When Louis-Victor-Pierre-Raymond de Broglie(1892-1987) wrote his doctoral thesis in 1923, he pro-posed a radical new idea with implications he, himself,did not fully appreciate. De Broglie, a graduate studentof noble French background, was guided by personalintuition and mathematical analogies rather than by anyexperimental evidence when he posed what turned outto be a very crucial question: Is it possible that parti-cles, such as electrons, exhibit wave characteristics?

De Broglie’s arguments led to the conclusion thatelectrons had a wavelength given by

wavelength ! Planck’s constant (mass of an electron)(speed of an electron)

.

Indeed, Planck’s constant is a very small number and,for reasonable speeds of an electron, the equationalways results in a very short wavelength (of the orderof atomic dimensions). In order for a wave to exhibitthe characteristic behavior of interference and diffrac-tion, it must pass through openings of about the samedimension as the wavelength, i.e., in the case of elec-trons, through openings about as wide as an atom or thespacing between atoms.

If one were to calculate from de Broglie’s relation-ship the wavelength associated with a baseball, onewould find a prediction of the order of 10–34 meters.This small number follows from de Broglie’s relation-ship because it has the mass of the object in the denom-inator (very large for a baseball) and Planck’s constantin the numerator (a very small number). Waves so shortwould exhibit diffraction and interference only if theywent through very narrow openings (about 10–34meters).Since the present limit of measurability is about 10–15

meters, we can conclude that for macroscopic objectslike baseballs, the waves might just as well not exist.

The next step was to somehow bring together Bohr’satom (with its success at predicting the atomic spectra) andde Broglie’s matter waves. The success of Bohr’s atomhinged on the idea that there were certain discrete orbitswith associated discrete energies. But what determineswhich orbits are allowed and which are not? Perhaps,someone thought, it is de Broglie’s wave. Perhaps the onlyorbits allowed are those that an integral number of deBroglie electron wavelengths would fit into! (See Fig.

16.1)

Figure 16.1. De Broglie’s matter waves “explain” whyonly certain orbits are possible in Bohr’s atom. Thewaves must just fit the orbit.

This idea was a great success. It predicted just theright orbits and just the right energy levels to explain thelight spectrum for hydrogen (though it was less suc-cessful for helium, lithium, etc.). Certainly it wasmathematically equivalent to the idea that Bohr hadoriginally used (which did not use waves), but it wasalso radically different because it introduced a powerfulnew idea: the electrons surrounding the nucleus of theatom are some kind of wave whose wavelength dependson the mass and speed of the electron.

But waves of what? When we think of the wavesof our everyday experience, we think of disturbancespropagating in a medium as, for example, waves on the

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16. Duality of Matter

De BroglieWave

AllowedBohr Orbit

De BroglieWave

DisallowedBohr Orbit

Nucleus

Nucleus

ocean moving through the water. We have introducedwaves, wavelengths, and frequencies without everaddressing the fundamental question: waves of what?

The Two-Slit Experiment

There is probably no stronger evidence for thewave nature of light and electrons than the so-calledtwo-slit experiment in which electrons (or photons) passthrough a double slit arrangement to produce an “inter-ference” pattern on the screen behind. The experimentis simple and straightforward and, if we can understandthat, then we ought to be able to understand wave-parti-cle duality (if it can be understood at all).

One of the most enlightening explanations of“quantum mechanics” (the name given to the generalarea of atomic modeling that we are discussing) wasgiven in a series of lectures by Nobel laureate RichardFeynman and reproduced in his book, The Character ofPhysical Law. He discusses the two-slit experiment bycontrasting the experiment using indisputable particles,then using indisputable waves, and finally using elec-trons.

