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Chapter 29 Particles and Waves, An Introduction to Quantum Physics Friday, January 14, 2011 10:03 AM Physical Theories How does science work? (At least in the case of physics and other mathematical sciences.) First you observe the world, and you also do experiments. You also abstract from the many observations and experiments the key quantities (such as position, velocity, force, etc.) that will appear in your theories. Then you create theories that relate the key quantities in ways that help you explain the phenomena that you observed and/or appeared in your experiments. You create the theories by guessing; that's right, guessing. (OK, if you want a fancy word for it, you can call it inductive logic. But it's still guessing.) Of course, I don't mean random guessing. It's a creative process that requires a deep knowledge of the current scientific understanding of the world, and it helps if you know the history of the development of science. What I'm trying to say is that you don't logically derive the laws of physics; you just create them. Then, once they are created, you test them using logic, and if they survive these tests, then you test them using observations and experiments. Ultimately, the vast majority of theories are discarded; few survive to form part of the ever-evolving currently generally accepted body of science. You test your theories against the phenomena that you observed/experimented on. If the equations of your theory predict results that agree with your observations or experiments, then good. If not, you will have to modify your theory, or maybe discard it and start from scratch. Then you use deductive logic to try to derive consequences of the theory that were not observed before. If you can do this, and if subsequent experiments or observations agree with the predictions of the theory, then that is very good. Otherwise, you will have to modify Ch29L Page 1 the theory, then that is very good. Otherwise, you will have to modify your theory, or maybe discard it and start from scratch. Logic plays a key role in testing physical theories. A theory of physics must be logically consistent; for example, it must not be possible to derive two contradictory predictions from the theory. But the creation of a theory is not necessarily a logical process, at least not in the same sense. Intuition, analogy, "feeling," play a greater role in creation; logic plays the primary role in testing the theory for consistency and for deriving consequences and predictions. But the ultimate test of a physical theory is observation and experimental verification. No amount of experimental or observational testing can ever prove a scientific theory correct. Though scientists sometimes use such terms (saying a theory is right or true or correct) colloquially, they are not meaningful because a scientific theory can never be proved correct, because it's impossible to test the theory at all points in space and at all times. Asking whether a scientific theory is correct is like asking whether your marriage is red or green (which would be truly confusing if you are married to Red Green). Or asking whether a sculpture by Modigliani is true or false. Such questions are meaningless. Although Picasso once said that "Art is a lie that helps you see the truth." Beautiful, isn't it? And a scientific theory is something like an art work as well: A human creation that is somehow false (has approximations built in, has oversimplifications, idealizations, has limited applicability, etc.), but yet helps us gain insight into our wonderful world. And yet there has to be some truth to physical theory; just look at all the bridges and buildings that exist without falling down, and the cars that move along the streets, and the computers that we all use, and the electrical power systems that bring electricity into our homes, and all the machines in our hospitals, etc. All of this wonderful technology is based on our understanding of science, and it would be extreme to suggest that there is no truth whatsoever in it. (And yes, there is a dark side to technology, but that we shall discuss another time.) Some people try to denigrate science using phrases such as "it's only a theory." That demonstrates a profound misunderstanding of science Ch29L Page 2 theory." That demonstrates a profound misunderstanding of science (or perhaps a willful attempt to mislead). There is a difference between the every-day use of the term "theory," to mean uninformed speculation, and the scientific use of the term theory. If science were like the Olympic games, then achieving the status of "theory" would be analogous to winning a gold medal. Becoming a theory (successfully tested by observation and experiment) is the pinnacle of achievement for a scientific idea. So, although scientific theories can't be proven correct, they are nevertheless precious. They represent the highest achievements in scientific thought. They represent the most successfully tested, hardened-by-trials products of the scientific enterprise. The vast majority of scientific ideas end up in the slag heap; the best theories are the survivors. Reflect on the words of Henri Poincare, which emphasize the role of creativity: "Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house." Also reflect on the words of Isaac Asimov: "Consider some of what the history of science teaches. First, since science originated as the product of men and not as a revelation, it may develop further as the continuing product of men. If a scientific law is not an eternal truth but merely a generalization which, to some man or group of men, conveniently described a set of observations, then to some other man or group of men, another generalization might seem even more convenient. Once it is grasped that scientific truth is limited and not absolute, scientific truth becomes capable of further refinement. Until that is understood, scientific research has no meaning." If he were writing today, Asimov would no doubt have used the word "person" instead of "man," but I'm sure you get the idea: Laws of physics are not directives to be obeyed, but are rather convenient generalizations describing nature's workings. The collection of all physical theories is like a vast work of art; nobody would call it Ch29L Page 3 physical theories is like a vast work of art; nobody would call it correct, but it's beautiful, and absolutely useful. The bridges engineers design using Newton's laws don't fall down, do they? And the MP3 players made using principles of electromagnetism and quantum theory are rather functional as well. So physical theories are not "true," but they are tightly constrained to apply very closely to this world. But some day, maybe tomorrow, maybe next century, someone (maybe one of you?) may create a new theory, that is somehow more beautiful, or more useful, or in some way of value, so that it may supersede or replace an existing theory of physics. **************** Classical Mechanics and Quantum Mechanics OK, now let's get down to some specifics about quantum mechanics (also called quantum theory, also called quantum physics). To put this in perspective, let's first say a few words about classical mechanics (also called Newtonian mechanics). Mechanics can be broadly divided into two branches, kinematics and dynamics. Kinematics is the description of motion, particularly the mathematical description of motion, and dynamics is an