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
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16. Duality of Matter
De BroglieWave
Allowed
Bohr Orbit
De BroglieWave
Disallowed
Bohr Orbit
Nucleus
Nucleus
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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
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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.
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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-
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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 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
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Figure 16.6. Electron source. Most electrons pass
through the grid.
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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 .
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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.
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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
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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 slit
were even narrower?
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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.
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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 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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