Duality of Matter

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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 student

of noble French background, was guided by personal

intuition and mathematical analogies rather than by any

experimental evidence when he posed what turned out

to 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 that

electrons 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 equation

always results in a very short wavelength (of the order

of atomic dimensions). In order for a wave to exhibit

the characteristic behavior of interference and diffrac-

tion, it must pass through openings of about the same

dimension as the wavelength, i.e., in the case of elec-

trons, through openings about as wide as an atom or the

spacing between atoms.

If one were to calculate from de Broglie’s relation-

ship the wavelength associated with a baseball, one

would 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 constant

in the numerator (a very small number). Waves so short

would exhibit diffraction and interference only if they

went through very narrow openings (about 10–34meters).

Since the present limit of measurability is about 10–15

meters, we can conclude that for macroscopic objects

like baseballs, the waves might just as well not exist.The next step was to somehow bring together Bohr’s

atom (with its success at predicting the atomic spectra) and

de Broglie’s matter waves. The success of Bohr’s atom

hinged on the idea that there were certain discrete orbits

with associated discrete energies. But what determines

which orbits are allowed and which are not? Perhaps,

someone thought, it is de Broglie’s wave. Perhaps the only

orbits allowed are those that an integral number of de

Broglie electron wavelengths would fit into! (See Fig.

16.1)

Figure 16.1. De Broglie’s matter waves “explain” why

only certain orbits are possible in Bohr’s atom. The

waves must just fit the orbit.

This idea was a great success. It predicted just the

right orbits and just the right energy levels to explain the

light spectrum for hydrogen (though it was less suc-

cessful for helium, lithium, etc.). Certainly it was

mathematically equivalent to the idea that Bohr had

originally used (which did not use waves), but it was

also radically different because it introduced a powerful

new idea: the electrons surrounding the nucleus of the

atom are some kind of wave whose wavelength depends

on the mass and speed of the electron.

But waves of what? When we think of the waves

of our everyday experience, we think of disturbances

propagating in a medium as, for example, waves on the

143

16. Duality of Matter

De BroglieWave

Allowed

Bohr Orbit

De BroglieWave

Disallowed

Bohr Orbit

Nucleus

Nucleus



ocean moving through the water. We have introduced

waves, wavelengths, and frequencies without ever

addressing the fundamental question: waves of what?

The Two-Slit Experiment

There is probably no stronger evidence for the

wave nature of light and electrons than the so-called

two-slit experiment in which electrons (or photons) passthrough a double slit arrangement to produce an “inter-

ference” pattern on the screen behind. The experiment

is simple and straightforward and, if we can understand

that, 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 general

area of atomic modeling that we are discussing) was

given in a series of lectures by Nobel laureate Richard

Feynman and reproduced in his book, The Character of

Physical Law. He discusses the two-slit experiment by

contrasting 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-slit

arrangement 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 do

not break up. The machine gun is a little shaky, so the

bullets sometimes go through one slit, sometimes the

other. They can ricochet from the edges of the slits, so

they can arrive at various positions behind them.

Positioned behind the slits is a bucket of sand to catch

the bullets. It is placed first in one position and the

number of bullets arriving in a given amount of time is

counted. The bucket is then moved to an adjacent posi-

tion, and the process is repeated until all the possible

arrival positions behind the slits have been covered.

Then a piece of paper is used to make a graph of the

number 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 of

high probability that lie directly behind the open slits.

When the experiment is repeated with one of the slits

closed, the curve has only one peak. We will refer to the

plotted curves as “probability” curves.

Now imagine an analogous experiment performed

with waves (Fig. 16.3). Visualize long straight waves

moving along the length of a pan of water. Into the path

of the waves we will place an obstacle with two slits. In

doing so we set up the classical demonstration of wave

interference. Behind the slits and along a straight line

paralleling the barrier but some distance behind, we will

observe the waves. Yet waves are not particles; it does-

n’t make any sense to measure the probability of arrival

of a wave at some particular point. In fact, the wave

arrives spread out over many points along the backdrop.

So rather than even trying to measure a probability

curve, we will observe the amplitude of the wave at var-

ious positions along the backdrop by placing a cork in

the water and observing the extent of the vertical motion

of the cork. Squaring the amplitude gives the “intensi-

ty” of the wave. The result is a multipeaked curve. The

peaks mark the regions of constructive interference; thevalleys mark the regions of destructive interference. If

one slit is closed, the interference largely disappears and

144

Figure 16.2. A two-slit experiment using particles (bul-

lets) fired by a rickety machine gun. The probability of

arrival of the bullets at the backdrop is described by a

double-peaked curve. Each peak is roughly behind one

of the open slits.