explanation for how the causes of motion (forces) create motion (that is, dynamics is a quantitative version of "everything happens for a reason"). So classical kinematics is all about describing motion in terms of position, velocity, acceleration, angles, and so on, and then understanding the relations among the variables. Classical dynamics consists of Newton's laws of motion and related conservation laws. Classical mechanics is a very successful theory. Using classical mechanics, we have built great cities, long bridges and tunnels, engines of all kinds, and aircraft and spacecraft that fly into the skies and into space. Supplementing classical mechanics with the classical theory of electricity and magnetism, we have created motors and generators, and communicate wirelessly across continents in an Ch29L Page 4 generators, and communicate wirelessly across continents in an instant. All of these applications are successful tests of the classical theories of mechanics and electromagnetism. We use the theories, do the math, and figure out how to build the rockets, how long to keep the engines on, when and in which direction to blast the engines to correct the course, and so on. And voila! The spacecraft actually makes it to the moon. The predictions of the theory are verified in practice, and this gives us confidence that the theory is useful. However, when we apply the classical theories of electricity and magnetism to atoms and their innards, they fail. Completely. And. Utterly. Fail. Does that mean the classical theories are wrong? Well, yes, I suppose so. But they worked so well for building the bridges, and for sending spacecraft to the moon, and for safely lighting our houses, and for sending TV and radio signals around the world, so it seems like a pity to throw the theories away just because they fail in the atomic and subatomic realms. So we don't throw them away; we just recognize their limitations along with the realms in which they are wonderfully useful. But we have to come up with theories that work in the atomic and subatomic realms. This was done by many physicists; it was a real team effort, led by Planck, Einstein, Bohr and many others in the early days (1900 to the 1920s), by Heisenberg, Schrödinger, Dirac, Born, and many others in the 1920s and 1930s, and by many others subsequently. Quantum mechanics is a theory that successfully describes motions within atoms. It forms the foundation for atomic and molecular physics (and chemistry), solid-state physics (also called condensed matter physics), lasers, fibre optics, and other photonic systems, and so on. Quantum physics is even being applied nowadays to understand microbiology! Quantum physics, together with modern theories of electromagnetism, have been applied to produce the basic devices Ch29L Page 5 electromagnetism, have been applied to produce the basic devices that underlie many of our neat modern technologies. The laser devices (CD and DVD players, optical memory drives, laser surgical devices, etc.), all the miniaturization that goes on in the computer world, the fancy new materials, the solar (photovoltaic) cells, and so on, all of it is possible thanks to quantum mechanics. In this course we'll have a very brief introduction to quantum ideas. If you want a more in-depth introduction, take Physics 2P50 (Modern Physics) next year, and you'll learn more about Einstein's theory of special relativity as a bonus! And if you want some great introductory books to read over the summer, try one or more of these: Thirty Years That Shook Physics, by George Gamow (full of funny stories about the great physicists of the early 20th century, told by someone who rubbed shoulders with them) The Strange Story of the Quantum, by Banesh Hoffmann Ch29L Page 6 The state of affairs in physics at about 1900 By the turn of the 20th century, classical mechanics (Newton and his successors) and the classical theory of electricity and magnetism (Faraday, Maxwell, and their contemporaries and successors) were well-established core theories of physics. Additionally, there was a large body of evidence that firmly established that light is a wave phenomenon, in contradiction to Newton's "corpuscular" theory of light, which he pioneered in the late 1600s and early 1700s. Historically, there were two simplified models for light, one in terms of particles and the other in terms of waves. Newton considered light to be a stream of particles (he called them "corpuscles," which is just a fancy word for particles, so his theory became known as the corpuscular theory), and his Ch29L Page 7 theory became known as the corpuscular theory), and his contemporary rival Huygens considered light to be made up of waves. Newton's great work on this subject, Opticks, which was published in 1704, reported on a large number of optical experiments that Newton had carried out, and he was able to successfully explain the phenomena by thinking of light as a stream of tiny particles. Huygens, on the other hand, published his great work on optics, Treatise on Light, in 1690, and he conceived of light as being formed from longitudinal waves. His theory was also very successful in terms of explaining basic reflection, refraction, etc., what we would call geometrical optics nowadays. Newton's theory was much more widely accepted in the early 1700s, probably because Newton was so famous. Another reason for opinion swaying to Newton's side was Huygens's longitudinal wave theory's failure to explain birefringence (double refraction): The wave theory of light was rehabilitated later in the 1700s by treating light as a transverse wave, but for most of the 1700s Ch29L Page 8 treating light as a transverse wave, but for most of the 1700s Newton's theory of light ruled. Another reason for the widespread acceptance of Newton's theory is that one doesn't notice wave properties of light with the naked eye. For example, light casts sharp shadows, so the Newtonians argued that light couldn't possibly be waves, because it would bend around corners and shadows would be blurry. Waves experience all kinds of interference effects, and none of them were apparent at the time of Newton: However, one of the problems with Newton's particle theory of light is the observation that two light beams can pass through each other unaffected. This is very hard to explain with the corpuscular theory (there ought to be an enormous amount of scattering going on as the little particles smash against each other), but very easy to explain with a wave theory of light. Nevertheless, on balance, Newton's particle theory of light was generally accepted. One of the strengths of science is that it is continually revised. Nothing is set in stone, and we don't accept something just because famous great scientist X said so. (Although that was certainly true in the past; witness the ideological devotion to Aristotle's theories throughout the Middle Ages, just because it was Aristotle. However, we know better since the time of Galileo than to slavishly accept a great person's opinions just because the person is great; that is unscientific.) There is a continual search for evidence that could support or question a Ch29L Page 9 a continual search for evidence that could support or question a theory, and in this way ideas are shaped, evolve, and improved, and science progresses. By the early 1800s, numerous experiments by Thomas Young, Augustin Fresnel, and others, made it very difficult to believe in the particle