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Chapter 29 Particles and Waves, An
Introduction to Quantum Physics
Friday, January 14, 2011
10:03 AM
Physical Theories
How does science work? (At least in the case of physics and other
mathematical sciences.)
First you observe the world, and you also do experiments.
You also abstract from the many observations and experiments the
key quantities (such as position, velocity, force, etc.) that will appear
in your theories.
Then you create theories that relate the key quantities in ways that
help you explain the phenomena that you observed and/or appeared
in your experiments. You create the theories by guessing; that's right,
guessing. (OK, if you want a fancy word for it, you can call it inductive
logic. But it's still guessing.)
Of course, I don't mean random guessing. It's a creative process that
requires a deep knowledge of the current scientific understanding of
the world, and it helps if you know the history of the development of
science. What I'm trying to say is that you don't logically derive the
laws of physics; you just create them. Then, once they are created,
you test them using logic, and if they survive these tests, then you test
them using observations and experiments. Ultimately, the vast
majority of theories are discarded; few survive to form part of the
ever-evolving currently generally accepted body of science.
You test your theories against the phenomena that you
observed/experimented on. If the equations of your theory predict
results that agree with your observations or experiments, then good.
If not, you will have to modify your theory, or maybe discard it and
start from scratch.
Then you use deductive logic to try to derive consequences of the
theory that were not observed before. If you can do this, and if
subsequent experiments or observations agree with the predictions of
the theory, then that is very good. Otherwise, you will have to modify
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the theory, then that is very good. Otherwise, you will have to modify
your theory, or maybe discard it and start from scratch.
Logic plays a key role in testing physical theories. A theory of physics
must be logically consistent; for example, it must not be possible to
derive two contradictory predictions from the theory. But the creation
of a theory is not necessarily a logical process, at least not in the same
sense. Intuition, analogy, "feeling," play a greater role in creation;
logic plays the primary role in testing the theory for consistency and
for deriving consequences and predictions. But the ultimate test of a
physical theory is observation and experimental verification.
No amount of experimental or observational testing can ever prove a
scientific theory correct. Though scientists sometimes use such terms
(saying a theory is right or true or correct) colloquially, they are not
meaningful because a scientific theory can never be proved correct,
because it's impossible to test the theory at all points in space and at
all times.
Asking whether a scientific theory is correct is like asking whether
your marriage is red or green (which would be truly confusing if you
are married to Red Green). Or asking whether a sculpture by
Modigliani is true or false. Such questions are meaningless. Although
Picasso once said that "Art is a lie that helps you see the truth."
Beautiful, isn't it? And a scientific theory is something like an art work
as well: A human creation that is somehow false (has approximations
built in, has oversimplifications, idealizations, has limited applicability,
etc.), but yet helps us gain insight into our wonderful world. And yet
there has to be some truth to physical theory; just look at all the
bridges and buildings that exist without falling down, and the cars that
move along the streets, and the computers that we all use, and the
electrical power systems that bring electricity into our homes, and all
the machines in our hospitals, etc. All of this wonderful technology is
based on our understanding of science, and it would be extreme to
suggest that there is no truth whatsoever in it. (And yes, there is a
dark side to technology, but that we shall discuss another time.)
Some people try to denigrate science using phrases such as "it's only a
theory." That demonstrates a profound misunderstanding of science
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theory." That demonstrates a profound misunderstanding of science
(or perhaps a willful attempt to mislead). There is a difference
between the every-day use of the term "theory," to mean uninformed
speculation, and the scientific use of the term theory. If science were
like the Olympic games, then achieving the status of "theory" would
be analogous to winning a gold medal. Becoming a theory
(successfully tested by observation and experiment) is the pinnacle of
achievement for a scientific idea.
So, although scientific theories can't be proven correct, they are
nevertheless precious. They represent the highest achievements in
scientific thought. They represent the most successfully tested,
hardened-by-trials products of the scientific enterprise. The vast
majority of scientific ideas end up in the slag heap; the best theories
are the survivors.
Reflect on the words of Henri Poincare, which emphasize the role of
creativity: "Science is built up of facts, as a house is built of stones; but
an accumulation of facts is no more a science than a heap of stones is
a house."
Also reflect on the words of Isaac Asimov:
"Consider some of what the history of science teaches. First, since
science originated as the product of men and not as a revelation, it
may develop further as the continuing product of men. If a scientific
law is not an eternal truth but merely a generalization which, to some
man or group of men, conveniently described a set of observations,
then to some other man or group of men, another generalization
might seem even more convenient. Once it is grasped that scientific
truth is limited and not absolute, scientific truth becomes capable of
further refinement. Until that is understood, scientific research has no
meaning."
If he were writing today, Asimov would no doubt have used the word
"person" instead of "man," but I'm sure you get the idea: Laws of
physics are not directives to be obeyed, but are rather convenient
generalizations describing nature's workings. The collection of all
physical theories is like a vast work of art; nobody would call it
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physical theories is like a vast work of art; nobody would call it
correct, but it's beautiful, and absolutely useful. The bridges engineers
design using Newton's laws don't fall down, do they? And the MP3
players made using principles of electromagnetism and quantum
theory are rather functional as well.
So physical theories are not "true," but they are tightly constrained to
apply very closely to this world. But some day, maybe tomorrow,
maybe next century, someone (maybe one of you?) may create a new
theory, that is somehow more beautiful, or more useful, or in some
way of value, so that it may supersede or replace an existing theory of
physics.
****************
Classical Mechanics and Quantum Mechanics
OK, now let's get down to some specifics about quantum mechanics
(also called quantum theory, also called quantum physics). To put this
in perspective, let's first say a few words about classical mechanics
(also called Newtonian mechanics).
Mechanics can be broadly divided into two branches, kinematics and
dynamics. Kinematics is the description of motion, particularly the
mathematical description of motion, and dynamics is an explanation
for how the causes of motion (forces) create motion (that is, dynamics
is a quantitative version of "everything happens for a reason").
So classical kinematics is all about describing motion in terms of
position, velocity, acceleration, angles, and so on, and then
understanding the relations among the variables. Classical dynamics
consists of Newton's laws of motion and related conservation laws.
