Aside from certain elementary particles generated at particle accelerators or in cosmic-ray showers, most bosons that have mass are composite particles such as the helium nucleus or Cooper pairs of electrons. There is, however, a very common massless particle that has an intrinsic angular momentum = ħ and obeys Bose-Einstein statistics—light!55
Recall our discussion in Chapter 1 of Max Planck and how his explanation of the spectrum of light emitted from hot objects, shown in Figure 2, ushered in the quantum age. Measurements of the intensity of light given off from an object, as a function of the frequency of the light, found that very little light is emitted at low and high frequencies and that the light intensity peaks at a frequency that depends only on the temperature of the object (Figure 2 showed the light spectrum for an object with you right now). Theoretical physics calculations prior to Planck indicated that the intensity should indeed be small at low frequencies but would grow without limit as the frequency increased. We now know enough quantum statistics to see what these theorists got wrong.
A box of molecules, such as a gas, will have some total amount of energy that is indicated by the gas’s temperature. A central principle of nonquantum thermodynamics is that as the gas molecules collide with one another, they share and transfer their energy, so that in equilibrium we would find that on average each molecule has an equal portion of the total energy of the gas. Of course there will be random fluctuations, so that one may see a molecule with a little more or a little less energy than the per-molecule average, but subsequent collisions with other molecules would tend to bring this molecule’s energy back toward the per-molecule average. When you add up the average energy per molecule for all trillion trillion molecules in the box, you get the total energy of the gas. This not only is perfectly reasonable, but is in fact what is observed in real gases (when the quantum nature of the molecules can be ignored, that is, at high temperatures and low pressures, so that the molecule’s de Broglie wavelengths do not overlap).
What if the box were filled with light, treated as extended electromagnetic waves? The atoms in the walls of the box are at some temperature and will jitter back and forth around their normal crystalline positions. It was known before quantum mechanics that oscillating electrons emit electromagnetic waves, which is the basic principle underlying radio and television broadcasting. If there is some dust in the box that absorbs and reemits light, serving the same role as the collisions between gas molecules described earlier, then each electromagnetic wave will have the same average energy per wave.
Nonquantum thermodynamics, which is the only kind that existed before Max Planck published his paper in 1900, would say that the average energy per wave is a constant multiplied by the temperature of the system. This analysis works very well for a box filled with gas molecules. For the case of the gas molecules, we add up the average energy per molecule for the trillion trillion molecules and find the total energy of the gas. A trillion trillion is a big number, but it’s just a number. However, there is no upper limit on the frequencies of waves that could possibly reside in a box filled with light. A clamped guitar string when plucked has a lowest possible frequency, but there is, in principle, no upper limit on the highest frequency that can be excited. If each possible wave has the same average energy per wave, and there are an infinite number of possible waves, then the total energy of the light in the box is infinite! Fortunately, this does not happen in real objects, or else all matter would emit an infinite amount of energy in the form of X-rays and gamma rays. This would be catastrophic, which is why physicists called it the ultraviolet catastrophe.
To resolve this contradiction between calculated and observed light-intensity spectra, Planck assumed that the atoms in the walls of the box can lose energy only in steps proportional to the frequency of the light, from which the relationship Energy = h × (Frequency) was proposed. At this stage we know more about quantum mechanics than Planck did in 1900, so we can use a simpler argument than his original one to understand the observed spectrum of light emitted by all glowing objects.
The box containing light can be considered a gas of photons, each of which has an intrinsic angular momentum of ħ. These photons are thus bosons and will obey the same Bose-Einstein statistics that we invoked for helium atoms and Cooper-paired electrons. For a gas of bosons, there is an enhanced probability of finding the particles in lower energy states. Most bosons will be in the lowest energy state, some will be in the next higher level, a few will be in the next higher energy state, and states very high in energy will have an exponentially small chance of being occupied.