To better understand the two-slit experiment, imag-ine a shaky machine gun that fires bullets at a two-slitarrangement fashioned out of battleship steel (Fig.16.2). Bullets are clearly “particles.” They are small,localized structures that can be modeled as tiny points.In our experiment we shall assume that the bullets donot break up. The machine gun is a little shaky, so thebullets sometimes go through one slit, sometimes theother. They can ricochet from the edges of the slits, sothey can arrive at various positions behind them.

Positioned behind the slits is a bucket of sand to catchthe bullets. It is placed first in one position and thenumber of bullets arriving in a given amount of time iscounted. The bucket is then moved to an adjacent posi-tion, and the process is repeated until all the possiblearrival positions behind the slits have been covered.Then a piece of paper is used to make a graph of thenumber of bullets which arrive as a function of the posi-tion of the bucket. The result is a double-peaked curvethat reflects the probability of catching bullets at vari-ous positions behind the slits.

The two peaks correspond to the two regions ofhigh probability that lie directly behind the open slits.When the experiment is repeated with one of the slitsclosed, the curve has only one peak. We will refer to theplotted curves as “probability” curves.

Now imagine an analogous experiment performedwith waves (Fig. 16.3). Visualize long straight wavesmoving along the length of a pan of water. Into the pathof the waves we will place an obstacle with two slits. Indoing so we set up the classical demonstration of waveinterference. Behind the slits and along a straight lineparalleling the barrier but some distance behind, we willobserve the waves. Yet waves are not particles; it does-n’t make any sense to measure the probability of arrivalof a wave at some particular point. In fact, the wavearrives spread out over many points along the backdrop.So rather than even trying to measure a probabilitycurve, we will observe the amplitude of the wave at var-ious positions along the backdrop by placing a cork inthe water and observing the extent of the vertical motionof the cork. Squaring the amplitude gives the “intensi-ty” of the wave. The result is a multipeaked curve. Thepeaks mark the regions of constructive interference; thevalleys mark the regions of destructive interference. Ifone slit is closed, the interference largely disappears and

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Figure 16.2. A two-slit experiment using particles (bul-lets) fired by a rickety machine gun. The probability ofarrival of the bullets at the backdrop is described by adouble-peaked curve. Each peak is roughly behind oneof the open slits.

Figure 16.3. A two-slit experiment using waves. Thegraph with the peaks represents the intensity of thewaves along the backdrop as might be observed bywatching a cork at various positions. The peaks in thecurve are points of constructive interference; the valleysare points of destructive interference.

a single peaked curve reminiscent of the single-peakedcurve for the bullets is observed. We refer to thesecurves as “interference” curves.

Now we can try the experiment with electrons (Fig.16.4). But which experiment? If the electrons are parti-cles, then we try to measure the arrival of little lumps atspots behind the screen and try to plot a probabilitycurve. If the electrons are waves, we try to measure theamplitude of some disturbance and plot an interferencecurve. What, then, is the electron?

Some experiments do indicate that electronsbehave like little particles. So we proceed to set up atwo-slit experiment for electrons and design a detectorto place behind the slits to play the role the bucket ofsand played for bullets. Note, however, that the wave-lengths of electrons (according to de Broglie’s formula)are very short and the slits will have to be similar to thespacing between layers of atoms in a crystal. AsRichard Feynman expressed, this is where nature pro-vides us with a very strange and unexpected result:

That is the phenomenon of nature, that she pro-duces the curve which is the same as you wouldget for the interference of waves. She producesthis curve for what? Not for the energy in awave but for the probability of arrival of one ofthese lumps (Richard Feynman, The Characterof Physical Law, p. 137).

Figure 16.4. A two-slit experiment using electrons. Theprobability of arrival of the particles (electrons) takesthe form of the graph that describes the intensity ofwaves.

