum, p:

E =!

p2c2 + m2oc

4. (1.9)

Although it was not originally a part of the plan, if a body has no rest mass, then the energy can be writtenas simply

E = pc = h# = hc/", so :

p =h#

c=

h

"

7A weakness that Millikan seized on in 1907, “I would like to tell you how pleased I am that you have given up yourlight-quantum hypothesis.”


Here we have the application of relativity to quantum mechanics for the first time. The momentum of a zeromass object is related to its frequency, or equivalently, its wavelength. This means that if the momentumof a photon changes, its frequency changes; if the frequency changes, that changes its momentum. So, ina collision in which a photon collides with a molecular target some of its momentum will be given to thetarget and a reduced amount would stay with the photon...so, that frequency of the recoiling photon wouldbe di!erent. The momentum of blue light is higher than the momentum of red light, the momentum ofChannel 2 photons is less than the momentum of your dentist’s X-rays.

By 1923—eight years after Einstein’s idea of the photon momentum, and seventeen years after his originalprediction of photons, Arthur Holly Compton, an American, succeeded in slamming the door tightly againstany doubt of the particle nature of light. He studied the reaction

X rays + Carbon $ X rays + Carbon' + e $ '" + e".

This is a standard notation, where the Greek letter gamma (') always represents a photon and the primesindicate here that the scattered particles have di!erent characteristics from the initial ones. The use of Xrays was in part to facilitate the measurements of the final state, as we’ll see. But, they are also su"cientlyhigh in energy that the electron in the Carbon target is essentially at rest relative to the incoming photon.This greatly simplifies the mathematics of what to expect. It is strictly “billiard ball” kinematics, albeitwith tiny, bizarre billiard balls.

Here is the scenario: imagine that the initial state photon has an energy of % and that the initial stateelectron has momentum p and energy E. The final state objects’ momenta and energy will be labeled withthe primes. So, a fancy way of writing the reaction is


'(%) + e(p, E) $ '"(%") + e"(p", E").

1.3.3.1 An Interlude: Feynman Diagrams

Although it advances the history a bit, there is a great deal of insight gained through the use of FeynmanDiagrams. Richard Feynman always saw things a little di!erently from most everyone else. Sometimes itwas an unprecedented insight to the actual science that he would be the first to uncover. Often, it wouldbe his truly unique way to make di"cult things simple. This was most characteristically demonstratedwhen, as a member of the Challenger Space Shuttle investigation committee after listening to all kinds oftechnical engineering and materials science discussion of what may or may not have happened, he took apiece of the sealer material that was suspect as a source of the fatal Oxygen leak...and dropped it in hisglass of ice water. It solidified and was easily cracked. Feynman knew that there had been an unusualfreeze the night before the launch. The sealer got cold. Quintessential Feynman.

One of the ways in which he revolutionized physics and shared the Nobel Prize, was in his brilliantmerging of quantum mechanics with the Special Theory of Relativity. The two together make a verycomplicated story, one which results in the production of elementary particles from energy and the vacuum.A calculation of the probability for such a reaction might literally take a dozen pages of handwrittenmathematics, and typically takes me a full 2 hours to do on a blackboard in a graduate physics class.Feynman could do it, he invented many of the mathematical tools. But, he also figured out a cute way todraw some simple pictures and attach rules to the lines in the pictures that, when laid out in a cookbookfashion, cut 9 of the 12 pages of calculation away. These little pictures are called “Feynman Diagrams”and they now are routinely programmed on computers and even are generated automatically by computersfor calculations that might involve hundreds of graphs, an impossibility with pencil and paper.


Figure 1.6: A cartoon of collisions between two equal mass billiard balls as viewed from above. The top isthe situation before the collision, the bottom after.

The other nice thing about them is...they are so sensible and simple. They carry with them the hiddenmathematical complexity for the experts, but they also carry with them the simplicity of the physics in acartoon-like way. We’ll use them from now on to indicate atomic, nuclear, and sub-nuclear processes.

First, a simple one: literally a billiard ball collision. Suppose we first have two billiard balls that areconstrained to move in one dimension...imagine a groove cut in the billiard table. Ball #1 is sitting still(the target) and ball #2 is shot toward it. The trick about Feynman Diagrams is that they are diagramsin spacetime.