Classical mechanics is a very successful theory. Using classical
mechanics, we have built great cities, long bridges and tunnels,
engines of all kinds, and aircraft and spacecraft that fly into the skies
and into space. Supplementing classical mechanics with the classical
theory of electricity and magnetism, we have created motors and
generators, and communicate wirelessly across continents in an
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generators, and communicate wirelessly across continents in an
instant.
All of these applications are successful tests of the classical theories of
mechanics and electromagnetism. We use the theories, do the math,
and figure out how to build the rockets, how long to keep the engines
on, when and in which direction to blast the engines to correct the
course, and so on. And voila! The spacecraft actually makes it to the
moon. The predictions of the theory are verified in practice, and this
gives us confidence that the theory is useful.
However, when we apply the classical theories of electricity and
magnetism to atoms and their innards, they fail. Completely. And.
Utterly. Fail.
Does that mean the classical theories are wrong? Well, yes, I suppose
so. But they worked so well for building the bridges, and for sending
spacecraft to the moon, and for safely lighting our houses, and for
sending TV and radio signals around the world, so it seems like a pity
to throw the theories away just because they fail in the atomic and
subatomic realms.
So we don't throw them away; we just recognize their limitations
along with the realms in which they are wonderfully useful. But we
have to come up with theories that work in the atomic and subatomic
realms. This was done by many physicists; it was a real team effort,
led by Planck, Einstein, Bohr and many others in the early days (1900
to the 1920s), by Heisenberg, Schrödinger, Dirac, Born, and many
others in the 1920s and 1930s, and by many others subsequently.
Quantum mechanics is a theory that successfully describes motions
within atoms. It forms the foundation for atomic and molecular
physics (and chemistry), solid-state physics (also called condensed
matter physics), lasers, fibre optics, and other photonic systems, and
so on. Quantum physics is even being applied nowadays to
understand microbiology!
Quantum physics, together with modern theories of
electromagnetism, have been applied to produce the basic devices
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electromagnetism, have been applied to produce the basic devices
that underlie many of our neat modern technologies. The laser
devices (CD and DVD players, optical memory drives, laser surgical
devices, etc.), all the miniaturization that goes on in the computer
world, the fancy new materials, the solar (photovoltaic) cells, and so
on, all of it is possible thanks to quantum mechanics.
In this course we'll have a very brief introduction to quantum ideas. If
you want a more in-depth introduction, take Physics 2P50 (Modern
Physics) next year, and you'll learn more about Einstein's theory of
special relativity as a bonus!
And if you want some great introductory books to read over the
summer, try one or more of these:
Thirty Years That Shook Physics, by George Gamow (full of funny
stories about the great physicists of the early 20th century, told by
someone who rubbed shoulders with them)
The Strange Story of the Quantum, by Banesh Hoffmann
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The state of affairs in physics at about 1900
By the turn of the 20th century, classical mechanics (Newton and his
successors) and the classical theory of electricity and magnetism
(Faraday, Maxwell, and their contemporaries and successors) were
well-established core theories of physics.
Additionally, there was a large body of evidence that firmly
established that light is a wave phenomenon, in contradiction to
Newton's "corpuscular" theory of light, which he pioneered in the
late 1600s and early 1700s. Historically, there were two simplified
models for light, one in terms of particles and the other in terms of
waves. Newton considered light to be a stream of particles (he called
them "corpuscles," which is just a fancy word for particles, so his
theory became known as the corpuscular theory), and his
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theory became known as the corpuscular theory), and his
contemporary rival Huygens considered light to be made up of waves.
Newton's great work on this subject, Opticks, which was published in
1704, reported on a large number of optical experiments that
Newton had carried out, and he was able to successfully explain the
phenomena by thinking of light as a stream of tiny particles. Huygens,
on the other hand, published his great work on optics, Treatise on
Light, in 1690, and he conceived of light as being formed from
longitudinal waves. His theory was also very successful in terms of
explaining basic reflection, refraction, etc., what we would call
geometrical optics nowadays.
Newton's theory was much more widely accepted in the early 1700s,
probably because Newton was so famous. Another reason for opinion
swaying to Newton's side was Huygens's longitudinal wave theory's
failure to explain birefringence (double refraction):
The wave theory of light was rehabilitated later in the 1700s by
treating light as a transverse wave, but for most of the 1700s
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treating light as a transverse wave, but for most of the 1700s
Newton's theory of light ruled. Another reason for the widespread
acceptance of Newton's theory is that one doesn't notice wave
properties of light with the naked eye. For example, light casts sharp
shadows, so the Newtonians argued that light couldn't possibly be
waves, because it would bend around corners and shadows would be
blurry. Waves experience all kinds of interference effects, and none of
them were apparent at the time of Newton:
However, one of the problems with Newton's particle theory of light is
the observation that two light beams can pass through each other
unaffected. This is very hard to explain with the corpuscular theory
(there ought to be an enormous amount of scattering going on as the
little particles smash against each other), but very easy to explain with
a wave theory of light. Nevertheless, on balance, Newton's particle
theory of light was generally accepted.
One of the strengths of science is that it is continually revised. Nothing
is set in stone, and we don't accept something just because famous
great scientist X said so. (Although that was certainly true in the past;
witness the ideological devotion to Aristotle's theories throughout the
Middle Ages, just because it was Aristotle. However, we know better
since the time of Galileo than to slavishly accept a great person's
opinions just because the person is great; that is unscientific.) There is
a continual search for evidence that could support or question a
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a continual search for evidence that could support or question a
theory, and in this way ideas are shaped, evolve, and improved, and
science progresses. By the early 1800s, numerous experiments by
Thomas Young, Augustin Fresnel, and others, made it very difficult to
believe in the particle theory of light. It was natural and satisfying to
explain the experiments in terms of a wave theory, and so the wave
theory of light soon came to dominate. It became clear that wave
aspects of light were not discovered sooner because the wavelength
of light is extremely small.
A few classic phenomena are Newton's rings, and interference as
shown by Young's double-slit experiment:
http://video.mit.edu/watch/thomas-youngs-double-slitexperiment-8432/
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By the 1800s the situation seemed clear beyond the shadow of a
doubt: light is a wave. The work of Maxwell in the 1860s, with
experimental verification by Hertz and others starting in 1887, made it
clear what kind of a wave light is: a transverse electromagnetic wave.