As Feynman wrote, this is very strange indeed.Look at one of the valleys in the probability curve, apoint where—with both slits open—there are no detect-ed electrons (see point A in Fig. 16.4). Then imagineslowly closing one of the slits so that the valley of prob-ability goes away and the detector begins to count the

arrival of electrons that must be coming through theremaining open slit. Now open the closed slit. Thedetector stops counting. How do electrons goingthrough one slit know whether the other slit is open orclosed? If the electrons are little lumps, how do thelumps get “canceled” so that nothing arrives at thedetector? You can slow down the rate of the electronsso that only one electron at a time is fired—and theystill “know.” The “interference” curve, in fact, becomesa composite of all the arrivals of the individual elec-trons. It is very puzzling.

Waves of Probability

The presently accepted explanation is that thewaves associated with electrons are waves of probabili-ty. They are not real disturbances in a medium. Thewaves themselves are nothing more than mathematicaldescriptions of probability of occurrence. And perhapswe wouldn’t take them too seriously if ErwinSchrödinger (1887-1961) hadn’t devised the equationthat bears his name.

Without some mathematical sophistication,Schrödinger’s equation doesn’t mean much. But it is anequation that describes how waves move through spaceand time. The waves it describes may have peaks andvalleys that correspond to the high or low probability offinding an electron at a particular place at a given time.The peaks and valleys move and the equation describestheir movements. When we imagine firing an electronat the two slits, we visualize such a wave being pro-duced, and we imagine it propagating toward andthrough the two slits and approaching the screen behind.All of this motion of the probability wave is describedby Schrödinger’s equation. The motion of the wave ofprobability is deterministic.

Now imagine the wave of probability positionedjust in front of the detecting screen with its peaks andvalleys spread out like an interference curve. At thispoint something almost mystical happens that no onecan predict. Out of all the possibilities represented bythe spread-out probability curve, one of them becomesreality and the electron is seen to strike the screen at aparticular, small spot. No one knows how this choice ismade. Theories that have tried to incorporate some wayof deciding how the probabilities become realities havealways failed to agree with experiments. It is as if somegiant dice-roller in the sky casts the dice and realityrests on the outcome.

Electron Microscope

Yet the diffraction and interference patterns dis-played by electrons seem to be real enough. If they arereal in at least some sense, they offer the potential tosolve a very important problem. Microscopes are limit-

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ed in what they can see by the effects of diffraction.Typical microscopes use visible light. When the wave-length of the light being used is about the same measureas the size of the objects being viewed, the diffraction ofthe light around the edges of the objects becomes sub-stantial and the images get fuzzy. The microscope can-not resolve the detail and has reached its fundamentallimit (see Fig. 16.5). The only way the problem can besolved is to use shorter wavelengths of viewing light.Short-wavelength x-rays would work, but nothing is ableto bend or focus short-wavelength x-rays in the way thata glass lens bends and focuses rays of visible light.

Figure 16.5. Upper: Two small objects that are to beviewed by a microscope. Lower: When the wavelengthof the viewing light is the same size as the objects, dif-fraction causes a loss of resolution.

This is where electron waves come to the rescue.According to de Broglie’s equation, electrons have veryshort wavelengths. Indeed, they can be made evenshorter by increasing the speed of the electrons. If the

electrons could be accelerated to shorten the wave-length, and if they could be focused, we would have themakings of a very high-resolution microscope—anelectron microscope.

The electron accelerator (or electron gun) is madeof two parts: an electrically heated wire and a grid,which is basically a small piece of window screen.When the grid is positively charged, it attracts electronsfrom the hot wire. The electrons are accelerated towardthe grid, and most of them pass through its holes, as dia-gramed in Figure 16.6. The electrons achieve morespeed (shorter wavelengths) as the grid is given morepositive charge.

In the microscope itself, the electron beam isfocused and its diameter is magnified by magnets, asshown in Figure 16.7. The image is made visible whenthe electrons strike a glass plate (the screen) that is coat-ed with a material that glows when struck by energeticparticles. Because the wavelengths of the electrons canbe made very short, the resolution of the electron micro-scope is much better than the resolution that can beachieved by visible light microscopes. (See Fig. 16.8;see also Fig. 5.7 and Color Plates 1 and 2.)

