The Amazing Story of Quantum Mechanics

by Kakalios, James

  When Einstein wrote his paper on the “photoelectric” effect, for that is the name used to describe Lenard’s experiment, he was an underemployed twenty-six-year-old patent clerk, third class. That would soon change, for Einstein’s paper was published in 1905, the same year he published his Special Theory of Relativity, followed by a paper describing the equivalence between energy and mass—E = mc2—and two other papers on an atomistic explanation for Brownian motion and diffusion processes that would have cemented his reputation as a theoretical physicist of the first order even if he’d done nothing else. Within a few years of the papers’ publication, Einstein would be offered professorships and honors. Most scientists would be thrilled to have their work permanently associated with Einstein, but not Lenard, for one simple reason. Einstein was Jewish, and Lenard was a rabid anti-Semite, to such an extent that Adolf Hitler named Lenard chief of Aryan physics.

  Thus did his experiments on the influence of ultraviolet light on metals cause a personal catastrophe for Lenard—he spent such effort denouncing Einstein and his interpretation of the photoelectric experiments that his own scientific reputation was ruined, and he is now remembered as much for his bigotry as for his talent as an experimentalist.

  Fortunately for Einstein, Lenard was not the only physicist who strongly disagreed with the hypothesis that light was comprised of discrete packets of energy. Nearly all physicists, including the American physicist Robert Millikan, were convinced that light was a continuous wave, and that Einstein’s suggestion (the title of his original paper described his proposal as a “heuristic viewpoint”—which is fancy talk for “not a rigorous solution, but I should get the credit if it turns out to be right”) could not possibly be correct. I say “fortunately for Einstein,” for he did indeed turn out to be right, and it was Millikan who proved it. Millikan was one of the most careful and gifted experimentalists of his day, and he spent ten years trying to show that Einstein was wrong. What he wound up showing, in fact, was that the only possible explanation for the photoelectric effect was Einstein’s hypothesis. Even though he still believed that Einstein’s photon idea was crazy, he stood by his data, which not only unambiguously supported Einstein’s claim but also won Millikan the Nobel Prize. The best advocate for your position is someone who initially doubts but is converted by overwhelming evidence. Einstein’s heuristic viewpoint is now everyone’s viewpoint, and Einstein received the Nobel Prize in Physics, not for his work on relativity or E = mc2, but for his theory of Lenard’s photoelectric effect and the introduction of the concept of the light photon (though Gilbert Lewis was the first to use the term “photon,” in 1926).

  Figure 4: Sketch of a light wave reflecting from the top and bottom surfaces of an oil slick. In (a) the wavelength and thickness of the slick result in constructive interference of the wave reflected from the top surface, and the wave that travels through the slick reflects from the bottom surface and then exits the slick. For constructive interference the light would be very bright when viewed from the top. In (b) the wavelength and thickness result in destructive interference, in which case no light would be observed from the top surface.

  Now, for a very long time, a lot of smart people firmly believed that light was a wave, based on a large body of compelling experimental evidence. To cite just one manifestation of the wavelike nature of light, consider the rainbow of colors one sees on an oil slick following a rainstorm. Anyone who lives in a city, or has a messy driveway, may be familiar with the spectrum of light reflected from a thin layer of oil when the ground is wet from a good soaking. In this case the oil is repelled by the water and sits as a thin, freestanding slab that may be only a fraction of a millimeter thick. Rarely will the surface be atomically uniform, so the thickness of the oil slick will vary from location to location. Some of the light striking the oil slick reflects from the top surface, while some penetrates through the oil and then bounces off the oil-water interface. As illustrated by the sketch in Figure 4, if the thickness at one particular spot on the oil slick is exactly equal to one-fourth9 of the wavelength of one particular color of light, then the light reflected from the top and the light that passed through the slick and bounced back out will be exactly in phase and will add up coherently. The light at that color will thus be particularly bright. Just as two pendulums, if they are swinging side by side with exactly the same frequency and the same phase (so that they both are at the top of the arc or both at the bottom of the swing at the same time), are said to be coherently oscillating, the light that is coherently reflected will add up and give a much more intense color. All other colors will have wavelengths that are not exact multiples of the oil-slick thickness at that location, and they will not add up constructively. This familiar phenomenon is termed “interference” and is a signature property of waves.

