The Amazing Story of Quantum Mechanics

by Kakalios, James

  “The Skylark of Space” leaves vague the exact nature of the “intra-atomic energy” released by the copper when catalyzed by Element X and an electrical current. A rival scientist of Seaton’s puts it as follows: “Chemists have known for years that all matter contains enormous stores of intra-atomic energy, but have always considered it ‘bound’—that is, incapable of liberation. Seaton has liberated it.” As chemists certainly knew, even in 1928, how to release the energy stored in chemical bonds between atoms in molecules such as nitroglycerin or TNT, the vast amounts of intra-atomic energy liberated by Element X may refer to the conversion of mass into energy through Einstein’s relationship E = mc2. This seems likely; when a spaceship propelled by Element X is accidently set to full thrust, the resulting acceleration becomes so great that no one on board can move to the control board to decrease their speed and the ship stops its motion only when the copper supplies are exhausted. While illustrating Einstein’s principle of the interrelation between energy and mass, this scene contradicts the Special Theory of Relativity when it reveals that this uncontrolled acceleration has resulted in the ship traveling many times the speed of light. When Seaton wonders how this can be reconciled with Einstein’s famous work, his companion replies, “That is a theory, this measurement of distance is a fact, as you know from our tests.” Like any good scientist, Seaton agrees that observation is the final arbiter of correctness and concludes of Einstein, “That’s right. Another good theory gone to pot.”

  The scientists in “The Skylark of Space” should not be so quick to abandon Einstein, for their X-plosive bullets of intra-atomic energy provide confirmation of another of his theories. This application of Element X, as well as the ray guns wielded by Buck Rogers, Flash Gordon, and other heroes of the science fiction pulps and comic strips, is not too far from the mark, as reflected in the first quantum principle, at the top of this chapter. As proposed by Albert Einstein the same year he developed his Special Theory of Relativity, all light consists of “bullets,” that is, discrete packets of energy, termed “photons.”

  Now that we have the answers to quantum mechanics—what were the questions that called for these new physical principles? The ultraviolet catastrophe alluded to earlier concerned the brightness of the light emitted by an object as a function of temperature. Certain objects, such as graphite and coal dust, are black, as they absorb nearly all light that shines on them. In equilibrium, the light energy absorbed is balanced by light given off. The spectrum of light of such blackbodies, that is, how much light is emitted at a given frequency, depends only on how hot it is and is the same for metals, insulators, gases, liquids, or people if they are at the same temperature.

  The theory of electromagnetic waves, developed by James Clerk Maxwell in the second half of the nineteenth century, was able to account for the energy emitted by a glowing object at low frequencies, such as infrared light, but at higher frequencies (above visible light) this theory predicted results that were nonsensical. Calculations indicated that the light from any heated object would become infinitely intense at high frequencies, above the ultraviolet portion of the spectrum. Thus, anyone looking at the glowing embers in a fireplace, or the interior of an oven, should be instantly incinerated with a lethal dose of X-rays. If this were true, most people would notice. This so-called ultraviolet catastrophe (which, as indicated, was a catastrophe more for theoreticians making the predictions than for anyone else) disappeared following Planck’s suggestion that when the atoms in a glowing object emit light, the atoms lose energy as if they were moving down the steps of a ladder, and that those atoms must always move from rung to rung of the ladder and cannot make any other transitions between rungs. Why this would resolve the ultraviolet catastrophe, we’ll explain in Section 4. For now let’s focus on this “ladder” of possible energy values.

  Figure 2: A plot of the light intensity given off from a “blackbody” object as a function of the frequency of light. The measured curve (solid line) shows that the total amount of light emitted is finite, while the pre-quantum mechanics calculated curve (dashed line) continues to rise as the frequency of light increases. That is, before quantum mechanics, physics predicted that even objects at room temperature would give off an infinite amount of light energy in the ultraviolet portion of the spectrum—a clearly ridiculous result.

  Planck firmly believed that light was a continuous electromagnetic wave, like ripples on the surface of a lake,6 as theoretical considerations and extensive experimental evidence indicated. His proposal of discreteness in atomic energy loss was fairly modest (or as modest as a revolution in scientific thought can be). It turns out that while Planck justly receives credit for letting the quantum genie out of the bottle, there were other experimental conundrums waiting in the wings regarding how atoms interacted with light that would require far bolder steps than Planck was willing to take. At the same time that scientists were measuring the light given off by hot objects, giving rise to the ultraviolet catastrophe, Philipp Lenard was studying the electrons emitted by metals exposed to ultraviolet light. This led to a different catastrophe, both personal and scientific.