Figure 16.7. Diagram of an electron microscope.Actual height is about 2 meters.

The (Heisenberg) Uncertainty Principle

The wave nature of matter raises another interest-ing problem. To what extent is it possible to determine

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Figure 16.6. Electron source. Most electrons passthrough the grid.

the position of a particle? The probability of locating aparticle at a particular point is high where the wavefunction is large and low where the wave function issmall. Imagine such a wave as depicted in Figure 16.9.This kind of wave might be called a “wave packet.” If

we were to look, we would expect to find the particlewithin the space occupied by the packet, but we don’tknow where with certainty. The uncertainty of the posi-tion, which we will designate as ∆x, is roughly thelength of the region in which there are large peaks andvalleys. If the region is broad (∆x is large), then theposition is quite uncertain; if the region is narrow (∆x issmall), then the position is less uncertain. In otherwords, the position of particles is specified more pre-cisely by narrow waves than by broad waves. This lim-iting precision is decreed by nature. Since the packetcan be no shorter than the wavelength, the wavelengthitself sets the limit of precision.

The speeds of particles have probabilities anduncertainties in just the same way that positions haveprobabilities and uncertainties. For technical reasonswe will multiply the uncertainty in speed by the mass ofthe electron and call the resulting quantity ∆p. The twouncertainties are related through the Schrödinger equa-tion in a relationship called the (Heisenberg)Uncertainty Principle:

(∆x) " (∆p) is greater than Planck’s constant .

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Figure 16.8. Scanning electron micrograph of a human whisker at "500 magnification. (Courtesy of W. M. Hess)

Figure 16.9. A moving electron might be represented asa localized wave or wave packet. In each case, the elec-tron can only be specified as being somewhere insidethe region bounded by the dashed lines.

What the relationship means is that experimentsdesigned to reduce the uncertainty in the position of aparticle always result in loss of certainty about the speedof the particle. The opposite is also true: Experimentsdesigned to reduce the uncertainty in the speed of a par-ticle always result in a loss of certainty about the posi-tion of the particle. Because Planck’s constant is small,these two uncertainties can still be small by everydaystandards; but when we get down to observations of theatomic world, the Uncertainty Principle becomes animportant factor.

Consider, for example, the electrons passingthrough a single slit as in Figure 16.10. The arrival pat-tern of the many electrons passing through the slit isbroad. Just as for light, the narrower the slit, the broad-er the pattern. When we make the slit narrow, we alsomake the uncertainty in the horizontal position of theelectron small, as it passes through the slit. We know theposition of the electron to within the width of the slit asit passes through. But making the slit narrower meansthat the uncertainty in the horizontal speed of the elec-tron must get larger so that the Uncertainty Principle issatisfied. We no longer know precisely where it is head-ed or how much horizontal motion it has acquired byinteracting with the slit. It is this large uncertainty in thehorizontal speed that makes it impossible to predict pre-cisely where the electron will land. This causes the pat-tern to spread. We can think of the spreading as a con-sequence of the Uncertainty Principle.

What is Reality?

The Newtonian clockwork was a distressinglydeterministic machine. The Second Law ofThermodynamics said the clock was running down.Quantum mechanics threatens to make the world morea slot machine than a clock.

The Uncertainty Principle prevents us from predict-ing with certainty the future of an individual particle. Inthe Newtonian view the future was exactly predictable.If the position, speed and direction of motion of a parti-

cle were known, the Newtonian laws of motion wouldpredict the future. Indeed, by following Newtonian ideasand using computer programs, scientists can predict themotions of planets for thousands of years into the future.But to make Newtonian physics work for electrons, wehave to know exactly where the electron is and, simulta-neously, its speed and direction of motion. This is pre-cisely what the Uncertainty Principle says we fundamen-tally cannot know. We can know one or the other, but notboth together. Thus, we cannot predict the future of theelectron. Therefore, when an electron is fired at the twoslits, we cannot predict exactly where it will land on thescreen behind. The best we can do is to know the proba-bilities associated with the wave function.