  We therefore must figure out why light exhibits phenomena that indicate that it is a wave, when it is in fact comprised of photons. It is tempting to argue that the wavelike properties of light are some sort of collective behavior when many photons interact with one another. Oh, if it were only this simple. Consider the following situation: If one sent not a continuous beam of light at the oil slick described above, but single photons, one at a time, then the photons would reflect from the slick, from either the top or bottom surface, and be detected by some sort of light sensor. We would see a single flash at a particular location on our detector when each photon had reflected from the slick. When many such photons had been reflected, the resulting pattern of flashes of light observed would be an interference pattern, identical to what we would see if a continuous beam had been used. That is, even though the photons saw the slick individually, they reflected in such a way that when added together, they yielded a wavelike constructive interference pattern.

  Technically, a photon is defined as a quantum of excitation of the radiation field. Well, that certainly clears that up! For our purposes, we will simply accept the notion of photons as discrete entities that move at the speed of light, that have a definite energy (and hence frequency, through E = h × f), definite momentum (and hence wavelength, through the relation wavelength = speed of light/ frequency—which we will discuss later on), and definite intrinsic angular momentum (the “spin” of a photon = h/2π, measured in the direction it is moving). The photon does not spread out as it travels, the way a wave on the ocean’s surface does, but propagates unchanged until it interacts with matter or other photons. There are no simple or satisfying answers to such questions as how big the photon is, or whether it is a wave or a particle. If you find this confusing—join the club! It’s a big club with some rather distinguished members. The president of this club would be the scientist who introduced the photon concept in the early days of quantum mechanics. As Albert Einstein reflected, “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.”

  CHAPTER THREE

  Fearful Symmetry

  Matter is comprised of discrete particles that

  exhibit a wavelike nature.

  Readers of the February 1930 issue of Science Wonder Stories were treated to thrilling tales of the “Streamers of Death” and “A Rescue from Jupiter”; they traveled to “The Land of the Bipos” and visited “The World of a Hundred Men.” The cover features a scene from the “Bipos” yarn. Two robbers who have broken into the home laboratory of a Dr. Sanborn, who was experimenting on methods to send living beings to another world (whether in this universe or an alternate one is never made clear), have been trapped in a large glass device. This cylinder, large enough to hold two grown men, is described in the story as a “cathode ray tube”—though its appearance is quite different from the cathode-ray tubes one finds in older-model television sets. Sanborn is shown moments before throwing a switch that will convert the two thieves into electricity. They will then travel at the speed of light to the land of the Bipos, where they will b
e reassembled into their human form. Bipos, apparently, are a race of intelligent three-foot-tall penguins. The means of transportation appears to be an early ancestor of Star Trek’s famed transporter. That Sanborn was able to construct such a fantastic scientific marvel, with no outside assistance and using his own financial resources, is perhaps not so surprising once we discover that in his day job Dr. Sanborn is . . . a druggist!

  Science Wonder Stories was not devoted solely to fantastic scienctifiction but also featured descriptions and discussions of real-world current scientific advances. This particular issue contained a “Symposium,” in which an essay on the question “Can Man Free Himself from Gravity?” was followed by letters from knowledgeable experts. The short essay by Th. Wolff of Berlin was translated for the pulp from the original German. Wolff tantalized readers with a report of an American physicist, Charles Brush, who claimed to have discovered a material made up of silicates (the exact composition known only to Brush) that exhibited an acceleration due to gravity of only 9.2 meters per second per second, rather than the larger value of 9.8 meters per second per second that all normal matter experiences. “If true, this would be a fine achievement,” allowed Wolff, for “by increasing the valuable property of these mysterious substances one might perhaps attain approximate or even complete freedom from gravity. Let us wait for it!”