However, as we shall see, the situation was complicated in the early
1900s when it became clear that light is also a particle! The evidence
for the particle-like nature of light is also very clear, and nowadays we
call particles of light "photons."
An important part of classical physics that we did not study in PHYS
1P22/1P92 or in PHYS 1P21/1P91 is statistical mechanics. If you
studied PHYS 1P23/1P93 you learned a bit about thermodynamics, the
science of the flow of thermal energy and its interactions with
mechanical forces. In the latter part of the 1800s a fundamental
theory of physics, called statistical mechanics, was developed by
Boltzmann (and others; Maxwell also made important contributions),
that explained thermodynamics in terms of the interactions of a
swarm of microscopic particles (molecules and atoms) described by
the laws of Newtonian mechanics. The theory had many successes
(although its early derisive detractors may have had a role in
Boltzmann's subsequent sad descent into insanity), and convinced
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Boltzmann's subsequent sad descent into insanity), and convinced
many scientists of the reality of atoms long before there was definitive
evidence.
The numerous successes of physics in describing the natural world
throughout the 19th century convinced some foolhardy physicists that
all the fundamental aspects of physics were already well understood,
and there was nothing new (in a fundamental sense) left to discover. A
notorious example of this variety of hubris is the following
pronouncement of A.A. Michelson, in 1903:
“The more important fundamental laws and facts of physical science
have all been discovered, and these are now so firmly established
that the possibility of their ever being supplanted in consequence of
new discoveries is exceedingly remote. Nevertheless, it has been
found that there are apparent exceptions to most of these laws, and
this is particularly true when the observations are pushed to a limit,
i.e., whenever the circumstances of experiment are such that extreme
cases can be examined. Such examination almost surely leads, not to
the overthrow of the law, but to the discovery of other facts and laws
whose action produces the apparent exceptions. As instances of such
discoveries, which are in most cases due to the increasing order of
accuracy made possible by improvements in measuring instruments,
may be mentioned: first, the departure of actual gases from the
simple laws of the so-called perfect gas, one of the practical results
being the liquefaction of air and all known gases; second, the
discovery of the velocity of light by astronomical means, depending on
the accuracy of telescopes and of astronomical clocks; third, the
determination of distances of stars and the orbits of double stars,
which depend on measurements of the order of accuracy of one-tenth
of a second-an angle which may be represented as that which a pin's
head subtends at a distance of a mile. But perhaps the most striking of
such instances are the discovery of a new planet or observations of
the small irregularities noticed by Leverrier in the motions of the
planet Uranus, and the more recent brilliant discovery by Lord
Rayleigh of a new element in the atmosphere through the minute but
unexplained anomalies found in weighing a given volume of nitrogen.
Many other instances might be cited, but these will suffice to justify
the statement that 'our future discoveries must be looked for in the
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the statement that 'our future discoveries must be looked for in the
sixth place of decimals.'”
However, the quantum revolution had already begun, with the
astonishing work of Planck in 1900; we'll get to this shortly. First,
though, let's discuss four physical phenomena that were rather
puzzling around this time:
1. blackbody radiation
2. atomic spectra
3. atomic structure
4. the photoelectric effect
We'll discuss each of these important situations, then we'll discuss
how they led to the introduction of the early ideas of quantum
physics, and we'll also discuss how these quantum ideas helped to
solve the puzzles.
Then we'll discuss some of the mysteries of quantum physics that still
remain, waiting for current or future researchers (maybe some of
you?) to provide further insight.
• blackbody radiation
Warm solid objects glow, as do warm liquids and warm samples of
gas. The warm objects emit electromagnetic radiation, with various
amounts emitted at various wavelengths; the precise amounts depend
on the material, the characteristics of its surface, and especially its
temperature. A graph of what is called "spectral radiance" (check the
units on the vertical axis of the graph below) vs. wavelength for an
idealized emitter called a blackbody is shown below:
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The blue, green, and red curves represent experimental data for
emitters that are excellent approximations to blackbodies. The blue
curve represents an emitter at a temperature of 5000 K, the green
curve represents an emitter at a temperature of 4000 K, and the red
curve represents an emitter at a temperature of 3000 K. The emitted
intensity is proportional to the area under each graph; note that the
intensity increases with increasing temperature.
The statistical mechanics of Boltzmann and Maxwell, so successful at
providing a microscopic explanation for so many thermodynamical
properties of matter, resulted in total nonsense (represented by the
black curve in the diagram) when applied to the problem of blackbody
radiation by Rayleigh and Jeans. They predicted that an infinite
amount of energy would be radiated, with an increasing amount of
energy emitted as you move towards the UV end of the spectrum. This
prediction catastrophically contradicts experimental results. (The
phrase "ultraviolet catastrophe" was coined by Ehrenfest.)
Trying to explain the spectrum of a blackbody was a puzzle at the turn
of the 20th century. Planck resolved this puzzle in December 1900
(published in 1901) with a truly ingenious idea, that is all the more
remarkable because it went exactly opposite to his understanding at
the time.
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Think about, for example, people locked in an argument. How often is
one of the parties convinced by the other to change his or her view? It
takes a lot of courage to do this.
Planck was one of the worldwide experts in thermodynamics at this
time, and he made a lot of progress by assuming that heat is a
continuous phenomenon, much like a wave or a field. The atomists
(led by Boltzmann) adopted the opposite stance, saying that all of the
macroscopic phenomena (such as heat, for example) are ultimately
the result of an enormous number of chaotic motions of microscopic
particles (atoms or molecules).
Planck decided to attempt to derive a family of formulas for the
blackbody radiation curves (in the graph above) as a way of proving
that his perspective was right and the atomists were wrong. A
microscopic explanation for the blackbody radiation curves was one of
the big outstanding problems at the time, and if Planck could solve it
this would give him powerful ammunition for his side of the argument.
He failed for six years.
Then one day, desperate to solve the problem, he tried an extravagant
mathematical trick, as a kind of wild guess. In trying to derive a
formula, he had been up to now doing the usual thing, which is to
determine the total power emitted by the blackbody by using calculus;
that is, by integrating the power in each mode of vibration of the
blackbody. This is the same method used by Rayleigh and Jeans (and
by Wien, and everyone else), with "catastrophic" results. Part of the
problem is that Rayleigh and Jeans had used the equipartition
principle (from Boltzmann), which stated that each mode of vibration
is equally likely. This means that very high-energy modes of vibration
are just as likely as low-energy modes of vibration, which resulted in
the UV catastrophe.