  In the late 1800s physicists had discovered that certain materials, such as radium and thorium, gave off energy in the form of what would eventually be termed “radiation.” Scientific researchers entered a “library phase,” cataloging all of the different types of radiation that different substances emitted. Using the Greek alphabet as labels (α, β, γ, etc., instead of a, b, c, and so on), they started with “alpha rays,” which turned out to be helium nuclei (two protons and two neutrons) ejected from atoms found near the end of the periodic table of the elements,7 then moved on to “beta rays,” (high-speed electrons), followed by “gamma rays” (very high-energy electromagnetic radiation).8 When William Roentgen discovered a form of radiation that would fog a photographic plate, passing through paper or flesh but not metal or bones, he termed this unknown ra diation “X-rays.” Roentgen’s discovery came before the Greek nomenclature tradition used for naming rays; he used the letter X, as it is the letter traditionally employed in math problems for the unknown quantity. (Roentgen’s “X-rays” were the forerunner of many science fiction X-based characters, such as the X-Men, Professor X, Planet X, Dimension X, Element X, and X the Unknown). It was later shown that X-rays are simply electromagnetic waves—that is, light—with more energy than visible and ultraviolet light but less energy than gamma rays.

  These varieties of radiation provided scientists at the end of the nineteenth century with new tools to study matter. By exposing different materials to these forms of radiation and observing their effect, they could probe the inner working of atoms. They were able, for the first time, to metaphorically take off the back plate and examine the mechanisms inside the atomic watches. Admittedly this tool was more like a hammer than a jeweler’s screwdriver, but you use what you have.

  Lenard was working at the University of Heidelberg and investigating the influence of light exposure on various metals. He discovered through a series of careful experiments that certain metals, when illuminated with ultraviolet light, give off beta rays, that is, electrons. It would turn out that the electrons he was observing originated from the sea of electrons that explain why all metals are good conductors of heat and electricity. This, in hindsight, is not that surprising. Light carries energy, and when an object absorbs energy it warms up. Some of the excess energy in the metal can be transferred to the electrons, and if they are sufficiently energetic they can fly free, not unlike the energetic water molecules leaving the liquid surface of a hot cup of coffee and forming a cloud of steam above the mug. Philipp Lenard set about systematically investigating how the number of electrons emitted from a given metal and their speed depended on the frequency and intensity of the ultraviolet light he used. Here the troubles began.

  Imagine a metal as a sandy beach at the ocean’s shore, and the electrons in the metal as small pebbles randomly scattered along the beach (see Figure 3). The ocean wave
s crashing onto the beach can be considered the ultraviolet electromagnetic waves shining on the metal in Lenard’s experiment. This allegorical beach has a gentle slope away from the water’s edge, so that one must do work against gravity to push the pebbles away from the shoreline. When the pebbles have advanced up the beach all the way to the boardwalk, we’ll consider them free, and as our stand-ins for electrons in a metal, they will represent those electrons that are ejected from the solid by the ultraviolet light in Lenard’s experiment. In this analogy it is reasonable that the energy responsible for picking up the pebbles and moving them up the beach comes from the water waves. The bigger the wave, the more energy the pebbles will have. The more frequently the waves arrive, the greater the number of pebbles will be advanced. If the waves have small amplitudes, one might have to patiently wait for several wave fronts to transfer sufficient energy to the pebbles before they can move up the beachfront. Perfectly reasonable—except this was not what Lenard observed.

  Figure 3: Cartoon sketch of pebbles on a beach, pushed up toward the top of the beach by either ocean waves (a) or by photon bullets (b).

  He found that the energy of the electron leaving the metal did not depend on the intensity of the light. Raising the light intensity did not affect the electron’s speed, though it did increase the number of electrons emitted per second. But the number of electrons ejected per second was supposed to depend on the frequency of the waves, that is, how many wave crests arrived per second, and not their amplitude. Moreover, there was a threshold for electron emission—if the frequency of the light were below a certain value (which varied for different metals), then no electrons came off, regardless of how bright the light. In the water analogy, this would suggest that if the number of wave peaks per second were below a given value, then even a tsunami would not push the pebbles up the beach. What did govern whether or not the electrons were emitted was the frequency of the light. For the beach analogy, this would be as if once the number of wave peaks per second rose above a certain threshold, then even a very low-amplitude, gentle wave would promote pebbles up to the boardwalk, just as long as the time between crests was short enough. Finally, if the frequency of the light was above this threshold, then electrons came off instantly, with no time delay, no matter how low the light’s intensity. These experimental results were a challenge to understand within the context of light being a continuous electromagnetic wave, but we already know the answer to this question, stated at the start of this chapter. Light is not a continuous wave but is composed of individual energy bullets, termed “photons.”

  How does light actually consisting of discrete packets of energy explain Lenard’s experimental results? What we took to be a continuous, uniform series of waves washing up onshore, gently pushing the pebbles up the beach, is actually comprised of bullets fired at the pebbles by a machine gun sweeping back and forth across the beach. With light being a collection of photons, the brightness of the light is determined by the number of photons passing through a given area per second. For our machine gun, this is equivalent to the rate at which bullets are fired; that is, more bullets per second leads to more intense illumination. One bullet a second is a weak light source, while a million shots per second is much brighter. The energy of the photons would be reflected in our analogy in the speed of the bullets (let’s suppose for the sake of argument that we have a magical machine gun that provides independent control over the speed of the bullets it fires).