theory of light. It was natural and satisfying to explain the experiments in terms of a wave theory, and so the wave theory of light soon came to dominate. It became clear that wave aspects of light were not discovered sooner because the wavelength of light is extremely small. A few classic phenomena are Newton's rings, and interference as shown by Young's double-slit experiment: http://video.mit.edu/watch/thomas-youngs-double-slitexperiment-8432/ Ch29L Page 10 By the 1800s the situation seemed clear beyond the shadow of a doubt: light is a wave. The work of Maxwell in the 1860s, with experimental verification by Hertz and others starting in 1887, made it clear what kind of a wave light is: a transverse electromagnetic wave. However, as we shall see, the situation was complicated in the early 1900s when it became clear that light is also a particle! The evidence for the particle-like nature of light is also very clear, and nowadays we call particles of light "photons." An important part of classical physics that we did not study in PHYS 1P22/1P92 or in PHYS 1P21/1P91 is statistical mechanics. If you studied PHYS 1P23/1P93 you learned a bit about thermodynamics, the science of the flow of thermal energy and its interactions with mechanical forces. In the latter part of the 1800s a fundamental theory of physics, called statistical mechanics, was developed by Boltzmann (and others; Maxwell also made important contributions), that explained thermodynamics in terms of the interactions of a swarm of microscopic particles (molecules and atoms) described by the laws of Newtonian mechanics. The theory had many successes (although its early derisive detractors may have had a role in Boltzmann's subsequent sad descent into insanity), and convinced Ch29L Page 11 Boltzmann's subsequent sad descent into insanity), and convinced many scientists of the reality of atoms long before there was definitive evidence. The numerous successes of physics in describing the natural world throughout the 19th century convinced some foolhardy physicists that all the fundamental aspects of physics were already well understood, and there was nothing new (in a fundamental sense) left to discover. A notorious example of this variety of hubris is the following pronouncement of A.A. Michelson, in 1903: “The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote. Nevertheless, it has been found that there are apparent exceptions to most of these laws, and this is particularly true when the observations are pushed to a limit, i.e., whenever the circumstances of experiment are such that extreme cases can be examined. Such examination almost surely leads, not to the overthrow of the law, but to the discovery of other facts and laws whose action produces the apparent exceptions. As instances of such discoveries, which are in most cases due to the increasing order of accuracy made possible by improvements in measuring instruments, may be mentioned: first, the departure of actual gases from the simple laws of the so-called perfect gas, one of the practical results being the liquefaction of air and all known gases; second, the discovery of the velocity of light by astronomical means, depending on the accuracy of telescopes and of astronomical clocks; third, the determination of distances of stars and the orbits of double stars, which depend on measurements of the order of accuracy of one-tenth of a second-an angle which may be represented as that which a pin's head subtends at a distance of a mile. But perhaps the most striking of such instances are the discovery of a new planet or observations of the small irregularities noticed by Leverrier in the motions of the planet Uranus, and the more recent brilliant discovery by Lord Rayleigh of a new element in the atmosphere through the minute but unexplained anomalies found in weighing a given volume of nitrogen. Many other instances might be cited, but these will suffice to justify the statement that 'our future discoveries must be looked for in the Ch29L Page 12 the statement that 'our future discoveries must be looked for in the sixth place of decimals.'” However, the quantum revolution had already begun, with the astonishing work of Planck in 1900; we'll get to this shortly. First, though, let's discuss four physical phenomena that were rather puzzling around this time: 1. blackbody radiation 2. atomic spectra 3. atomic structure 4. the photoelectric effect We'll discuss each of these important situations, then we'll discuss how they led to the introduction of the early ideas of quantum physics, and we'll also discuss how these quantum ideas helped to solve the puzzles. Then we'll discuss some of the mysteries of quantum physics that still remain, waiting for current or future researchers (maybe some of you?) to provide further insight. • blackbody radiation Warm solid objects glow, as do warm liquids and warm samples of gas. The warm objects emit electromagnetic radiation, with various amounts emitted at various wavelengths; the precise amounts depend on the material, the characteristics of its surface, and especially its temperature. A graph of what is called "spectral radiance" (check the units on the vertical axis of the graph below) vs. wavelength for an idealized emitter called a blackbody is shown below: Ch29L Page 13 The blue, green, and red curves represent experimental data for emitters that are excellent approximations to blackbodies. The blue curve represents an emitter at a temperature of 5000 K, the green curve represents an emitter at a temperature of 4000 K, and the red curve represents an emitter at a temperature of 3000 K. The emitted intensity is proportional to the area under each graph; note that the intensity increases with increasing temperature. The statistical mechanics of Boltzmann and Maxwell, so successful at providing a microscopic explanation for so many thermodynamical properties of matter, resulted in total nonsense (represented by the black curve in the diagram) when applied to the problem of blackbody radiation by Rayleigh and Jeans. They predicted that an infinite amount of energy would be radiated, with an increasing amount of energy emitted as you move towards the UV end of the spectrum. This prediction catastrophically contradicts experimental results. (The phrase "ultraviolet catastrophe" was coined by Ehrenfest.) Trying to explain the spectrum of a blackbody was a puzzle at the turn of the 20th century. Planck resolved this puzzle in December 1900 (published in 1901) with a truly ingenious idea, that is all the more remarkable because it went exactly opposite to his understanding at the time. Ch29L Page 14 Think about, for example, people locked in an argument. How often is one of the parties convinced by the other to change his or her view? It takes a lot of courage to do this. Planck was one of the worldwide experts in thermodynamics at this time, and he made a lot of progress by assuming that heat is a continuous phenomenon, much like a wave or a field. The atomists (led by Boltzmann) adopted the opposite stance, saying that all of the macroscopic phenomena (such as heat, for example) are ultimately the result of an enormous number of chaotic motions of microscopic particles (atoms or molecules). Planck decided to attempt to derive a family of formulas for the blackbody radiation curves (in the graph above) as a way of proving that his perspective was right and the atomists were wrong. A microscopic explanation for the blackbody