Figure 16.3. A two-slit experiment using waves. The

graph with the peaks represents the intensity of the

waves along the backdrop as might be observed by

watching a cork at various positions. The peaks in the

curve are points of constructive interference; the valleys

are points of destructive interference.



a single peaked curve reminiscent of the single-peaked

curve for the bullets is observed. We refer to these

curves 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 at

spots behind the screen and try to plot a probability

curve. If the electrons are waves, we try to measure the

amplitude of some disturbance and plot an interferencecurve. What, then, is the electron?

Some experiments do indicate that electrons

behave like little particles. So we proceed to set up a

two-slit experiment for electrons and design a detector

to place behind the slits to play the role the bucket of

sand 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 the

spacing between layers of atoms in a crystal. As

Richard 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 would

get for the interference of waves. She produces

this curve for what? Not for the energy in a

wave but for the probability of arrival of one of

these lumps (Richard Feynman, The Character

of Physical Law, p. 137).

Figure 16.4. A two-slit experiment using electrons. The

probability of arrival of the particles (electrons) takesthe form of the graph that describes the intensity of

waves.

As Feynman wrote, this is very strange indeed.

Look at one of the valleys in the probability curve, a

point where—with both slits open—there are no detect-

ed electrons (see point A in Fig. 16.4). Then imagine

slowly 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 the

remaining open slit. Now open the closed slit. The

detector stops counting. How do electrons going

through one slit know whether the other slit is open or

closed? If the electrons are little lumps, how do the

lumps get “canceled” so that nothing arrives at the

detector? You can slow down the rate of the electrons

so that only one electron at a time is fired—and they

still “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 the

waves associated with electrons are waves of probabili-

ty. They are not real disturbances in a medium. The

waves themselves are nothing more than mathematical

descriptions of probability of occurrence. And perhaps

we wouldn’t take them too seriously if Erwin

Schrödinger (1887-1961) hadn’t devised the equation

that bears his name.

Without some mathematical sophistication,

Schrödinger’s equation doesn’t mean much. But it is an

equation that describes how waves move through space

and time. The waves it describes may have peaks and

valleys that correspond to the high or low probability of

finding an electron at a particular place at a given time.

The peaks and valleys move and the equation describes

their movements. When we imagine firing an electron

at the two slits, we visualize such a wave being pro-

duced, and we imagine it propagating toward and

through 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 of

probability is deterministic.

Now imagine the wave of probability positioned

just in front of the detecting screen with its peaks and

valleys spread out like an interference curve. At this

point something almost mystical happens that no one

can predict. Out of all the possibilities represented by

the spread-out probability curve, one of them becomes

reality and the electron is seen to strike the screen at a

particular, small spot. No one knows how this choice is

made. Theories that have tried to incorporate some way

of deciding how the probabilities become realities have

always failed to agree with experiments. It is as if somegiant dice-roller in the sky casts the dice and reality

rests on the outcome.

Electron Microscope

Yet the diffraction and interference patterns dis-

played by electrons seem to be real enough. If they are

real in at least some sense, they offer the potential to

solve a very important problem. Microscopes are limit-

145



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 measure

as the size of the objects being viewed, the diffraction of

the 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 fundamental

limit (see Fig. 16.5). The only way the problem can be

solved is to use shorter wavelengths of viewing light.Short-wavelength x-rays would work, but nothing is able

to bend or focus short-wavelength x-rays in the way that

a glass lens bends and focuses rays of visible light.

Figure 16.5. Upper: Two small objects that are to be

viewed by a microscope. Lower: When the wavelength

of 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 very

short wavelengths. Indeed, they can be made even

shorter 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 the

makings of a very high-resolution microscope—an

electron microscope.

The electron accelerator (or electron gun) is made

of 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 electrons

from 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 more

speed (shorter wavelengths) as the grid is given more

positive charge.

In the microscope itself, the electron beam is

focused and its diameter is magnified by magnets, as

shown in Figure 16.7. The image is made visible when

the electrons strike a glass plate (the screen) that is coat-

ed with a material that glows when struck by energetic

particles. Because the wavelengths of the electrons can

be made very short, the resolution of the electron micro-

scope is much better than the resolution that can be

achieved 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

146

Figure 16.6. Electron source. Most electrons pass

through the grid.



the position of a particle? The probability of locating aparticle at a particular point is high where the wave

function is large and low where the wave function is

small. 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’t

know where with certainty. The uncertainty of the posi-

tion, which we will designate as ∆x, is roughly the

length of the region in which there are large peaks and

valleys. If the region is broad (∆x is large), then the

position is quite uncertain; if the region is narrow (∆x is

small), then the position is less uncertain. In other

words, 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 packet

can be no shorter than the wavelength, the wavelength

itself sets the limit of precision.