Bohr saw the position and the speed as comple-mentary descriptors of the electron. But no one couldknow both with precision at the same time. He, there-fore, denied reality to a description that specifies both atthe same time.

Recall the two-slit experiment. Imagine again awave function that describes an electron fired by a guntoward the slits and a screen behind. Imagine the wavefunction with its hills and valleys undulating throughspace. The wave function (probability curve) containsall the information about the possibilities of where theelectron might land for any electron fired through theslits to a screen behind. Not only does the wave functioncontain the possibilities about where the electron shouldfinally land on the screen, but it also contains a descrip-tion of the probabilities to assign to those possibilities.

Now imagine that the wave function reaches thescreen. What we would see is a tiny spot where theelectron strikes the screen. The spot is much smallerthan the space over which the wave was actuallyextended. The wave is spread over the entire pattern ofspots that eventually becomes the “interference pat-tern.” When the electron is revealed as it strikes thescreen, the pattern of probabilities changes drasticallyand immediately. Suddenly the probability becomes 1.0at the spot of observation and 0.0 everywhere else. Ofall the possibilities, only one has become reality. For

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Figure 16.10. A beam of electrons diffracts as it passes through a narrow slit. How would the pattern change if the slitwere even narrower?

the next electron through the slits, a different possibili-ty will become reality. This sudden change is called the“collapse of the wave function.”

The collapse of the wave function is a great mys-tery. One interpretation (by Max Born in 1920) is thatthe collapse is not a physical change but, rather, “ourknowledge of the system suddenly changes.” Humans,in their capacity as observers, become an essential fea-ture of physics. In science, reality is what we see ordetect with the help of instruments. As long as the wavefunction moves unobserved through space with all of itspossibilities and probabilities, we cannot attribute reali-ty to the electron’s position, its speed, or even its exis-tence. Is it the seeing and observing itself which turnsthe possibilities into a single reality? If this view is cor-rect (and it is surely debatable), then reality is literallycreated by the act of conscious observation! Withoutthe observation, the world remains unrealized possibili-ty and probability.

In a sense, we can say that the electron moves fromplace to place as a wave of probability. Yet it is alwaysobserved as a particle. When we don’t observe it, wecannot say what its nature is. We cannot ascribe realityto any of the usual descriptions of position, speed, ener-gy, and so forth.

Niels Bohr defended and enlarged the above view-point into what has come to be called the CopenhagenInterpretation of quantum mechanics. To Einstein, whobelieved in a world that existed completely indepen-dently of man’s consciousness, the interpretation wasanathema; he argued and fought against it with tenacityin what came to be a losing battle. The whole idea is inviolent contradiction to the rigidly deterministicNewtonian view we discussed earlier.

The Role of the Newtonian Laws

The Newtonian view has merit even though wenow know it is incomplete. Its amazing success whenapplied to a wide variety of phenomena should indicatethat it is at least close to being an accurate descriptionof nature. How can we reconcile the quantum mechan-ical view of change in the universe with the older, butsuccessful, exact determinism of Newtonian physics?

Quantum mechanics and Newtonian laws giveexactly the same results in the domain in which theNewtonian laws have been successful. Suppose wewere to use the usual laws of motion and gravitation topredict the position of the earth in its motion about thesun. We would assign a specific location and motion forthe earth at the present time, and then predict exact loca-tions and motions for later times. Now suppose wewere to use quantum mechanics and the modern view todescribe the same motion. We could then predict prob-abilities for the arrival of the earth at certain locations atparticular times. The result would be a high probabili-

ty, essentially 100 percent, of finding the earth at thelocations predicted by the Newtonian laws. We wouldfind low probability, nearly zero percent, of finding itanywhere else. The two views predict exactly the samefuture locations.