Planck tried, as a guess, what must have been unthinkable to
everyone else: he used a regular sum (albeit an infinite sum) instead of
an integral. That is, he avoided calculus. And, surprisingly, he obtained
a family of formulas that seemed to work! He presented his work at a
regular meeting of the German Physical Society in October 1900, and
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regular meeting of the German Physical Society in October 1900, and
very soon after he received confirmation from experimental physicists
that his formula worked extremely well for all available experimental
data.
But what does the trick mean? Planck could not be satisfied with
merely coming up with a "rabbit-out-of-a-hat" formula, he wanted to
explain the fundamental physical idea behind the formula. (In effect,
he had figured out the answer, but now he had to "show his work!")
This he set about doing, in what he later described as the most
strenuous period of work in his life.
Alas, he found that his mathematical trick kept leading him towards
Boltzmann's atomistic view of microscopic physics, and away from his
own continuous perspective. Rather than fight the inevitable, he gave
in, gave Boltzmann credit for being right, and converted to the atomist
perspective.
Planck's physical explanation for his mathematical trick is that the
energy of the elementary vibrators in the glowing solid, which are
responsible for emitting electromagnetic radiation, can have not just
any energy, but rather only whole-number multiples of a certain unit
of energy, which he called an elementary quantum of energy.
(Nowadays we would interpret "elementary vibrators" as vibrating
atoms, but then it was unclear whether atoms even existed, so they
simply spoke in terms of modes of vibrations without being more
specific.)
Thus, according to Planck, each mode of vibration of the glowing
object could have only a discrete amount of energy, which is related
to its frequency through the formula E = hf, where h is a universal
constant now known as Planck's constant. The value of Planck's
constant is extremely tiny, which explains why we don't notice it in
everyday life.
Of course, according to Planck, once the vibrators actually emitted
electromagnetic radiation, the emitted radiation was nice and
continuous, electromagnetic waves just as Maxwell had described. But
while the vibrators were collecting up the energy needed for emission
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while the vibrators were collecting up the energy needed for emission
of electromagnetic waves, the amount of energy collected was
quantized.
Planck neatly explained away the ultraviolet catastrophe by relying on
Boltzmann's statistical mechanics perspective: High-energy quanta of
radiated energy are far less likely than low-energy quanta, because it
takes time to amass the necessary energy. In that time, it's far more
likely for the excess energy to be radiated away in lower-energy
quanta. The explanation made sense to Planck and his
contemporaries, and did much to sway the physics community to the
atomist perspective, simultaneously lending support to Boltzmann.
Planck was a conciliator by nature, and one wonders whether a more
egocentric scientist would have been quite so willing to embrace and
then publicly champion his opponent's ideas.
Planck's revolutionary idea, which he presented at another regular
meeting of the German Physical Society in December 1900 (the paper
was published in 1901) ushered in the quantum age. As counterintuitive as Planck's idea was, everyone (including Planck) still
understood that once a blackbody emitted electromagnetic radiation,
even if it was emitted in blobs, the electromagnetic radiation would
move as an electromagnetic wave. After all, there were decades of
solid experimental evidence that light is a wave phenomenon, and
growing experimental evidence in the past 15 years or so that light is
an electromagnetic wave.
It would take Einstein to further overturn this neat view of
electromagnetic radiation in 1905, just a few years later. But we'll get
to that part of the story soon; first let's continue to talk about light
emitted by matter.
• atomic spectra
By the late 1800s there was an enormous amount of experimental
data on light emitted by matter. Besides the continuous spectra of the
blackbody type, discussed above, glowing objects also emitted what
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blackbody type, discussed above, glowing objects also emitted what
are called discrete spectra, or equivalently, line spectra. Often these
two types of spectra are superimposed, so that it takes some work to
separate the two types. Compare the following diagram (of the Sun's
emission) with the continuous blackbody spectra studied previously:
Notice that the Sun's emission spectrum in the previous diagram is
similar to blackbody radiation curves, except that the intensity is less at
certain wavelengths. What is decreasing the intensity at certain
wavelengths?
Here are some examples of discrete spectra. They are produced by
passing the electromagnetic radiation emitted by an object through a
prism, so that its various wavelengths are separated. (The
electromagnetic radiation is first passed through a slit, which results in
lines, rather than dots or other shapes.)
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Note that the solar spectrum contains certain dark lines against a
continuous background. The continuous background represents
(nearly) blackbody radiation produced because of the Sun's
temperature (about 5800 K). The dark lines in the Sun's spectrum are
known as absorption lines; apparently something in the Sun's
atmosphere is absorbing light of certain wavelengths that had been
emitted from the Sun's surface, so that these wavelengths are now
missing from the spectrum.
The bright lines in the other spectra in the diagram are known as
emission lines. (The other spectra are from gaseous samples of the
given elements.) Notice the bright yellow line in sodium's spectrum;
there are other emission lines in sodium's spectrum, but only a portion
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there are other emission lines in sodium's spectrum, but only a portion
of the range of wavelengths is shown in the diagram, and in this
portion the only emission line is the one shown.
The emission lines in the other spectra have the same character: Only
certain wavelengths appear, and they are separated by gaps that vary
in size.
By the late 1800s, there was an enormous amount of spectroscopic
data, but no explanation for why discrete spectra occur. Why is
electromagnetic radiation emitted only at certain wavelengths? Is
there any pattern? What is the physical explanation?
In 1885, Johann Balmer guessed a formula that described part of the
spectrum of hydrogen; that is, one could calculate the wavelengths by
substituting certain values into the formula. This was quite a
spectacular achievement given how little data Balmer had to work
with: Just FOUR wavelengths! Other workers, particularly Johannes
Rydberg, generalized Balmer's formula, and stimulated the search for
(and discovery of) further spectral lines of hydrogen. We'll check out
Rydberg's formula in the next chapter.
So a pattern was discovered in the wavelengths of the spectral lines of
hydrogen, but is there a similar pattern in the spectra of other
elements? And, above all, what is the physical explanation behind this?
