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Chapter 28 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
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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 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
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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.
Some people try to denigrate science using phrases such as
"it's only a 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: "A science is no more a collection of facts
than a house is a heap of stones."
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
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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 to be obeyed, but are rather
convenient generalizations of nature's workings. The
collection of all 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,
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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
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 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
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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, and many others
in the 1920s and 1930s, and by many others subsequently.
Quantum mechanics is the 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 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 about
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2P50 (Modern Physics) next year, and you'll learn 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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Consider the photoelectric effect, first observed by the same
Hertz whose experiments confirmed that light is an
electromagnetic wave. (Irony alert!) 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.
This encouraged Hertz and others to study more carefully
this photoelectric phenomenon (that light shining on a metal
helps electrons to jump out of the metal). Here's a typical
setup:
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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
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There should be a time delay,
during which enough light is
absorbed by electrons, before
electrons begin to be ejected
from the metal. The time delay
should depend on the intensity
of the light; the greater the
intensity, the smaller the time
delay.
If electrons are ejected for light
of a certain frequency, then
keeping the frequency the same
and increasing the intensity of
the light should increase the
energy of the ejected electrons.
(Classically, the intensity of a
wave is a measure of its energy.)
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.
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.
If light of a certain frequency is
There is a threshold
able to eject electrons from a
frequency f0; for light of
metal, then light of any frequency frequency below the
should also be able to eject
threshold, no electrons
electrons from the same metal; are ejected, no matter
there might be a time delay if the how great the light
intensity is low.
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 intensity
of the incident light is.
The stopping potential
does depend on the
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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 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). Note how
Einstein's proposal brilliantly explains the strange results of
photoelectric effect experiments.
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 be absorbed (if at all) by a single electron.
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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.
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
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.
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
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.
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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.
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
The stopping potential is equal of the incident light is.
to Kmax/e, which depends on
The stopping potential
the frequency according to the does depend on the
equation just below.
frequency of the light.
The maximum kinetic energy
The energy of ejected
of an ejected electron is
electrons increases
linearly with the
Kmax = hf E0
where E0 is the work function of frequency of the incident
light. (Millikan, 1915)
the metal.
Applications of the photoelectric effect
• photovoltaic cells (solar energy) (strictly speaking, this involves
semi-conductors 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 the robbery movies)..
• 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)
Exercises:
Chapter 25, CP 34 Determine the energy (in eV) of a photon
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of visible light that has a wavelength of 500 nm.
Chapter 25, CP 35 Determine the energy (in eV) of an x-ray
photon that has a wavelength of 1.0 nm.
Chapter 25, CP 41 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.
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Chapter 28, CP 7 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.
Chapter 28, CP 11 Zinc has a work function of 4.3 eV. (a)
Determine the longest wavelength of light that will release
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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.
Chapter 28, CP 21 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
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emits each second at maximum power.
Wave-Particle Duality
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!
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Matter Waves
In 1924, Louis de Broglie introduced the idea that each
moving matter particles is guided by a mysterious wave.
(Nowadays we say that they are waves, and therefore also
exhibit wave-particle duality.) He was perhaps guided 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
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.
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hypothesis.
Irony alert: 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 properties. Wave-particle
duality within one family! (de Broglie got the Nobel Prize in
1929.)
Application of matter waves: electron microscope.
Dr. Quantum and the double-slit experiment:
http://www.youtube.com/watch?v=DfPeprQ7oGc
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-captures by the strange theory of quantum
mechanics. Richard Feynman considered the double-slit experiment
to encompass the central quantum mystery.
Exercises:
CP 26 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.
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CP 31 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?
Energy Quantization For Bound Particles
Consider a string tied at both ends, such as a guitar string, or a
piano string. When the string is plucked, standing waves are
set up on the string. That is, the string vibrates in a pattern
such that the number of half-cycles in the pattern is a whole
number. That is, if L is the length of the string, then:
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Exercises:
CP 34 Determine the length of a box in which the minimum
energy of an electron is 1.5 × 10-18 J.
CP 37 The nucleus of a typical atom is 5.0 fm in diameter. A
very simple model of the nucleus is a one-dimensional box in
which protons are confined. Estimate the energy of a proton in
the nucleus by determining the first three allowed energies of a
proton in a box 5.0 fm long.
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Energy Levels and Quantum Jumps
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Heisenberg's Uncertainty Principle (also known as
Heisenberg's Indeterminacy Principle)
Some applets:
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
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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
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