radiation curves was one of the big outstanding problems at the time, and if Planck could solve it this would give him powerful ammunition for his side of the argument. He failed for six years. Then one day, desperate to solve the problem, he tried an extravagant mathematical trick, as a kind of wild guess. In trying to derive a formula, he had been up to now doing the usual thing, which is to determine the total power emitted by the blackbody by using calculus; that is, by integrating the power in each mode of vibration of the blackbody. This is the same method used by Rayleigh and Jeans (and by Wien, and everyone else), with "catastrophic" results. Part of the problem is that Rayleigh and Jeans had used the equipartition principle (from Boltzmann), which stated that each mode of vibration is equally likely. This means that very high-energy modes of vibration are just as likely as low-energy modes of vibration, which resulted in the UV catastrophe. Planck tried, as a guess, what must have been unthinkable to everyone else: he used a regular sum (albeit an infinite sum) instead of an integral. That is, he avoided calculus. And, surprisingly, he obtained a family of formulas that seemed to work! He presented his work at a regular meeting of the German Physical Society in October 1900, and Ch29L Page 15 regular meeting of the German Physical Society in October 1900, and very soon after he received confirmation from experimental physicists that his formula worked extremely well for all available experimental data. But what does the trick mean? Planck could not be satisfied with merely coming up with a "rabbit-out-of-a-hat" formula, he wanted to explain the fundamental physical idea behind the formula. (In effect, he had figured out the answer, but now he had to "show his work!") This he set about doing, in what he later described as the most strenuous period of work in his life. Alas, he found that his mathematical trick kept leading him towards Boltzmann's atomistic view of microscopic physics, and away from his own continuous perspective. Rather than fight the inevitable, he gave in, gave Boltzmann credit for being right, and converted to the atomist perspective. Planck's physical explanation for his mathematical trick is that the energy of the elementary vibrators in the glowing solid, which are responsible for emitting electromagnetic radiation, can have not just any energy, but rather only whole-number multiples of a certain unit of energy, which he called an elementary quantum of energy. (Nowadays we would interpret "elementary vibrators" as vibrating atoms, but then it was unclear whether atoms even existed, so they simply spoke in terms of modes of vibrations without being more specific.) Thus, according to Planck, each mode of vibration of the glowing object could have only a discrete amount of energy, which is related to its frequency through the formula E = hf, where h is a universal constant now known as Planck's constant. The value of Planck's constant is extremely tiny, which explains why we don't notice it in everyday life. Of course, according to Planck, once the vibrators actually emitted electromagnetic radiation, the emitted radiation was nice and continuous, electromagnetic waves just as Maxwell had described. But while the vibrators were collecting up the energy needed for emission Ch29L Page 16 while the vibrators were collecting up the energy needed for emission of electromagnetic waves, the amount of energy collected was quantized. Planck neatly explained away the ultraviolet catastrophe by relying on Boltzmann's statistical mechanics perspective: High-energy quanta of radiated energy are far less likely than low-energy quanta, because it takes time to amass the necessary energy. In that time, it's far more likely for the excess energy to be radiated away in lower-energy quanta. The explanation made sense to Planck and his contemporaries, and did much to sway the physics community to the atomist perspective, simultaneously lending support to Boltzmann. Planck was a conciliator by nature, and one wonders whether a more egocentric scientist would have been quite so willing to embrace and then publicly champion his opponent's ideas. Planck's revolutionary idea, which he presented at another regular meeting of the German Physical Society in December 1900 (the paper was published in 1901) ushered in the quantum age. As counterintuitive as Planck's idea was, everyone (including Planck) still understood that once a blackbody emitted electromagnetic radiation, even if it was emitted in blobs, the electromagnetic radiation would move as an electromagnetic wave. After all, there were decades of solid experimental evidence that light is a wave phenomenon, and growing experimental evidence in the past 15 years or so that light is an electromagnetic wave. It would take Einstein to further overturn this neat view of electromagnetic radiation in 1905, just a few years later. But we'll get to that part of the story soon; first let's continue to talk about light emitted by matter. • atomic spectra By the late 1800s there was an enormous amount of experimental data on light emitted by matter. Besides the continuous spectra of the blackbody type, discussed above, glowing objects also emitted what Ch29L Page 17 blackbody type, discussed above, glowing objects also emitted what are called discrete spectra, or equivalently, line spectra. Often these two types of spectra are superimposed, so that it takes some work to separate the two types. Compare the following diagram (of the Sun's emission) with the continuous blackbody spectra studied previously: Notice that the Sun's emission spectrum in the previous diagram is similar to blackbody radiation curves, except that the intensity is less at certain wavelengths. What is decreasing the intensity at certain wavelengths? Here are some examples of discrete spectra. They are produced by passing the electromagnetic radiation emitted by an object through a prism, so that its various wavelengths are separated. (The electromagnetic radiation is first passed through a slit, which results in lines, rather than dots or other shapes.) Ch29L Page 18 Note that the solar spectrum contains certain dark lines against a continuous background. The continuous background represents (nearly) blackbody radiation produced because of the Sun's temperature (about 5800 K). The dark lines in the Sun's spectrum are known as absorption lines; apparently something in the Sun's atmosphere is absorbing light of certain wavelengths that had been emitted from the Sun's surface, so that these wavelengths are now missing from the spectrum. The bright lines in the other spectra in the diagram are known as emission lines. (The other spectra are from gaseous samples of the given elements.) Notice the bright yellow line in sodium's spectrum; there are other emission lines in sodium's spectrum, but only a portion Ch29L Page 19 there are other emission lines in sodium's spectrum, but only a portion of the range of wavelengths is shown in the diagram, and in this portion the only emission line is the one shown. The emission lines in the other spectra have the same character: Only certain wavelengths appear, and they are separated by gaps that vary in size. By the late 1800s, there was an enormous amount of spectroscopic data, but no explanation for why discrete spectra occur. Why is electromagnetic radiation emitted only at certain wavelengths? Is there any pattern? What