This remained a prominent puzzle at the turn of the 20th century.
Connected to this puzzle was the puzzle of atomic structure. Now that
more and more scientists were becoming convinced about the reality
of atoms, the question of what atoms are like came to the forefront.
• atomic structure
Some of the ancients believed that the world consists of indivisible
little parts, which they called atoms. The idea was revived in the 1700s
and 1800s, and supported by the work of John Dalton (through his
1805 law of definite proportions, which described chemical reactions).
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1805 law of definite proportions, which described chemical reactions).
Dalton built on the earlier (1789) work of Antoine Lavoisier, who came
up with the idea of a chemical element and the law of conservation of
mass for chemical reactions.
A series of experiments by a number of workers, culminating in the
work of J.J. Thomson in 1897, identified electrons as negatively
charged constituents of atoms. But atoms are normally electrically
neutral, so there must be separate positively charges parts of an atom;
this line of reasoning suggests that atoms have structure, and are not
the simple, indivisible particles conceived of by the ancients.
OK, so atoms seem to contain separate bits of positive and negative
charges; what exactly is the structure of the interior of an atom?
This was in important scientific issue in the early 1900s, and the
quantum revolution provided dramatic insights in the first quarter of
the 20th century. We'll continue this part of the story in Chapter 30.
• the photoelectric effect
As we've already discussed, by the 1800s it was very well established
by a number of decisive experiments that light is a wave phenomenon.
(Think back to the earlier part of this course, where we discussed wave
interference, diffraction, and so on.) Maxwell hypothesized in the
1860s that light is an electromagnetic wave and by the late 1880s
Heinrich Hertz had confirmed Maxwell's theory with a series of careful
experiments.
In an ironic twist, the very experiment in which Hertz confirmed that
light is a wave also planted seeds of doubt in its validity! In the
experiment where he used electromagnetic waves to induce a spark
across a small gap in a loop of wire, he noticed that the sparks came a
little more readily if ultraviolet light was shining on the loop of wire.
Hertz died young (before his 37th birthday) and had no chance to
follow up on this puzzling phenomenon, but others did study this
photoelectric phenomenon more carefully (that light shining on a
metal helps electrons to jump out of the metal). Here's a typical
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metal helps electrons to jump out of the metal). Here's a typical
setup:
The applied voltage could be varied; experiments showed that if the
voltage exceeded a certain amount (called the stopping potential, or
stopping voltage, Vstop), then no electrons reached the collector plate
of the tube.
The wave theory of light predicts that the results of the experiments
should be as follows. (Imagine that light is like waves on an ocean,
and that the electrons are like buoys floating on it.)
Predictions based on the wave
Experimental results
theory of light
There should be a time delay, during The time delay is very tiny, and it
which enough light is absorbed by
is independent of the light
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which enough light is absorbed by
is independent of the light
electrons, before electrons begin to intensity. No matter how low you
be ejected from the metal. The time make the light intensity, the time
delay should depend on the intensity delay does not increase.
of the light; the greater the intensity,
the smaller the time delay.
If electrons are ejected for light of a Increasing the intensity increases
certain frequency, then keeping the the number of electrons ejected,
frequency the same and increasing but has no effect on the energy
the intensity of the light should
of individual electrons. The rate
increase the energy of the ejected
at which electrons are ejected
electrons. (Classically, the intensity of (i.e. current) is proportional to
a wave is a measure of its energy
the light intensity.
density.)
If light of a certain frequency is
There is a threshold frequency f0;
able to eject electrons from a
for light of frequency below the
metal, then light of any frequency
threshold, no electrons are
should also be able to eject
ejected, no matter how great the
light intensity is.
electrons from the same metal;
there might be a time delay if the
intensity is low.
The stopping potential depends
on the metal. For a particular
metal, and for a particular light
frequency, the stopping potential
is the same no matter what the
intensity of the incident light is.
The stopping potential does
depend on the frequency of the
light.
The energy of ejected electrons
increases linearly with the
frequency of the incident
light. (Millikan, 1915)
Nobody was able to explain the experimental results of Hertz,
Hallwachs, Lenard, Stoletov, J.J. Thomson, and others, in a
satisfactory way. That is, until Einstein came on the scene in 1905
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satisfactory way. That is, until Einstein came on the scene in 1905
with a very radical proposal: The photon hypothesis. Inspired by the
work of Planck (1900), Einstein proposed that light exists in little
bundles, which became known as photons. The energy of each bundle
is proportional to the frequency of the light:
E = hf
where h is Planck's constant (h = 6.63 × 10-34 Js). We’ll discuss shortly
how Einstein's proposal brilliantly explains the strange results of
photoelectric effect experiments. Also notice how much further
Einstein pushed Planck's hypothesis beyond what Planck first
proposed; Planck said that elementary vibrators within a blackbody
collected energy in quanta, but once the energy was emitted as
electromagnetic radiation, it travelled in waves. Einstein went much
further and postulated that light EXISTS as little particles, which were
later called photons (Einstein called them "light quanta"). Einstein
said that light is emitted, absorbed, and exists as particles.
Stop for a moment and marvel at the boldness of creative scientists
such as Planck and Einstein; they were able to entertain "what if"
games of thought, and were not afraid to publicly communicate their
conclusions even though they directly contradicted well-established
scientific theories. In Einstein's case, his bold proposal was made in
the face of a century of very solid evidence that light is a WAVE.
But how does Einstein's photon hypothesis square with all the
evidence that light is a wave? Einstein proposed that the connection
is statistical; that is, light consists of photons (little particles of light)
that in aggregate behave as a wave.
In Einstein's viewpoint (light is composed of photons), the intensity of
the light is a measure of the number of photons per second falling on
the metal per unit area. The basic idea is that typically each electron
will absorb a single photon (absorbing two or more at once will be a
very rare event). It's apparently not possible for a photon to be partly
absorbed; it's either completely absorbed, or not at all. Similarly, it's
apparently not possible for a photon to have some of its energy
absorbed by one electron and some by another; all of its energy must
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be absorbed (if at all) by a single electron.
If the single absorbed photon imparts enough energy to the electron,
then it will escape the metal. If not, then the electron's extra energy is
likely to be exchanged with other electrons or the atoms in the metal
through collisions before the electron has a chance to absorb another
photon.
