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 Chapter 28 An Introduction to Quantum Physics Friday, January 14, 2011 10:03 AM Ch28P Page 1
Transcript
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
Chapter 28 An Introduction to Quantum PhysicsFriday, January 14, 201110:03 AM
Ch28P Page 1
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
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
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,
Ch28P Page 4
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
Ch28P Page 5
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
Ch28P Page 6
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
Ch28P Page 7
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:
Ch28P Page 8
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 wavetheory of light
Experimental results
Ch28P Page 9
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.
The time delay is verytiny, and it is independentof the light intensity. No matter how low youmake the light intensity,the time delay does not increase.
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 awave is a measure of its energy.)
Increasing the intensityincreases the number ofelectrons ejected, but has no effect on the energy ofindividual electrons. Therate at which electronsare ejected (i.e. current) is proportional to thelight intensity.
If light of a certain frequency isable to eject electrons from ametal, then light of any frequencyshould also be able to eject electrons from the same metal; there might be a time delay if theintensity is low.
There is a thresholdfrequency f0; for light of frequency below the threshold, no electrons are ejected, no matterhow great the light intensity is.
The stopping potential depends on the metal.For a particular metal,and for a particular lightfrequency, the stoppingpotential is the same no matter what the intensityof the incident light is.The stopping potential does depend on the
Ch28P Page 10
does depend on the frequency of the light.
The energy of ejectedelectrons increases linearly with thefrequency of the incidentlight. (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.
Ch28P Page 11
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
Experimental results
The ejection of an electronoccurs because it absorbs a singlephoton. Low-intensity light hasfew photons, but the time delay will not depend on the number of photons.
The time delay is verytiny, and it is independentof the light intensity. No matter how low youmake the light intensity,the time delay does not increase.
Increasing the intensity of thelight increases the rate at whichphotons arrive at the metal, butit does not increase the energyof each photon. More photons means more electrons areejected, but each photon still passes on the same amount of energy to each electron.
Increasing the intensityincreases the number ofelectrons ejected, but has no effect on the energy ofindividual electrons. Therate at which electronsare ejected (i.e. current) is proportional to thelight intensity.
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.
There is a thresholdfrequency f0; for light of frequency below the threshold, no electrons are ejected, no matterhow great the light intensity is.
Each electron absorbs one The stopping potential
Ch28P Page 12
Each electron absorbs one photon. It's like a person tryingto jump over a step. If you don'tmake it over the step, then you fall back down and have to tryagain.
The stopping potential is equalto Kmax/e, which depends onthe frequency according to the equation just below.
The stopping potential depends on the metal.For a particular metal,and for a particular lightfrequency, the stoppingpotential is the same no matter what the intensityof the incident light is.The stopping potential does depend on the frequency of the light.
The maximum kinetic energy of an ejected electron is
Kmax = hf E0
where E0 is the work function of the metal.
The energy of ejectedelectrons increases linearly with thefrequency of the incidentlight. (Millikan, 1915)
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
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.
Ch28P Page 14
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
Ch28P Page 15
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
Ch28P Page 16
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!
Ch28P Page 17
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.
Ch28P Page 18
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.
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:
Ch28P Page 20
Ch28P Page 21
Ch28P Page 22
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.
Ch28P Page 23
Energy Levels and Quantum Jumps
Ch28P Page 24
Heisenberg's Uncertainty Principle (also known as Heisenberg's Indeterminacy Principle)
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