is the physical explanation? In 1885, Johann Balmer guessed a formula that described part of the spectrum of hydrogen; that is, one could calculate the wavelengths by substituting certain values into the formula. This was quite a spectacular achievement given how little data Balmer had to work with: Just FOUR wavelengths! Other workers, particularly Johannes Rydberg, generalized Balmer's formula, and stimulated the search for (and discovery of) further spectral lines of hydrogen. We'll check out Rydberg's formula in the next chapter. So a pattern was discovered in the wavelengths of the spectral lines of hydrogen, but is there a similar pattern in the spectra of other elements? And, above all, what is the physical explanation behind this? This remained a prominent puzzle at the turn of the 20th century. Connected to this puzzle was the puzzle of atomic structure. Now that more and more scientists were becoming convinced about the reality of atoms, the question of what atoms are like came to the forefront. • atomic structure Some of the ancients believed that the world consists of indivisible little parts, which they called atoms. The idea was revived in the 1700s and 1800s, and supported by the work of John Dalton (through his 1805 law of definite proportions, which described chemical reactions). Ch29L Page 20 1805 law of definite proportions, which described chemical reactions). Dalton built on the earlier (1789) work of Antoine Lavoisier, who came up with the idea of a chemical element and the law of conservation of mass for chemical reactions. A series of experiments by a number of workers, culminating in the work of J.J. Thomson in 1897, identified electrons as negatively charged constituents of atoms. But atoms are normally electrically neutral, so there must be separate positively charges parts of an atom; this line of reasoning suggests that atoms have structure, and are not the simple, indivisible particles conceived of by the ancients. OK, so atoms seem to contain separate bits of positive and negative charges; what exactly is the structure of the interior of an atom? This was in important scientific issue in the early 1900s, and the quantum revolution provided dramatic insights in the first quarter of the 20th century. We'll continue this part of the story in Chapter 30. • the photoelectric effect As we've already discussed, by the 1800s it was very well established by a number of decisive experiments that light is a wave phenomenon. (Think back to the earlier part of this course, where we discussed wave interference, diffraction, and so on.) Maxwell hypothesized in the 1860s that light is an electromagnetic wave and by the late 1880s Heinrich Hertz had confirmed Maxwell's theory with a series of careful experiments. In an ironic twist, the very experiment in which Hertz confirmed that light is a wave also planted seeds of doubt in its validity! In the experiment where he used electromagnetic waves to induce a spark across a small gap in a loop of wire, he noticed that the sparks came a little more readily if ultraviolet light was shining on the loop of wire. Hertz died young (before his 37th birthday) and had no chance to follow up on this puzzling phenomenon, but others did study this photoelectric phenomenon more carefully (that light shining on a metal helps electrons to jump out of the metal). Here's a typical Ch29L Page 21 metal helps electrons to jump out of the metal). Here's a typical setup: The applied voltage could be varied; experiments showed that if the voltage exceeded a certain amount (called the stopping potential, or stopping voltage, Vstop), then no electrons reached the collector plate of the tube. The wave theory of light predicts that the results of the experiments should be as follows. (Imagine that light is like waves on an ocean, and that the electrons are like buoys floating on it.) Predictions based on the wave Experimental results theory of light There should be a time delay, during The time delay is very tiny, and it which enough light is absorbed by is independent of the light Ch29L Page 22 which enough light is absorbed by is independent of the light electrons, before electrons begin to intensity. No matter how low you be ejected from the metal. The time make the light intensity, the time delay should depend on the intensity delay does not increase. of the light; the greater the intensity, the smaller the time delay. If electrons are ejected for light of a Increasing the intensity increases certain frequency, then keeping the the number of electrons ejected, frequency the same and increasing but has no effect on the energy the intensity of the light should of individual electrons. The rate increase the energy of the ejected at which electrons are ejected electrons. (Classically, the intensity of (i.e. current) is proportional to a wave is a measure of its energy the light intensity. density.) If light of a certain frequency is There is a threshold frequency f0; able to eject electrons from a for light of frequency below the metal, then light of any frequency threshold, no electrons are should also be able to eject ejected, no matter how great the light intensity is. electrons from the same metal; there might be a time delay if the intensity is low. The stopping potential depends on the metal. For a particular metal, and for a particular light frequency, the stopping potential is the same no matter what the intensity of the incident light is. The stopping potential does depend on the frequency of the light. The energy of ejected electrons increases linearly with the frequency of the incident light. (Millikan, 1915) Nobody was able to explain the experimental results of Hertz, Hallwachs, Lenard, Stoletov, J.J. Thomson, and others, in a satisfactory way. That is, until Einstein came on the scene in 1905 Ch29L Page 23 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 Js). We’ll discuss shortly how Einstein's proposal brilliantly explains the strange results of photoelectric effect experiments. Also notice how much further Einstein pushed Planck's hypothesis beyond what Planck first proposed; Planck said that elementary vibrators within a blackbody collected energy in quanta, but once the energy was emitted as electromagnetic radiation, it travelled in waves. Einstein went much further and postulated that light EXISTS as little particles, which were later called photons (Einstein called them "light quanta"). Einstein said that light is emitted, absorbed, and exists as particles. Stop for a moment and marvel at the boldness of creative scientists such as Planck and Einstein; they were able to entertain "what if" games of thought, and were not afraid to publicly communicate their conclusions even though they directly contradicted well-established scientific theories. In Einstein's case, his bold proposal was made in the face of a century of very solid evidence that light is a WAVE. But how does Einstein's photon hypothesis square with all the evidence that light is a wave? Einstein proposed that the connection is statistical; that is, light consists of photons (little particles of light) that in aggregate behave as a wave. In Einstein's viewpoint (light is composed of photons), the intensity of the light is a measure of the number of photons per second falling on the metal per unit area. The basic idea is that typically each electron will absorb a single photon (absorbing two or more at once will be a very rare event). It's apparently