Here is a summary of how Einstein's hypothesis neatly explains the
photoelectric effect; also see page 929 of the textbook for a nice
summary.
Einstein's explanation based on
the photon theory of light
The ejection of an electron occurs
because it absorbs a single photon.
Low-intensity light has few photons,
but the time delay will not depend
on the number of photons.
Experimental results
Increasing the intensity of the light
increases the rate at which photons
arrive at the metal, but it does not
increase the energy of each photon.
More photons means more
electrons are ejected, but each
photon still passes on the same
amount of energy to each electron.
If the energy of each individual
photon is not enough, then no
electron will be ejected, no matter
how many photons arrive at the
metal, because each electron
absorbs one photon at a time.
Each electron absorbs one photon.
It's like a person trying to jump over
a step. If you don't make it over the
step, then you fall back down and
have to try again.
Increasing the intensity increases
the number of electrons ejected,
but has no effect on the energy of
individual electrons. The rate at
which electrons are ejected (i.e.
current) is proportional to the
light intensity.
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The time delay is very tiny, and it is
independent of the light intensity.
No matter how low you make the
light intensity, the time delay does
not increase.
There is a threshold frequency f0;
for light of frequency below the
threshold, no electrons are
ejected, no matter how great the
light intensity is.
The stopping potential depends on
the metal. For a particular metal,
and for a particular light
frequency, the stopping potential
is the same no matter what the
have to try again.
The stopping potential is equal
to Kmax/e, which depends on
the frequency according to the
equation just below.
The maximum kinetic energy of an
ejected electron is
Kmax = hf  E0
where E0 is called the work function
of the metal.
is the same no matter what the
intensity of the incident light is.
The stopping potential does
depend on the frequency of the
light.
The energy of ejected electrons
increases linearly with the
frequency of the incident light.
(Millikan, 1915)
Applications of the photoelectric effect
• photovoltaic cells (solar energy) (strictly speaking, this involves semiconductors and therefore is more complicated than the simple
photoelectric effect)
• charge-coupled devices (used in digital cameras)
• some smoke detectors (and the same principle is used for those fancy
laser-alarms that you see in robbery movies); safety switches on
automatic garage-door openers/closers work on the same principle)
• photocopy machines; see http://en.wikipedia.org/wiki/Photocopier (strictly
speaking, this involves semi-conductors and therefore is more
complicated than the simple photoelectric effect)
• (photosynthesis is not an example of the photoelectric effect, but it's
a process by which photons of the "right" energy are absorbed to
induce a chemical reaction; similarly, vision in the eye involves the
absorption of photons by retinal cells and the consequent creation of
an electrical current in the optic nerve)
___________________________________________________
Problem: Determine the energy (in eV) of a photon of visible light
that has a wavelength of 500 nm.
Solution:
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________________________________________________________
Problem: Determine the energy (in eV) of an X-ray photon that has
a wavelength of 1.0 nm.
Solution:
_____________________________________________________
Problem: The intensity of electromagnetic radiation from the sun
reaching the earth's upper atmosphere is 1.37 kW/m2. Assuming
an average wavelength of 680 nm for this radiation, determine the
number of photons per second that strike a 1.00 m2 solar panel
directly facing the sun on an orbiting satellite.
Solution:
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________________________________________________________
Problem: Electrons are emitted when a metal is illuminated by
light with a wavelength less than 388 nm but for no greater
wavelength. Determine the metal's work function.
Solution:
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______________________________________________________
Problem: Zinc has a work function of 4.3 eV.
(a) Determine the longest wavelength of light that will release an
electron from a zinc surface.
(b) A 4.7 eV photon strikes the surface and an electron is emitted.
Determine the maximum possible speed of the electron.
Solution:
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______________________________________________________
Problem: Station KAIM in Hawaii broadcasts on the AM dial at
870 kHz, with a maximum power of 50,000 W. Determine how
many photons the transmitting antenna emits each second at
maximum power.
Solution:
______________________________________________________
The Compton Effect
Despite the spectacular success of Einstein's photon hypothesis in
explaining the photoelectric effect, the photon hypothesis was not met
with widespread acceptance. It's hard for people to accept
contradictory facts; on the one hand, by the early 20th century an
enormous amount of evidence had been collected confirming that light
is a wave. Now this upstart is saying that light is also a particle? How
can that be??
In particular, one aspect of the photon hypothesis is that photons carry
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In particular, one aspect of the photon hypothesis is that photons carry
momentum as well as energy. Well? Where was the experimental
evidence that photons carry momentum? In the absence of this kind of
evidence, it was natural for physicists to be skeptical about the photon
hypothesis.
Yet another line of evidence for the photon hypothesis came from what
we now call Compton scattering. At that time J.J. Thomson explained
the scattering of electromagnetic waves from free charged particles
using Maxwell's theory of electromagnetism, and the theory was
supported experimentally for incident light of low intensity. The
classical theory predicts NO change in wavelength of the scattered
electromagnetic waves.
However, in 1923 Compton hypothesized that if light were really
composed of photons, then high-energy photons scattering from
charged particles should experience a shift in their wavelength upon
scattering; some of a photon's energy would be transferred to the
charged particle in the collision, and because the energy of a photon is
related to its wavelength, the scattered photon would have a different
wavelength than before it collided with the charged particle. This is
now called the Compton effect, and it ought to be noticeable for
relatively high energy photons. (For low energy photons, the
wavelength change is not very great, and so there might not be enough
difference to distinguish between the predictions of Thomson and
Compton.)
Compton observed this wavelength shift in a series of experiments
beginning in 1923. The wavelength shift was in accord with the photon
hypothesis, and this triggered widespread acceptance of the photon
hypothesis.