  It turns out that if you imagine the light striking the metal as analogous to a spray of machine-gun bullets rather than continuous ocean waves, Lenard’s results are completely reasonable. He found that the brighter the light, the more electrons were emitted from the metal. In the machine-gun analogy, brighter light means more bullets per second, which will push more pebbles per second. There will be no time lag between light coming into the metal and electrons coming out, for once a bullet hits a pebble, and it has sufficient energy to knock it up the beach, then of course the effect will be instantaneous. The faster the bullets are traveling, the more energy will be imparted to the pebbles, and the faster they will move up the beach. This corresponds to saying that the greater the energy of each light photon, the more kinetic energy the ejected electrons will have. If the speed of the bullets is too low, they may move the pebbles a little bit but not knock them significantly up the beach. So the threshold effect Lenard observed is also explained. The only snag is that Lenard did not control the energy of his light to vary the energy of the emitted electrons, but rather its frequency. And here we come to the personal catastrophe for Lenard, for the resolution of this last remaining puzzle would cause him distress of a decidedly nonscientific nature. (More on this in a bit.)

  If light does indeed consist of discrete packets of energy, what determines the energy of each packet? Planck’s solution to the spectrum of glowing, hot objects was to propose that atoms could lose energy only in finite jumps. In order to get the equations to work out, he assumed that the energy of the jump was proportional to the frequency of the light. That is, the bigger the frequency, the larger the energy of the “quantum step.” Again, I’ll explain why this worked later on. For now the important point is that if the energy is proportional to the frequency, then we can say that the energy is equal to the frequency when multiplied by a constant.

  We often deal with situations involving simple proportions, such as the relationship that the longer you drive your automobile at a constant speed, the greater the distance you travel. But you do not measure distance in hours, so to figure out how far you have driven, you need to multiply the hours in the car by a “constant of proportionality,” namely, your constant speed (say 60 miles per hour). Then the product of the time driven (2 hours) and the uniform speed (60 miles per hour) will determine the distance traveled (120 miles). In the same way, the energy of the photon is proportional to the frequency, so that when the frequency is multiplied by a constant of proportionality, it is converted into a quantity of energy. Planck used the letter h to represent this constant of proportionality, and everyone who has followed has stuck with that convention, so that h is referred to as “Planck’s constant.” The equation for the energy lost by an atom in a glowing object by the emission of light is as follows:Energy = h × (Frequency)

  Let’s plug some numbers into this simple equation, which is mathematically no different from “distance = (speed) × (time).” One way to measure energy is in a unit termed Joules, named after James Joule, a Scottish physicist who demonstrated the clear equivalence between heat and mechanical work, thereby providing a foundation for the field of thermodynamics. For point of reference, a major-league baseball thrown at 60 miles per hour has a kinetic energy of 53 Joules, while an automobile traveling at 60 miles per hour has a kinetic energy of 600,000 Joules. Frequency is a measure of how many times a periodic function repeats a complete cycle in a given unit of time and is most naturally measured in terms of number of cycles per second. A child’s playground swing that takes 2 seconds to go all the way back and forth will complete only one half of a cycle in a second, so it has a frequency of ½ cycle/second. A much faster playground swing that goes back and forth in 1/10 of a second will complete ten loops in 1 second, and thus has a frequency of 10 cycles/second. Visible light has a frequency of one thousand trillion cycles per second. In order to fit his equation for the spectrum of light given off by a hot, glowing object to the experimentally measured curves, Planck had to set the value of his constant h to be h = 660 trillionth trillionth trillionth of a Joule-sec, which may seem very small but is in fact very, very, very supersmall.

  Planck argued, when justifying his proposed equation, that atoms could lose energy only in finite steps. The closest these energy levels could be was E = h × f. For light with a frequency of a thousand trillion cycles per second, this equation gives a spacing between adjacent energy levels of 0.66 millionth trillionths of a Joule. If you compare this energy of a photon to the 50 Joules of kinetic energy of a fastball, you can
see that the separation is so small compared to the energy scale of our everyday activities that it’s nearly impossible to observe. Consequently, to take a quantum jump involves a very, very tiny baby step and not some enormous leap. But the size of the step is, in fact, irrelevant, as the important and revolutionary concept introduced by Planck is that there is a step.

  As mentioned, Planck was actually somewhat conservative in his quantum hypothesis. For him light was still a continuous electromagnetic wave, like the metaphoric ocean waves washing onto the shore. While he argued that atoms could lose energy only in discrete steps, he was not so bold as to suggest that when they did so the light emitted was also a discrete packet of energy. But as the humorist James Thurber once wrote, “Fools rush in where angels fear to tread. And all the angels are in heaven, but few of the fools are dead.” Perhaps Planck’s hesitancy to extend the quantum graininess to the light itself came from the caution of age. He was a grand old man of forty-two, after all, when he developed his quantum hypothesis. It was left to a younger man of twenty-six to suggest that Lenard’s results could be qualitatively and quantitatively accounted for if light itself were discrete packets, the machine-gun bullets in our beach metaphor, where the energy of each bullet is determined by E = h × f. That young man was Albert Einstein.