  Figure 5: Dr. Sanborn about to test his homemade transporter device (that looks like an overgrown vacuum tube), which will send two intruders to the Land of the Bipos in 1930’s Science Wonder Stories.

  But Wolff did not think we should hold our breaths while waiting, for he went on to correctly point out that such a material would represent an “irreconcilable contradiction” to the Newtonian law of gravity, which indicates that the acceleration of a falling object is the same for all matter, regardless of composition. Brush’s report, Wolff informed readers, “must with absolute assurance be relegated to the realm of fiction. If there were exceptions and deviations from the general law of gravity, these would certainly have appeared before now in manifold and various ways, and it would not need the discovery of mysterious substances to bring them to our knowledge.” So much for flying cars—even back in 1930! But then Wolff goes too far—and dismisses space travel when he incorrectly calculates that the chemical fuels of the time would limit any rocket ship to heights no greater than 400 kilometers above the Earth’s surface, a mere fraction of the 384,000 kilometers from the Earth to the moon.

  This last point was challenged in letters from members of the Science Wonder Stories Board of Associate Editors, notably Robert H. Goddard of Clark University in Worcester, Massachusetts. Goddard pointed out that in 1919 he had authored a scientific publication in the Miscellaneous Collections of the Institute (namely, the Smithsonian Institute, which was funding his rocket research), stating that a multistage rocket, essentially of the design employed by NASA fifty years later, would indeed be able to exceed this 400-kilometer limit. Thus, while hopes of flying cars and perpetual motion10 were dashed, the promise of rocket trips to the moon and beyond were affirmed in the science fiction pulps.

  Goddard was an early example of a prominent scientist whose research would inspire many science fiction tales and whose choice of field and research subject was, in turn, inspired by science fiction. In a fan letter sent to H. G. Wells, the sixteen-year-old Goddard extolled the influence that reading The War of the Worlds had on him, such that no more than a year later, he “decided that what might conservatively be called ‘high altitude research’ was the most fascinating problem in existence.” Goddard was not the first scientist, of course, to find a muse in science fiction. Hermann Oberth, the Transylvanian-born scientist who is considered the “father of modern rocketry,” had an encounter at age eleven with Jules Verne’s From the Earth to the Moon that set the trajectory of his scientific career. Both Oberth and his pupil Wernher von Braun would serve as technical advisers for Woman in the Moon, a 1929 Fritz Lang science fiction motion picture that featured the first countdown to launch a rocket, in film or in the real world.

  Real science, as opposed to fiction, was also imparted in Science Wonder Stories’ regular features “What Is Your Science Knowledge?,” “Science Questions and Answers,” and “Science News of the Month.” Here, in this latter section, a brief item entitled “Electron Found to Have Dual Character” read, in its entirety:G. P. Thompson, British scientist, has made a new discovery in the field of physics. He states that the electron acts like a flying particle and also behaves like a wave. He rolled gold, nickel, aluminum and other metals, each to about one-tenth the thickness of gold leaf, and shot electrons through them. After passing through the films the electrons came in contact with a photographic film, and were recorded as concentric circles and other circular patterns.

  If the magazine had contained a detailed description of the chemical composition of an actual antigravity shield, it would not have presented a more profound or revolutionary report than this brief blurb regarding the electron’s “dual character.”

  The second quantum principle listed at the top of this chapter states that, just as there is a particle aspect to light, there is a corresponding wavelike nature to matter. Unlike the case of the photoelectric effect in the last chapter, this strange symmetrical hypothesis about the nature of matter was not proposed in order to resolve a mysterious experimental observation that contradicted expectations of classical physical theory—but was suggested precisely because it was a strange symmetrical hypothesis.