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How much momentum does a photon carry? Recall from our
introduction to special relativity that the energy and momentum of a
particle of mass m satisfy the following relations:
Also recall that the Lorentz-"gamma" factor is related to the speed of
the particle by
We can derive a relation between the energy and momentum of a
particle of mass m by first squaring the momentum relation above,
then solving for gamma, and substituting the resulting expression into
the energy relation:
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This is a fundamental relationship between the energy and momentum
for a particle of mass m. But might we not guess that the relationship is
also valid for a massless particle, such as a photon? In this case, the
relation simplifies to
Now, one may guess whatever one wishes, but the ultimate tests of
validity in science are, first, internal logical consistency of the theory,
and second, experimental verification. Playing with the momentum
of a photon requires a kind of logical leap, because the definition of
momentum for a particle of mass m is
If we wish to describe the momentum of a massless particle, this
clearly doesn't make sense, because then every massless particle
would have zero momentum. But, playing along for a moment, if we
take the newly-developed formula E = cp seriously, then we can
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take the newly-developed formula E = cp seriously, then we can
combine it with Einstein's relation for the energy of a photon to
obtain an expression for the momentum of a photon in terms of its
wavelength:
Thus, we have derived an expression for the momentum of a photon in
terms of its wavelength. Now, as we said earlier, you can guess whatever
you want in science, but your guess had better be consistent with other
theories, and it had better be verified by experiments. How are we to
test this guess about photon momentum? Well, this is exactly what
Compton did. Let's treat the scattering of light from an electron as a
collision problem, and use the principles of conservation of momentum
and energy to analyze the collision. Then we'll test the results of the
analysis against experiment. Compton did this, and his experimental
results were in accord with the calculations; this is strong support for the
photon hypothesis and the other guesses made here.
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Here's Compton's calculation:
Conservation of momentum in the x-direction:
Conservation of momentum in the y-direction:
Conservation of energy:
Compton's experiment was designed to measure the angle of the
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Compton's experiment was designed to measure the angle of the
scattered photon, as well as the wavelengths of the incoming and
scattered photons. Thus, we'll begin by eliminating the angle phi from
equations 1 and 2:
Squaring and adding equations 4 and 5, we obtain an expression with phi
eliminated:
Now manipulate and then square equation 3 to obtain:
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Now manipulate and then square equation 3 to obtain:
Now subtract equation 7 from equation 6 to obtain the desired relation:
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Thus,
This is Compton's relation. His experiments, the results of which were
in good agreement with the predictions of the formula, provided
decisive support for the photon hypothesis, and led to Compton
receiving the Nobel prize.
Joseph A. Gray, an Australian-Canadian physicist who worked
at Queen's University from 1924 to 1952, also did extensive
work on the scattering of photons from charged particles, and
some of my older physics profs (I was a student at Queen's in
the late 1970s and early 1980s) think he ought to have shared
credit with Compton, going so far as to call it the "ComptonGray" effect, instead of the Compton effect.
_________________________________________________________
Problem: A photon of red light (wavelength = 720 nm) and a Ping-Pong
ball (mass = 2.2 g) have the same momentum. At what speed is the ball
Ch29L Page 39
ball (mass = 2.2 g) have the same momentum. At what speed is the ball
moving?
Solution: The momentum of the photon is
The Ping-Pong ball has the same momentum, so
This speed is about 13 nm per billion years, so it's pretty, pretty,
pretty slow.
________________________________________________________
Problem: A sample is bombarded by incident X-rays, and free electrons
in the sample scatter some of the X-rays at an angle of 122 degrees
with respect to the incident X-rays (see the figure). The scattered Xrays have a momentum whose magnitude is 1.856 × 10-24 kg.m/s.
Determine the wavelength (in nm) of the incident X-rays.
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Solution: First determine the wavelength of the scattered X-rays;
then use Compton's formula.
Note the very small percentage difference between the incident and
scattered wavelengths for the X-ray, and hence the need for accurate
calculation (i.e., input data to at least four significant figures):
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_____________________________________________________
Problem: An incident X-ray photon of wavelength 0.2750 nm is
scattered from an electron that is initially at rest. The photon is
scattered at an angle of 180.0 degrees in the figure and has a
wavelength of 0.2825 nm. Use the principle of conservation of linear
momentum to find the momentum gained by the electron.
Solution: It's a head-on collision, and the X-ray is scattered straight
back. Thus, all the action takes place along a line, which we can call the
x-axis.
Before collision:
After collision:
By the principle of conservation of linear momentum,
where p is the momentum of the electron after the collision.
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where p is the momentum of the electron after the collision.
______________________________________________________
Problem: What is the maximum amount by which the wavelength of an
incident photon could change when it undergoes Compton scattering
from a nitrogen molecule (N2)?
Solution: Physically, the greatest wavelength change occurs when the
incoming photon transfers the maximum amount of momentum to the
nitrogen molecule; this happens in a head-on collision. This also makes
sense mathematically; consider Compton's formula:
How can you maximize the right side of the equation? You only have
control over the parenthesis, and its maximum value is clearly 2, when
the cosine of theta is -1, which occurs when theta is 180 degrees. Thus,
whether you argue physically or mathematically, the conclusion is the
same.
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This is an extremely small wavelength shift. If you wish to study the
Compton effect experimentally, you'd have an easier time measuring
the shift (i.e., the shift would be larger) if you used a less massive
particle, such as an electron.
________________________________________________________
Problem: An X-ray photon is scattered at an angle of 180.0 degrees
from an electron that is initially at rest. After scattering, the electron
has a speed of 4.67 × 106 m/s. Find the wavelength of the incident Xray photon.
Solution: It's a head-on collision, and the X-ray is scattered straight
back. Thus, all the action takes place along a line, which we can call the
x-axis.
Before collision:
After collision:
By the principle of conservation of linear momentum,
where p is the momentum of the electron after the collision.
By the principle of conservation of energy,
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Add equations 1 and 2 to eliminate the scattered wavelength:
_______________________________________________________
Wave-Particle Duality
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So now we have two theories of light, the wave theory and
the photon theory. Each works well in some circumstances,
fails in others. What gives? What is light, really? A particle or
a wave?
Answer: We don't know. Light is something a little
mysterious. We try to describe it using concepts that we
have abstracted from our macroscopic experience, and we
find that we can do pretty well if sometimes we use the
wave model, sometimes the particle model. But our clumsy
human models have not yet grasped the essentially sublime
character of light. Oh well, maybe someday one of you will
do better.
Wave-particle duality has been described as being somewhat
similar to a coin. A coin has two sides, but you can only see
one at a time. Similarly, light has these two aspects, but only
one aspect seems to come out in a single experiment.