not possible for a photon to be partly absorbed; it's either completely absorbed, or not at all. Similarly, it's apparently not possible for a photon to have some of its energy absorbed by one electron and some by another; all of its energy must Ch29L Page 24 be absorbed (if at all) by a single electron. If the single absorbed photon imparts enough energy to the electron, then it will escape the metal. If not, then the electron's extra energy is likely to be exchanged with other electrons or the atoms in the metal through collisions before the electron has a chance to absorb another photon. Here is a summary of how Einstein's hypothesis neatly explains the photoelectric effect; also see page 929 of the textbook for a nice summary. Einstein's explanation based on the photon theory of light The ejection of an electron occurs because it absorbs a single photon. Low-intensity light has few photons, but the time delay will not depend on the number of photons. Experimental results Increasing the intensity of the light increases the rate at which photons arrive at the metal, but it does not increase the energy of each photon. More photons means more electrons are ejected, but each photon still passes on the same amount of energy to each electron. If the energy of each individual photon is not enough, then no electron will be ejected, no matter how many photons arrive at the metal, because each electron absorbs one photon at a time. Each electron absorbs one photon. It's like a person trying to jump over a step. If you don't make it over the step, then you fall back down and have to try again. Increasing the intensity increases the number of electrons ejected, but has no effect on the energy of individual electrons. The rate at which electrons are ejected (i.e. current) is proportional to the light intensity. Ch29L Page 25 The time delay is very tiny, and it is independent of the light intensity. No matter how low you make the light intensity, the time delay does not increase. There is a threshold frequency f0; for light of frequency below the threshold, no electrons are ejected, no matter how great the light intensity is. The stopping potential depends on the metal. For a particular metal, and for a particular light frequency, the stopping potential is the same no matter what the have to try again. The stopping potential is equal to Kmax/e, which depends on the frequency according to the equation just below. The maximum kinetic energy of an ejected electron is Kmax = hf E0 where E0 is called the work function of the metal. is the same no matter what the intensity of the incident light is. The stopping potential does depend on the frequency of the light. The energy of ejected electrons increases linearly with the frequency of the incident light. (Millikan, 1915) Applications of the photoelectric effect • photovoltaic cells (solar energy) (strictly speaking, this involves semiconductors and therefore is more complicated than the simple photoelectric effect) • charge-coupled devices (used in digital cameras) • some smoke detectors (and the same principle is used for those fancy laser-alarms that you see in robbery movies); safety switches on automatic garage-door openers/closers work on the same principle) • photocopy machines; see http://en.wikipedia.org/wiki/Photocopier (strictly speaking, this involves semi-conductors and therefore is more complicated than the simple photoelectric effect) • (photosynthesis is not an example of the photoelectric effect, but it's a process by which photons of the "right" energy are absorbed to induce a chemical reaction; similarly, vision in the eye involves the absorption of photons by retinal cells and the consequent creation of an electrical current in the optic nerve) ___________________________________________________ Problem: Determine the energy (in eV) of a photon of visible light that has a wavelength of 500 nm. Solution: Ch29L Page 26 ________________________________________________________ Problem: Determine the energy (in eV) of an X-ray photon that has a wavelength of 1.0 nm. Solution: _____________________________________________________ Problem: The intensity of electromagnetic radiation from the sun reaching the earth's upper atmosphere is 1.37 kW/m2. Assuming an average wavelength of 680 nm for this radiation, determine the number of photons per second that strike a 1.00 m2 solar panel directly facing the sun on an orbiting satellite. Solution: Ch29L Page 27 ________________________________________________________ Problem: Electrons are emitted when a metal is illuminated by light with a wavelength less than 388 nm but for no greater wavelength. Determine the metal's work function. Solution: Ch29L Page 28 ______________________________________________________ Problem: Zinc has a work function of 4.3 eV. (a) Determine the longest wavelength of light that will release an electron from a zinc surface. (b) A 4.7 eV photon strikes the surface and an electron is emitted. Determine the maximum possible speed of the electron. Solution: Ch29L Page 29 ______________________________________________________ Problem: Station KAIM in Hawaii broadcasts on the AM dial at 870 kHz, with a maximum power of 50,000 W. Determine how many photons the transmitting antenna emits each second at maximum power. Solution: ______________________________________________________ The Compton Effect Despite the spectacular success of Einstein's photon hypothesis in explaining the photoelectric effect, the photon hypothesis was not met with widespread acceptance. It's hard for people to accept contradictory facts; on the one hand, by the early 20th century an enormous amount of evidence had been collected confirming that light is a wave. Now this upstart is saying that light is also a particle? How can that be?? In particular, one aspect of the photon hypothesis is that photons carry Ch29L Page 30 In particular, one aspect of the photon hypothesis is that photons carry momentum as well as energy. Well? Where was the experimental evidence that photons carry momentum? In the absence of this kind of evidence, it was natural for physicists to be skeptical about the photon hypothesis. Yet another line of evidence for the photon hypothesis came from what we now call Compton scattering. At that time J.J. Thomson explained the scattering of electromagnetic waves from free charged particles using Maxwell's theory of electromagnetism, and the theory was supported experimentally for incident light of low intensity. The classical theory predicts NO change in wavelength of the scattered electromagnetic waves. However, in 1923 Compton hypothesized that if light were really composed of photons, then high-energy photons scattering from charged particles should experience a shift in their wavelength upon scattering; some of a photon's energy would be transferred to the charged particle in the collision, and because the energy of a photon is related to its wavelength, the scattered photon would have a different wavelength than before it collided with the charged particle. This is now called the Compton effect, and it ought to be noticeable for relatively high energy photons. (For low energy photons, the wavelength change is not very great, and so there might not be enough difference to distinguish between the predictions of Thomson and Compton.) Compton observed this wavelength shift in a series of experiments beginning in 1923. The wavelength shift was in accord with the photon