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.

Chapter 28 An Introduction to Quantum PhysicsFriday, January 14, 201110:03 AM

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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 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

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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 (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

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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 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.

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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

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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 relatedconservation 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 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

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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 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

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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

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; a notable

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established that light is a wave phenomenon; a notable milestone is Thomas Young's double-slit experiment (1800), and we studied many other examples of the wave aspect of light (various examples of interference, diffraction through slits and gratings, etc.). This part of our story culminated in Maxwell's dramatic proposal (1860s) that light is an electromagnetic wave, and the subsequent experimental confirmation by Hertz in 1887.

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 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

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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 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:

blackbody radiation1.atomic spectra2.atomic structure3.

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atomic structure3.the photoelectric effect4.

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•

As was mentioned in the previous chapter, 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 irradiance" (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

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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.

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

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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 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.)

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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, a wave just as Maxwell had described. But 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 ego-centric 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 counter-intuitive 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

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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 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

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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.)

Note that the solar spectrum contains certain dark lines against a continuous background. The continuous background

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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 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

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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). 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

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.

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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 setup:

the quantum revolution provided dramatic insights in the first quarter of the 20th century. We'll continue this part of the story in Chapter 29.

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.

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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 wavetheory of light

Experimental results

There should be a time delay, during which enough light is

The time delay is verytiny, and it is independent

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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.

of 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 energydensity.)

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 frequency of the light.

The energy of ejectedelectrons increases linearly with the

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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). 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, called photons. 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

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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 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

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 energy

Increasing the intensityincreases the number ofelectrons ejected, but has no effect on the energy of

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it 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.

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 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 called 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 •

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those fancy laser-alarms that you see in the 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)

•

Exercises:

Chapter 25, CP 34 Determine the energy (in eV) of a photon of visible light that has a wavelength of 500 nm.

Solution:

Chapter 25, CP 35 Determine the energy (in eV) of an X-ray photon that has a wavelength of 1.0 nm.

Solution:

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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.

Solution:

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.

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function.

Solution:

Chapter 28, CP 11 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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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 emits each second at maximum power.

Solution:

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.

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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 pages 935-936 of the textbook).

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

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mechanics. Richard Feynman considered the double-slit experiment 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.

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.

Solution:

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?

Solution:

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Solution:

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.

Solution:

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Energy Levels and Quantum Jumps

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.

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 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

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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

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.

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

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course, today every rascal thinks he knows the answer, but he is deluding himself." A. Einstein, late in his life

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