The speeds of particles have probabilities and

uncertainties in just the same way that positions have

probabilities and uncertainties. For technical reasons

we will multiply the uncertainty in speed by the mass of

the electron and call the resulting quantity ∆p. The two

uncertainties are related through the Schrödinger equa-

tion in a relationship called the (Heisenberg)

Uncertainty Principle:

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

147

Figure 16.8. Scanning electron micrograph of a human whisker at500 magnification. (Courtesy of W. M. Hess)

Figure 16.9. A moving electron might be represented as

a localized wave or wave packet. In each case, the elec-

tron can only be specified as being somewhere inside

the region bounded by the dashed lines.



What the relationship means is that experiments

designed to reduce the uncertainty in the position of a

particle always result in loss of certainty about the speed

of the particle. The opposite is also true: Experiments

designed 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 everyday

standards; but when we get down to observations of the

atomic world, the Uncertainty Principle becomes an

important factor.

Consider, for example, the electrons passing

through a single slit as in Figure 16.10. The arrival pat-

tern of the many electrons passing through the slit is

broad. Just as for light, the narrower the slit, the broad-

er the pattern. When we make the slit narrow, we also

make the uncertainty in the horizontal position of the

electron small, as it passes through the slit. We know the

position of the electron to within the width of the slit asit passes through. But making the slit narrower means

that the uncertainty in the horizontal speed of the elec-

tron must get larger so that the Uncertainty Principle is

satisfied. We no longer know precisely where it is head-

ed or how much horizontal motion it has acquired by

interacting with the slit. It is this large uncertainty in the

horizontal 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 distressingly

deterministic machine. The Second Law of

Thermodynamics said the clock was running down.

Quantum mechanics threatens to make the world more

a slot machine than a clock.

The Uncertainty Principle prevents us from predict-

ing with certainty the future of an individual particle. In

the 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 would

predict the future. Indeed, by following Newtonian ideas

and using computer programs, scientists can predict the

motions of planets for thousands of years into the future.

But to make Newtonian physics work for electrons, we

have 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 not

both together. Thus, we cannot predict the future of the

electron. Therefore, when an electron is fired at the two

slits, we cannot predict exactly where it will land on the

screen 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 could

know both with precision at the same time. He, there-

fore, denied reality to a description that specifies both at

the same time.Recall the two-slit experiment. Imagine again a

wave function that describes an electron fired by a gun

toward the slits and a screen behind. Imagine the wave

function with its hills and valleys undulating through

space. The wave function (probability curve) contains

all the information about the possibilities of where the

electron might land for any electron fired through the

slits to a screen behind. Not only does the wave function

contain the possibilities about where the electron should

finally 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 the

screen. What we would see is a tiny spot where theelectron strikes the screen. The spot is much smaller

than the space over which the wave was actually

extended. The wave is spread over the entire pattern of

spots that eventually becomes the “interference pat-

tern.” When the electron is revealed as it strikes the

screen, the pattern of probabilities changes drastically

and immediately. Suddenly the probability becomes 1.0

at the spot of observation and 0.0 everywhere else. Of

all the possibilities, only one has become reality. For

148

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

were even narrower?



However, quite by accident, the surface of his piece of

nickel became oxidized, and he was forced to interrupt

the experiment to heat the piece of metal to restore its

condition. In doing so, and without knowing, some

areas of the nickel on the surface crystallized, forming

the regular layered structure of a crystal. The spaces

between the layers became perfect “slits” of just the right

dimension for electron waves to diffract and interfere;

this demonstrated their wave nature. But for Davissonthe interference pattern that appeared was a puzzle, “an

irritating failure” as he put it. Nevertheless, he was alert

to a possible discovery and tried with theory after theo-

ry to explain the results, until he was led by discussions

with European physicists to de Broglie’s work.

De Broglie won the Nobel Prize for his “French

comedy” in 1929 and Davisson for elaborations and

refinements of his “irritating failure” in 1937. (Adapt-

ed from Barbara Lovett Cline, Men Who Made A New

Physics, pp. 152-156.)