However, the situation is quite different if we try topredict the motion of an electron inside an atom. Herewe cannot even begin by using the Newtonian laws. Wecannot assign an exact position to the electron becauseof its wave nature. Since its position is not known, thereis no way to calculate accurately the force acting on itand thus no way to predict its acceleration or its futureposition. In this domain, quantum mechanics providesthe only way to predict the motion and future course ofthe electron wave.

Consider Figure 16.11. The sizes of things are listedalong a horizontal axis; their speeds are listed along thevertical axis. The things we normally encounter wouldoccur near the middle of the range of possible sizes andnear the low end of possible speeds. Such objects andsuch motions form the basis of our intuitive feelings aboutnature. The Newtonian laws accurately describe themotions of such objects that seem reasonable and proper.

Since we rarely deal directly with small things,such as atoms and their constituent particles, we shouldnot be surprised that their rules of behavior are differentfrom those to which we are accustomed.

The laws governing atoms represent refinements ofthe Newtonian laws. We could use any or all of the lawsof quantum mechanics to calculate and predict themotions of objects we normally encounter and find thesame answer as if we had used Newton’s laws. Theadditional sophistication is simply not necessary for anaccurate description of the things with which we ordi-narily deal. Only when we wish to understand thebehavior of things exceedingly small or fast do we needthe additional insights of the extended laws. In thissense, the Newtonian laws are approximations of themore complete laws; however, as approximations, theyare totally adequate when limited to objects with appro-priate size and speed.

Summary

Matter has a dual nature—particle and wave. Itsparticulate nature dominates the behavior of large-scaleobjects; the wave nature dominates small-scale (submi-croscopic) behavior. The wave properties of submicro-scopic matter limit the motion of particles in such a waythat the particles cannot have unique values for bothposition and speed simultaneously. This limitation isformalized by the Uncertainty Principle. The wavenature of matter also implies that changes in any part ofthe universe are statistical processes, and the laws wehave discovered allow us to predict the probability thatcertain kinds of changes will occur. Newton’s laws of

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motion are ways to calculate with high probability thekinds of motion that large objects will undergo.

Electrons in atoms behave almost entirely aswaves. These wave properties determine the structureof atoms and the ways that each atom interacts with itsneighbors. Electron wave interactions thus regulate thekinds of collections of atoms that can occur, and alsodetermine the properties of bulk matter.

Historical Perspectives

Some scoffed at de Broglie’s thesis, labeled it a“French comedy,” and expressed reservations aboutaccepting it. But by chance de Broglie’s thesis came tothe attention of Albert Einstein, who in 1905 had pro-posed the revolutionary idea of the dual nature of lightin order to explain the photoelectric effect. Therefore,it was natural for Einstein to be drawn to de Broglie’sproposal for a dual nature of matter. By 1923 Einsteinwas quite famous, and the scientific world hung on hisevery word. Einstein drew attention to de Broglie’s the-

sis in one of his own papers, restated the argument in aforceful way, and bolstered the conjecture with addi-tional arguments. Thus, the idea came to the attentionof Erwin Schrödinger, who developed a wave model ofthe atom based on the wave equation that bears hisname and is synonymous with “quantum mechanics.”Yet de Broglie still had not one shred of solid experi-mental evidence that electrons behaved like waves.

By 1925 the crucial experiments had already beendone by an American, Clinton Davisson, and, indepen-dently, by George Thomson. (Ironically, GeorgeThomson’s father, J. J. Thomson, received a NobelPrize [1906] for crucial experiments in demonstratingthe particle nature of the electron, and his son, George,later received a Nobel Prize [1937] for showing theentirely opposite wave nature of the electron.) But in1925, Davisson had not heard of de Broglie’s idea andwas struggling to explain puzzling results that he hadacquired quite by accident.

In his experiments he was observing the scatteringof electrons as they encountered a piece of nickel.