To make things more interesting, it seems that matter also
exhibits the same wave-particle duality!
Matter Waves
In 1924, Louis de Broglie introduced the idea that each
moving matter particle is guided by a mysterious wave.
(Nowadays we say that they are waves, and therefore also
exhibit wave-particle duality.) He was perhaps inspired by his
love of music; he viewed an atom as a kind of symphony of
vibrating energy.
The wavelength of the wave guiding a moving particle's
motion depends on the mass and velocity of the particle:
 = h/mv
de Broglie's proposal was met with quite a lot of skepticism.
However, Einstein was an enthusiastic supporter of the idea
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However, Einstein was an enthusiastic supporter of the idea
of matter waves, and even created independent arguments
in favour of matter waves. (In fact, de Broglie submitted a
thesis based on his ideas for his Ph. D., but his thesis
supervisor, Paul Langevin, was uncertain whether such
outlandish ideas were valid, so he sent a copy of de Broglie's
thesis to Einstein for his opinion. Einstein gave the thumbs
up, and de Broglie was allowed to proceed to his thesis
defence.)
Perhaps because of Einstein's support of de Broglie's bold
idea, experimenters were encouraged to test it. In 1927,
George Thomson, and (independently) Davisson and Germer
showed that electrons diffracted from the surface of a
crystal, and from the resulting diffraction pattern, they were
able to calculate the wavelength of the electrons. The results
were consistent with de Broglie's hypothesis. Subsequently,
Otto Stern repeated the experiment using atoms, with
results that also supported de Broglie's matter wave
hypothesis.
George Thomson (who shared the Nobel Prize with Davisson in
1937) performed experiments that showed that electrons are
waves. His father, J.J. Thomson got the Nobel Prize in 1906 for his
1897 discovery of the electron as a particle, and for measuring its
particle properties. Wave-particle duality within one family! (de
Broglie got the Nobel Prize in 1929.)
Application of matter waves: electron microscope (see page
895 of the textbook for an image produced by a SEM).
The double-slit experiment using electrons (or photons) is a
compelling illustration of wave-particle duality (in both what we
traditionally consider "matter particles" and "wave phenomena"
such as light), and really highlights the strange nature of microscopic
reality, which is well-captured by the strange theory of quantum
mechanics. Richard Feynman considered the double-slit experiment
to encompass the central quantum mystery.
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to encompass the central quantum mystery.
A double-slit pattern created using very low-intensity light:
Diffraction patterns produced by X-rays, electrons, and neutrons:
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Whatever this strange dual nature is that light embodies, matter has
it too.
_______________________________________________________
Problem: Estimate your de Broglie wavelength when walking at a
speed of 1 m/s. Repeat for an electron moving at a speed of
100,000 m/s.
Solution:
_____________________________________________________
Problem: The diameter of an atomic nucleus is about 10 fm.
What is the kinetic energy, in MeV, of a proton with a de
Broglie wavelength of 10 fm?
Solution:
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_____________________________________________________
Heisenberg's Uncertainty Principle (also known as
Heisenberg's Indeterminacy Principle)
The conclusions of Heisenberg's uncertainty principle are
startling. One conclusion is that it's not possible for a subatomic
particle to be "at rest," for then the uncertainty in its position
would be zero, violating the principle. This leads to the concept
of "zero-point-energy;" that is, no matter how much you cool a
molecule, atom, or subatomic particle, there will always be some
residual kinetic energy, even at "absolute zero."
Another consequence of Heisenberg's uncertainty principle is
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Another consequence of Heisenberg's uncertainty principle is
that it calls into question the classical idea of the "clockwork
universe;" that is, the idea that the world is deterministic.
(Laplace said (1814) that if we only knew the position and
velocity of every particle in the universe at some moment, then
we could (in principle) calculate the position and velocity of
every particle in the universe at all later times, using Newtonian
mechanics.)
But causality is a bedrock principle of science ("things happen for
a reason"), whereas determinism is not.
Quantum theories are the best, most accurately verified theories
in physics. Some quantities (energy levels of atoms, lifetimes of
excited states, etc.) can be predicted (and are verified) with
extraordinary precision. However, other quantities cannot be
predicted in principle, but still in some of these cases
probabilities can be predicted extremely accurately. A good
example of this is radioactive decay, where the time at which an
individual radioactive atom transforms is unpredictable
(according to our current understanding), and yet the proportion
of a sample of radioactive atoms that will transform in a certain
time (which is equivalent to the probability that an individual
atom will transform in a certain time) can be predicted very
accurately.
A final conclusion of Heisenberg's uncertainty principle is that
subatomic particles have a wavelike essence.
How this very strange microscopic quantum behaviour gives rise
to macroscopic classical behaviour is still not well-understood,
and you might like to think more about it. It's not even clear if
the terms in the previous sentence are well-defined, because
some macroscopic objects (such as lasers and the
semiconductors in miniaturized electronic devices) are
essentially quantum devices.
____________________________________________________
Problem: A proton is confined to a nucleus that has a diameter of
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Problem: A proton is confined to a nucleus that has a diameter of
5.5 × 10-15 m. If this distance is considered to be the uncertainty
in the position of the proton, what is the minimum uncertainty in
its momentum?
Solution:
____________________________________________________
Problem: A subatomic particle created in an experiment exists in
a certain state for a time of 7.4 × 10-30 s before decaying into
other particles. Apply both the Heisenberg uncertainty
principle and the equivalence of energy and mass to determine
the minimum uncertainty involved in measuring the mass of this
short-lived particle.
Solution: Besides the original Heisenberg's uncertainty principle,
there are other similar relationships between certain pairs of
physical quantities. One such relationship is
Using the relationship E = mc2, we can write
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Thus,
____________________________________________________
Some applets that may be of interest:
Fourier synthesis: http://www.falstad.com/fourier/
http://phet.colorado.edu/en/simulation/fourier
http://eve.physics.ox.ac.uk/Personal/artur/Keble/Quanta/Applets/quantum/heisenbergmain.html
Concluding remark on wave-particle duality
Q: So, really, what is a photon?
A: "All the fifty years of conscious brooding have brought me no
closer to the answer to the question, 'what are light quanta?' Of
course, today every rascal thinks he knows the answer, but he is
deluding himself." A. Einstein, late in his life
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