STUDY GUIDE

Chapter 16: Duality of Matter

A. FUNDAMENTAL PRINCIPLES

1. 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 of

probability.

2. Wave-Particle Duality of Electromagnetic

Radiation: See Chapter 14.

3. The (Heisenberg) Uncertainty Principle: The

product of the uncertainty in the position of an

object 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 fixed

mass 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 closely

spaced slits in a metal sheet?

3. What pattern would be observed if water waves

were allowed to pass through two openings in adike?

4. What pattern would be observed if electrons were

allowed to pass through two very closely spaced

slits? What would the pattern be like if the elec-

trons were sent through the device one at a time? If

one of the holes were closed, what pattern would

develop?

5. Is there evidence to support the position that matter

has a particle aspect?

6. Is there evidence to support the position that matter

has a wave aspect?

7. What is uncertain in the Uncertainty Principle, and

can the uncertainty be eliminated with more careful

experiments?

C. GLOSSARY

1. Interference Curve: In the context of this chapter,

we mean a mathematical graph which is a measureof the squared amplitude of waves in a region

where 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 famed

bell-shaped curve that describes the probability of

having a particular IQ) which describes the proba-

bility of finding an electron (or other particle) at

various positions in space.

4. Probability Wave: A probability curve which is

changing in time and space in a manner that is like

the 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 bounds

within which the quantity is known with high prob-

ability. If you knew that your friend was on the

freeway somewhere between Provo and Orem, the

uncertainty in your knowledge of exactly where

he/she was on the freeway might be about 10 miles

since the two towns are about 10 miles apart.

D. FOCUS QUESTIONS

1. A single electron is sent toward a pair of very close-ly spaced slits. The electron is later detected by a

screen placed on the opposite side. Then, a great

many electrons are sent one at a time through the

same device.

a. Describe the pattern produced on the screen by

the single electron and later the total pattern of the

many electrons.

b. Name and state a fundamental principle that

can 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 closely

spaced slits. The photon is later detected by ascreen placed on the opposite side. Then, a great

many photons are sent one at a time through the

same device.

a. Describe the pattern produced on the screen by

the single photon, and later the total pattern of the

many photons.

b. Name and state in your own words the funda-

mental principle that can account for the observa-

tions.

151



c. Explain the observations in terms of the funda-

mental principle.

3. A single electron is sent through a tiny slit. Later it

is 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 to

know exactly where the electron is when it passes

through the slit, what else will happen? Will we

now be able to predict where the electron will be

seen on the screen? Explain in terms of the

Heisenberg Uncertainty Principle.

4. How does the Newtonian Model of motion of

things in the world differ from the Uncertainty

Principle? To what extent is the future determined

by the present according to the Newtonian Model?

To what extent is the future determined according

to a model that includes wave-particle duality?

E. EXERCISES

16.1. Why do we think of matter as particles?

Carefully describe some experimental evidence that

supports this view.

16.2. Why do we think of matter as waves?

Carefully describe some experimental evidence that

supports this view.

16.3. Is matter wavelike or particlelike? Carefully

describe some experimental evidence that supports your

conclusion.

16.4. What is meant by the term “wave-particle

duality?” Does it apply to matter, or to electromagnet-

ic radiation, or to both?

16.5. One possible explanation of the interference

effects of electrons is to presume that the wavelike

behavior is due to the cooperative effect of groups of

electrons acting together. What experimental evidence

is there for believing the opposing view that electron

waves are associated with individual particles?

16.6. Consider an experiment to test the diffraction of electrons as illustrated in Figure 16.10. Why would it be

important to place a charged rod or a magnet near the

beam 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 into

account?

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 it

is to be confined within a very small region of space?

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

speed?

16.12. How does the Newtonian model differ from

the Uncertainty Principle?

16.13. Explain the meaning of the Uncertainty

Principle.

16.14. Why is it that the Uncertainty Principle is

important 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 does

not permit objects to be completely at rest, even when

at the temperature of absolute zero.

16.16. To what extent is the future determined by

the present according to (a) Newtonian physics and (b)

quantum physics?

16.17. Illustrate the statistical nature of physical

processes by describing the motion of individual parti-

cles in the one- or two-slit experiments.

16.18. How does the Uncertainty Principle modify

our view that the universe is “deterministic?”

16.19. Why should we not be surprised when the

rules governing very small or very fast objects do not

seem “reasonable?”

16.20. In what situations would you expect both the

Newtonian laws and wave mechanics to accurately pre-

dict the motions of objects? In what situations would

the 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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