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Figure 16.11. The Newtonian laws are adequate for describing the motions of objects whose size and speed are with-in our range of experience. More precise laws must be used outside that range.

However, quite by accident, the surface of his piece ofnickel became oxidized, and he was forced to interruptthe experiment to heat the piece of metal to restore itscondition. In doing so, and without knowing, someareas of the nickel on the surface crystallized, formingthe regular layered structure of a crystal. The spacesbetween the layers became perfect “slits” of just the rightdimension for electron waves to diffract and interfere;this demonstrated their wave nature. But for Davissonthe interference pattern that appeared was a puzzle, “anirritating failure” as he put it. Nevertheless, he was alertto a possible discovery and tried with theory after theo-ry to explain the results, until he was led by discussionswith European physicists to de Broglie’s work.

De Broglie won the Nobel Prize for his “Frenchcomedy” in 1929 and Davisson for elaborations andrefinements of his “irritating failure” in 1937. (Adapt-ed from Barbara Lovett Cline, Men Who Made A NewPhysics, pp. 152-156.)

STUDY GUIDEChapter 16: Duality of Matter

A. FUNDAMENTAL PRINCIPLES1. Wave-Particle Duality of Matter: Matter in its

finest state is observed as particles (electrons, pro-tons, quarks), but when unobserved (such as mov-ing from place to place) is described by waves ofprobability.

2. Wave-Particle Duality of ElectromagneticRadiation: See Chapter 14.

3. The (Heisenberg) Uncertainty Principle: Theproduct of the uncertainty in the position of anobject and the uncertainty in its momentum isalways larger than Planck’s constant. (The momen-tum of an object is its mass times its speed; thus,uncertainty in momentum of an object of fixedmass is an uncertainty in speed.)

B. MODELS, IDEAS, QUESTIONS, OR APPLICA-TIONS

1. What was the de Broglie hypothesis?2. What pattern would be observed if a rickety

machine gun fired bullets through two closelyspaced slits in a metal sheet?

3. What pattern would be observed if water waveswere allowed to pass through two openings in adike?

4. What pattern would be observed if electrons wereallowed to pass through two very closely spacedslits? What would the pattern be like if the elec-trons were sent through the device one at a time? Ifone of the holes were closed, what pattern woulddevelop?

5. Is there evidence to support the position that matterhas a particle aspect?

6. Is there evidence to support the position that matterhas a wave aspect?

7. What is uncertain in the Uncertainty Principle, andcan the uncertainty be eliminated with more carefulexperiments?

C. GLOSSARY1. Interference Curve: In the context of this chapter,

we mean a mathematical graph which is a measureof the squared amplitude of waves in a regionwhere wave interference is taking place.

2. Planck’s Constant: See Chapter 14.3. Probability Curve: In the context of this chapter,

we mean a mathematical graph (like the famedbell-shaped curve that describes the probability ofhaving a particular IQ) which describes the proba-bility of finding an electron (or other particle) atvarious positions in space.

4. Probability Wave: A probability curve which ischanging in time and space in a manner that is likethe movement of a wave in space and time.

5. Quantum Mechanics: The set of laws and princi-ples that govern wave-particle duality.

6. Uncertainty: For a quantity that is not known pre-cisely, the uncertainty is a measure of the boundswithin which the quantity is known with high prob-ability. If you knew that your friend was on thefreeway somewhere between Provo and Orem, theuncertainty in your knowledge of exactly wherehe/she was on the freeway might be about 10 milessince the two towns are about 10 miles apart.

D. FOCUS QUESTIONS1. A single electron is sent toward a pair of very close-

ly spaced slits. The electron is later detected by ascreen placed on the opposite side. Then, a greatmany electrons are sent one at a time through thesame device.a. Describe the pattern produced on the screen bythe single electron and later the total pattern of themany electrons.b. Name and state a fundamental principle thatcan account for all of these observations.c. Explain the observations in terms of the funda-mental principle.