Obviously, if the billiard balls are of the same mass, then the collision is simple. The one at rest gets kickedand the one doing the kicking, stops. A Feynman Diagram of this situation is di!erent. First, rememberthat the spacetime diagram for a ball just sitting still corresponds to the ball not moving in space, andmoving in time at a constant velocity, namely zero. The spacetime diagram for an object moving with aconstant velocity in one dimension is a slanted line, representing an increasing distance with an increasingtime. Here is the Feynman Diagram for the process we’re discussing:


Figure 1.7: The Feynman Diagram for the process shown in Fig. 1.3.3.1. The position of the axes isarbitrary.

The grey circle is meant to indicate that what exactly happens at the instant that the (finite-sized) ballscollide is not of interest...and will have a serious meaning when we actually consider elementary particleslater.

So, for our Compton Scattering process, the situation is shown in Fig.1.8: Notice that it is the same as thebilliard ball circumstance, with the electron moving so slowly relative to the photon that it’s essentially atrest. A part of the Feynman diagram rule-set governs the kind of things that can happen when lines cometogether, the vertex. For electrons, the lines have to be continuous from the initial state into the finalstate: an electron line cannot just disappear. In practice, this implies that the photon strikes the electronand then is absorbed by it for a finite time before it is emitted. So, there is an intermediate period where a


Figure 1.8: Compton Scattering, the laboratory rest frame. The horizontal line defines the “beam” directionin the z direction and momenta will be defined with respect to it.

photon has annihilated and then is re-created. The Feynman Diagram is then shown in Fig.1.9. Hopefully,it’s clear just how much this simple diagram conveys about the mechanism of a photon scattering from anelectron.

Now, the actual calculation of what is observable proceeds as a “regular” momentum/energy conservationcalculation. The momentum conservation relationship is simple...remembering that we’re dealing in aframe of reference in which the electron is at rest (and hence has zero momentum):

h(k = (p" + h(k" (1.10)


Figure 1.9: Feynman Diagram for Compton Scattering.

Where |k| % #/c and is a standard way of representing the momentum of a photon. Refer to Fig.1.8 forthe definition of the coordinates. Here, h(k = h|k|z and the momentum must be conserved along thataxis, initial to final, as well as the direction perpendicular to the z axis, which I’ll call T for “transverse.”Here, we can see that the momentum of the photon in the final state which is transverse to the z axis iskTf = k"sin()) and it must be balanced by the transverse momentum of the electron in the final state.

The energy conservation equation is a little simpler:


h# + m0c2 = h# " + E" (1.11)

h# + m0c2 = h# " + (p"2c2 + m2

0c4)1/2 (1.12)

Here, the relativistic energy relation is used for the last equation and, as a relativistically correct equation,the rest mass of the electron must be included throughout. The simultaneous solution of Eqs.1.10 and 1.12gives the change in the frequency between the initial photon and the final photon. It’s actually a littleeasier to do it in terms of wavelength. Notice that by rearranging Eq.1.12 that the frequency di!erence,h(# # # ") is a positive number (the energy of the photon goes down) and so the final frequency is smallerthan the initial one, or that the wavelength of the final state photon is longer than the initial one. Theincrease in wavelength is predicted to be:

!" =h

m0c(1# cos()).

Compton carefully measured the scattering of K!X rays from the element Molybdenum. Where theprediction was a di!erence of 0.084Å, he found 0.089Å. Figure 1.10 shows Compton’s original results.This is a result that could not occur if the photon was not, for practical purposes, a relativistic billiardball of zero mass and that it scatters, one by one, from another particle. As he said in his classic paper:

“The present theory depends essentially up on the assumption that each electron which is e!ective in the scatteringscatters a complete quantum. It involves also the hypothesis that the quanta of radiation are received from definitedirections and are scattered in definite directions. The experimental support of the theory indicates very convincinglythat a radiation quantum carries with it directed momentum as well as energy.”

It could not be clearer than that: photons behave like waves and they behave like particles.


Figure 1.10: Wavelengths of the initial and the scattered X rays from Carbon in Compton’s 1923 experi-ment.

1.3.4 What’s the Meaning of This?

Alas, his enthusiasm for what he hath wrought did not last and he became the most conservative amongthe group of physicists who eventually broke free of the common sense that warmed to classical physics,leaving Einstein philosophically behind.


1.4 Quanta Show Up in Other Ways

Documents

Ea rly Quantum T heo ry - Michigan State University · Ea rly Quantum T heo ry In sum , one can sa y that there is hardly one among the great pr oblem s in w hi ch m o de rn ph ys