hypothesis, and this triggered widespread acceptance of the photon hypothesis. Ch29L Page 31 How much momentum does a photon carry? Recall from our introduction to special relativity that the energy and momentum of a particle of mass m satisfy the following relations: Also recall that the Lorentz-"gamma" factor is related to the speed of the particle by We can derive a relation between the energy and momentum of a particle of mass m by first squaring the momentum relation above, then solving for gamma, and substituting the resulting expression into the energy relation: Ch29L Page 32 Ch29L Page 33 This is a fundamental relationship between the energy and momentum for a particle of mass m. But might we not guess that the relationship is also valid for a massless particle, such as a photon? In this case, the relation simplifies to Now, one may guess whatever one wishes, but the ultimate tests of validity in science are, first, internal logical consistency of the theory, and second, experimental verification. Playing with the momentum of a photon requires a kind of logical leap, because the definition of momentum for a particle of mass m is If we wish to describe the momentum of a massless particle, this clearly doesn't make sense, because then every massless particle would have zero momentum. But, playing along for a moment, if we take the newly-developed formula E = cp seriously, then we can Ch29L Page 34 take the newly-developed formula E = cp seriously, then we can combine it with Einstein's relation for the energy of a photon to obtain an expression for the momentum of a photon in terms of its wavelength: Thus, we have derived an expression for the momentum of a photon in terms of its wavelength. Now, as we said earlier, you can guess whatever you want in science, but your guess had better be consistent with other theories, and it had better be verified by experiments. How are we to test this guess about photon momentum? Well, this is exactly what Compton did. Let's treat the scattering of light from an electron as a collision problem, and use the principles of conservation of momentum and energy to analyze the collision. Then we'll test the results of the analysis against experiment. Compton did this, and his experimental results were in accord with the calculations; this is strong support for the photon hypothesis and the other guesses made here. Ch29L Page 35 Here's Compton's calculation: Conservation of momentum in the x-direction: Conservation of momentum in the y-direction: Conservation of energy: Compton's experiment was designed to measure the angle of the Ch29L Page 36 Compton's experiment was designed to measure the angle of the scattered photon, as well as the wavelengths of the incoming and scattered photons. Thus, we'll begin by eliminating the angle phi from equations 1 and 2: Squaring and adding equations 4 and 5, we obtain an expression with phi eliminated: Now manipulate and then square equation 3 to obtain: Ch29L Page 37 Now manipulate and then square equation 3 to obtain: Now subtract equation 7 from equation 6 to obtain the desired relation: Ch29L Page 38 Thus, This is Compton's relation. His experiments, the results of which were in good agreement with the predictions of the formula, provided decisive support for the photon hypothesis, and led to Compton receiving the Nobel prize. Joseph A. Gray, an Australian-Canadian physicist who worked at Queen's University from 1924 to 1952, also did extensive work on the scattering of photons from charged particles, and some of my older physics profs (I was a student at Queen's in the late 1970s and early 1980s) think he ought to have shared credit with Compton, going so far as to call it the "ComptonGray" effect, instead of the Compton effect. _________________________________________________________ Problem: A photon of red light (wavelength = 720 nm) and a Ping-Pong ball (mass = 2.2 g) have the same momentum. At what speed is the ball Ch29L Page 39 ball (mass = 2.2 g) have the same momentum. At what speed is the ball moving? Solution: The momentum of the photon is The Ping-Pong ball has the same momentum, so This speed is about 13 nm per billion years, so it's pretty, pretty, pretty slow. ________________________________________________________ Problem: A sample is bombarded by incident X-rays, and free electrons in the sample scatter some of the X-rays at an angle of 122 degrees with respect to the incident X-rays (see the figure). The scattered Xrays have a momentum whose magnitude is 1.856 × 10-24 kg.m/s. Determine the wavelength (in nm) of the incident X-rays. Ch29L Page 40 Solution: First determine the wavelength of the scattered X-rays; then use Compton's formula. Note the very small percentage difference between the incident and scattered wavelengths for the X-ray, and hence the need for accurate calculation (i.e., input data to at least four significant figures): Ch29L Page 41 _____________________________________________________ Problem: An incident X-ray photon of wavelength 0.2750 nm is scattered from an electron that is initially at rest. The photon is scattered at an angle of 180.0 degrees in the figure and has a wavelength of 0.2825 nm. Use the principle of conservation of linear momentum to find the momentum gained by the electron. Solution: It's a head-on collision, and the X-ray is scattered straight back. Thus, all the action takes place along a line, which we can call the x-axis. Before collision: After collision: By the principle of conservation of linear momentum, where p is the momentum of the electron after the collision. Ch29L Page 42 where p is the momentum of the electron after the collision. ______________________________________________________ Problem: What is the maximum amount by which the wavelength of an incident photon could change when it undergoes Compton scattering from a nitrogen molecule (N2)? Solution: Physically, the greatest wavelength change occurs when the incoming photon transfers the maximum amount of momentum to the nitrogen molecule; this happens in a head-on collision. This also makes sense mathematically; consider Compton's formula: How can you maximize the right side of the equation? You only have control over the parenthesis, and its maximum value is clearly 2, when the cosine of theta is -1, which occurs when theta is 180 degrees. Thus, whether you argue physically or mathematically, the conclusion is the same. Ch29L Page 43 This is an extremely small wavelength shift. If you wish to study the Compton effect experimentally, you'd have an easier time measuring the shift (i.e., the shift would be larger) if you used a less massive particle, such as an electron. ________________________________________________________ Problem: An X-ray photon is scattered at an angle of 180.0 degrees from an electron that is initially at rest. After scattering, the electron has a speed of 4.67 × 106 m/s. Find the wavelength of the incident Xray photon. Solution: It's a head-on collision, and the X-ray is scattered straight back. Thus, all the action takes place along a line, which we can call the x-axis. Before collision: After collision: By the principle of conservation of linear momentum, where p is the momentum of the electron after the collision. By the principle of conservation of energy, Ch29L Page 44 Add equations 1 and 2 to eliminate the scattered wavelength: _______________________________________________________ Wave-Particle Duality Ch29L Page 45 So now we have two theories