2. A single photon is sent toward a pair of closelyspaced slits. The photon is later detected by ascreen placed on the opposite side. Then, a greatmany photons are sent one at a time through thesame device.a. Describe the pattern produced on the screen bythe single photon, and later the total pattern of themany photons.b. Name and state in your own words the funda-mental principle that can account for the observa-tions.

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c. Explain the observations in terms of the funda-mental principle.

3. A single electron is sent through a tiny slit. Later itis detected by a screen placed on the opposite side.It is possible to change the width of the slit.a. What is observed on the screen?b. Is it possible to predict exactly where the elec-tron will be seen when it arrives at the screen?c. State the Heisenberg Uncertainty Principle.d. If the slit is made narrower in an attempt toknow exactly where the electron is when it passesthrough the slit, what else will happen? Will wenow be able to predict where the electron will beseen on the screen? Explain in terms of theHeisenberg Uncertainty Principle.

4. How does the Newtonian Model of motion ofthings in the world differ from the UncertaintyPrinciple? To what extent is the future determinedby the present according to the Newtonian Model?To what extent is the future determined accordingto a model that includes wave-particle duality?

E. EXERCISES16.1. Why do we think of matter as particles?

Carefully describe some experimental evidence thatsupports this view.

16.2. Why do we think of matter as waves?Carefully describe some experimental evidence thatsupports this view.

16.3. Is matter wavelike or particlelike? Carefullydescribe some experimental evidence that supports yourconclusion.

16.4. What is meant by the term “wave-particleduality?” Does it apply to matter, or to electromagnet-ic radiation, or to both?

16.5. One possible explanation of the interferenceeffects of electrons is to presume that the wavelikebehavior is due to the cooperative effect of groups ofelectrons acting together. What experimental evidenceis there for believing the opposing view that electronwaves are associated with individual particles?

16.6. Consider an experiment to test the diffraction ofelectrons as illustrated in Figure 16.10. Why would it beimportant to place a charged rod or a magnet near thebeam between the diffracting hole and photographic film?

16.7. How does one describe the motion of elec-trons when their wave properties must be taken intoaccount?

16.8. What is the meaning of the term “quantum

mechanics”?

16.9. Why are electron microscopes used for view-ing atoms instead of regular light microscopes?

16.10. Why must a particle have a high speed if itis to be confined within a very small region of space?

16.11. What does the Uncertainty Principle sayabout simultaneous measurements of position andspeed?

16.12. How does the Newtonian model differ fromthe Uncertainty Principle?

16.13. Explain the meaning of the UncertaintyPrinciple.

16.14. Why is it that the Uncertainty Principle isimportant in dealing with small particles such as elec-trons, but unimportant when dealing with ordinary-sized objects such as billiard balls and automobiles?

16.15 Explain why the Uncertainty Principle doesnot permit objects to be completely at rest, even whenat the temperature of absolute zero.

16.16. To what extent is the future determined bythe present according to (a) Newtonian physics and (b)quantum physics?

16.17. Illustrate the statistical nature of physicalprocesses by describing the motion of individual parti-cles in the one- or two-slit experiments.

16.18. How does the Uncertainty Principle modifyour view that the universe is “deterministic?”

16.19. Why should we not be surprised when therules governing very small or very fast objects do notseem “reasonable?”

16.20. In what situations would you expect both theNewtonian laws and wave mechanics to accurately pre-dict the motions of objects? In what situations wouldthe two predictions be significantly different?

16.21. Which of the following would form an inter-ference pattern?

(a) electrons(b) blue light(c) radio waves(d) sound waves(e) all of the above

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16.22. The Uncertainty Principle(a) is an outcome of Newtonian mechanics(b) applies mainly to subatomic particles(c) conflicts with wave-particle duality(d) supports strict determinism(e) all of the above

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