  In 1923, Prince Louis de Broglie (yes, he actually was a French prince as well as a physicist), struck by the counterintuitive suggestion that light was comprised of corpuscular particles, proposed that there was a wave—originally termed a “pilot wave”—associated with the motion of real particles, such as electrons, protons, and atoms. De Broglie had an answer for why this “pilot wave” had not been previously observed—its wavelength varied inversely with the momentum of the moving object, so the larger the object (which is easier to observe), the smaller the wavelength of its pilot wave.

  How to test the proposal that there is a wave associated with the motion of matter? As mentioned in the last chapter, interference effects, such as when white light creates a spectrum of reflected colors from an oil slick suspended on a wet surface, are an excellent test of the existence of waves. To recap, when the thickness of the slick is exactly equal to specific fractions of a given color’s wavelength, the waves corresponding to this color reflected from the top and those that have traveled through the slick, bounced off the bottom, and passed again through the slick and exited from the top surface add together coherently. When this happens, the color is brighter to us due to this constructive interference. Other waves corresponding to other colors at this location add up incoherently, out of phase, and the net effect is that from the white light shining on the oil, one color is primarily reflected from the slick from a given point on the slick. As the thickness of the slick can vary from point to point, we observe different colors across its surface.

  The thickness of an oil slick can be several thousand nanometers (one nanometer is approximately the length of three carbon atoms, stacked one atop the other), while the wavelength of visible light ranges from 650 nanometers for red light to 400 nanometers for violet light. Thus, only very thin oil slicks, whose thickness is no more than a few times the wavelength of light, exhibit the interference pattern described above (if the slick is too thick, then the light traveling through the oil has too great a chance to be absorbed and won’t make it back through the top surface). If we want to use a similar interference effect to verify the wavelike nature of the motion of matter as proposed by de Broglie, we first need to know how large or small the “matter wavelength” will be. De Broglie proposed that the connection between the wavelength of the “pilot wave” for any moving object and its momentum is given by the following expression:Momentum × Wavelength = h

  This equation indicates that the larger the momentum, the smaller th
e wavelength. The product of the two quantities is a constant, and de Broglie suggested that it should be Planck’s constant. Again, this equation is mathematically no different from the relationship described in the last chapter connecting distance traveled and time driving, that is, distance = (speed) × (time). In order to determine how long a car trip to Chicago from Madison, Wisconsin, may take, we note that the distance is a constant, approximately 120 miles, and not open to alteration. If our average speed is 60 miles per hour, then this equation indicates that the trip will last 2 hours. A slower speed will lead to a longer trip, and to shorten the trip to 1 hour, we must look to a speed of 120 miles per hour.11 In principle, the trip may last as short or as long as we like, as long as we vary our average velocity so that, when multiplied by the travel time, it yields a distance of 120 miles.

  The momentum of an object is defined as the product of its mass and its velocity. The bigger an object, the more momentum it has at a given speed, and the harder it would be to stop. Which would you rather have collide with you: a linebacker or a ballerina, both running at the same speed? If we use the mass and speed of a major league fastball in de Broglie’s equation above, we find that its de Broglie wavelength is smaller than a millionth trillionth of the diameter of an atomic nucleus. There is no structure that can be conceived of that would exhibit interference effects of a baseball.

  One way to increase the size of the de Broglie wavelength is to decrease the momentum of the object, as their product is a constant, and the simplest way to do that is to consider smaller objects. That is, the smaller the object, the lower its momentum (just as the ballerina has a smaller momentum than the football player), and consequently the larger its de Broglie wavelength. An electron obviously has a much smaller mass than a baseball, and a correspondingly smaller momentum. Even for an electron traveling at a speed of nearly 1 percent of the speed of light, its momentum is a trillion trillion times smaller than the baseball’s, and its corresponding de Broglie wavelength is a trillion trillion times larger. For just such an electron the de Broglie wavelength turns out to be about one-fourth of a nanometer, or roughly the diameter of an atom. In order to observe interference effects that would reflect the wavelike nature of matter, we would thus need to send a beam of electrons at an “oil slick” that is only a few atoms thick. That’s still pretty small, but fortunately nature provides us with just such “slicks”—we call them crystals.