of light, the wave theory and the photon theory. Each works well in some circumstances, fails in others. What gives? What is light, really? A particle or a wave? Answer: We don't know. Light is something a little mysterious. We try to describe it using concepts that we have abstracted from our macroscopic experience, and we find that we can do pretty well if sometimes we use the wave model, sometimes the particle model. But our clumsy human models have not yet grasped the essentially sublime character of light. Oh well, maybe someday one of you will do better. Wave-particle duality has been described as being somewhat similar to a coin. A coin has two sides, but you can only see one at a time. Similarly, light has these two aspects, but only one aspect seems to come out in a single experiment. To make things more interesting, it seems that matter also exhibits the same wave-particle duality! Matter Waves In 1924, Louis de Broglie introduced the idea that each moving matter particle is guided by a mysterious wave. (Nowadays we say that they are waves, and therefore also exhibit wave-particle duality.) He was perhaps inspired by his love of music; he viewed an atom as a kind of symphony of vibrating energy. The wavelength of the wave guiding a moving particle's motion depends on the mass and velocity of the particle: = h/mv de Broglie's proposal was met with quite a lot of skepticism. However, Einstein was an enthusiastic supporter of the idea Ch29L Page 46 However, Einstein was an enthusiastic supporter of the idea of matter waves, and even created independent arguments in favour of matter waves. (In fact, de Broglie submitted a thesis based on his ideas for his Ph. D., but his thesis supervisor, Paul Langevin, was uncertain whether such outlandish ideas were valid, so he sent a copy of de Broglie's thesis to Einstein for his opinion. Einstein gave the thumbs up, and de Broglie was allowed to proceed to his thesis defence.) Perhaps because of Einstein's support of de Broglie's bold idea, experimenters were encouraged to test it. In 1927, George Thomson, and (independently) Davisson and Germer showed that electrons diffracted from the surface of a crystal, and from the resulting diffraction pattern, they were able to calculate the wavelength of the electrons. The results were consistent with de Broglie's hypothesis. Subsequently, Otto Stern repeated the experiment using atoms, with results that also supported de Broglie's matter wave hypothesis. George Thomson (who shared the Nobel Prize with Davisson in 1937) performed experiments that showed that electrons are waves. His father, J.J. Thomson got the Nobel Prize in 1906 for his 1897 discovery of the electron as a particle, and for measuring its particle properties. Wave-particle duality within one family! (de Broglie got the Nobel Prize in 1929.) Application of matter waves: electron microscope (see page 895 of the textbook for an image produced by a SEM). The double-slit experiment using electrons (or photons) is a compelling illustration of wave-particle duality (in both what we traditionally consider "matter particles" and "wave phenomena" such as light), and really highlights the strange nature of microscopic reality, which is well-captured by the strange theory of quantum mechanics. Richard Feynman considered the double-slit experiment to encompass the central quantum mystery. Ch29L Page 47 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: Ch29L Page 48 Whatever this strange dual nature is that light embodies, matter has it too. _______________________________________________________ Problem: Estimate your de Broglie wavelength when walking at a speed of 1 m/s. Repeat for an electron moving at a speed of 100,000 m/s. Solution: _____________________________________________________ Problem: The diameter of an atomic nucleus is about 10 fm. What is the kinetic energy, in MeV, of a proton with a de Broglie wavelength of 10 fm? Solution: Ch29L Page 49 _____________________________________________________ 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 Ch29L Page 50 Another consequence of Heisenberg's uncertainty principle is that it calls into question the classical idea of the "clockwork universe;" that is, the idea that the world is deterministic. (Laplace said (1814) that if we only knew the position and velocity of every particle in the universe at some moment, then we could (in principle) calculate the position and velocity of every particle in the universe at all later times, using Newtonian mechanics.) But causality is a bedrock principle of science ("things happen for a reason"), whereas determinism is not. Quantum theories are the best, most accurately verified theories in physics. Some quantities (energy levels of atoms, lifetimes of excited states, etc.) can be predicted (and are verified) with extraordinary precision. However, other quantities cannot be predicted in principle, but still in some of these cases probabilities can be predicted extremely accurately. A good example of this is radioactive decay, where the time at which an individual radioactive atom transforms is unpredictable (according to our current understanding), and yet the proportion of a sample of radioactive atoms that will transform in a certain time (which is equivalent to the probability that an individual atom will transform in a certain time) can be predicted very accurately. A final conclusion of Heisenberg's uncertainty principle is that subatomic particles have a wavelike essence. How this very strange microscopic quantum behaviour gives rise to macroscopic classical behaviour is still not well-understood, and you might like to think more about it. It's not even clear if the terms in the previous sentence are well-defined, because some macroscopic objects (such as lasers and the semiconductors in miniaturized electronic devices) are essentially quantum devices. ____________________________________________________ Problem: A proton is confined to a nucleus that has a diameter of Ch29L Page 51 Problem: A proton is confined to a nucleus that has a diameter of 5.5 × 10-15 m. If this distance is considered to be the uncertainty in the position of the proton, what is the minimum uncertainty in its momentum? Solution: ____________________________________________________ Problem: A subatomic particle created in an experiment exists in a certain state for a time of 7.4 × 10-30 s before decaying into other particles. Apply both the Heisenberg uncertainty principle and the equivalence of energy and mass to determine the minimum uncertainty involved in measuring the mass of this short-lived particle. Solution: Besides the original Heisenberg's uncertainty principle, there are other similar relationships between certain pairs of physical quantities. One such relationship is Using the relationship E = mc2, we can write Ch29L Page 52 Thus, ____________________________________________________ Some applets that may be of interest: Fourier synthesis: http://www.falstad.com/fourier/ http://phet.colorado.edu/en/simulation/fourier http://eve.physics.ox.ac.uk/Personal/artur/Keble/Quanta/Applets/quantum/heisenbergmain.html Concluding remark on wave-particle duality Q: So, really, what is a photon? A: "All the fifty years of conscious brooding have brought me no closer to the answer to the question, 'what are light quanta?' Of course, today every rascal thinks he knows the answer, but he is deluding himself." A. Einstein, late in his life Ch29L Page 53