The energy of the photon gas as a function of the frequency of light is the energy of the photon (E = h × f) multiplied by the average number of photons with this energy. Most of the photons are in the low-energy states that carry very little energy. Higher-energy photon states are exponentially less likely to be occupied, so the contribution to the average energy from these photons will also be low. The resulting product of an increasing energy for a photon with a decreasing number of photons with that particular energy yields an average energy per frequency that is very low at low frequencies, peaks at some intermediate frequency, and is again very small at higher frequencies, exactly as observed.
This is akin to the payouts for a Powerball or Lotto lottery system. There the ticket holder must match all six randomly selected numbers in order to claim the grand prize jackpot. But even if no one matches all six numbers, smaller awards are possible. Those matching only three of the selected numbers will win a smaller prize, say ten dollars. Those with four matching numbers might win ten thousand dollars, and five matches would garner one hundred thousand dollars. The payout amount starts off small—many people may match one or two numbers, but they do not win any money; some will match three numbers, but the amount they win is low—fewer will match four numbers, but they have a larger payout, and very few will have selected five of the winning numbers, so there the total payout will also be lower (a large prize but with few winners). A graph of the amount paid out by the lottery agency against the size of the prize would start off small, reach some peak value, and then drop back down.
The number of gas molecules in a box is fixed when we set up the container, but the number of photons can vary, depending on the temperature. Hot objects emit very bright light (that is, give off a large number of photons), while cooler objects emit a lower number of photons. At higher temperatures the exponential tail in the number of photons extends to higher energies. The peak in the spectrum of light energy emitted by a glowing object as a function of frequency will thus depend on the object’s temperature. Cold objects will have their peak at lower frequencies, and the hotter the object, the higher the frequency at which the curve peaks.
Measurements of the light spectrum of objects that can be considered blackbodies therefore provide a way to determine the temperature of very hot objects, such as the interior of a blast furnace or the surface temperature of the sun. But this technique works for cold objects as well. Space is infused with microwave radiation that is the remnant energy from the big bang creation of the universe. Measurements of the spectrum of this radiation as a function of frequency find that it beautifully fits the Planck expression if the characteristic temperature of the universe is 2.7 degrees above absolute zero. Figure 2 from Chapter 2 in fact shows the measured blackbody spectrum of the cosmic microwave background radiation, present at every point in the universe, even where you, Fearless Reader, are right now! From the measured expansion rate of the universe, we can determine that it took approximately fifteen billion years for the universe to grow and cool to its presently measured temperature. In Section 3, I showed how quantum mechanics, developed to account for the manner by which atoms interact with light, enables, through radioactive isotope decays, a determination of the age of the Earth. Now we see that quantum physics also provides an age for the oldest thing in the universe—the universe itself!
SECTION 5
MODERN MECHANICS AND I
NVENTIONS
CHAPTER FOURTEEN
Quantum Invisible “Ink”
Light is an electromagnetic wave that is actually
comprised of discrete packets of energy.
New York City in 1933 boasted many skyscrapers, but only one had an eighty-sixth floor. In our world the eighty-sixth floor of the Empire State Building is dedicated to the Observatory deck, but in the world of the pulps, this entire floor was rented to one man, who made it his residential home, complete with an extensive library and advanced chemical, medical, and electronic laboratories. This man, who excelled in all pursuits intellectual and physical, was frequently joined by his five close associates, each an expert in a different field of the practical and mechanical arts, such as chemistry, law, electronics, engineering, and archeology, on adventures that spanned the globe. The leader of this team, not content to rely solely on his amazing mental capabilities and his imposing physical prowess, would also employ a host of seemingly miraculous inventions and gadgets. Many of these exotic devices would not be realized in our world until years later, when nonfictional scientists and engineers had mastered the principles of quantum mechanics I’ve described, and managed to catch up to the achievements of one of pulp fiction’s greatest heroes, Clark Savage, Jr. Though he had the equivalent of several Ph.D.s, owing to his M.D. from Johns Hopkins and several years studying brain surgery and neurology in Vienna, his friends and the public knew him as “Doc.”
Doc Savage’s adventures were described in the pulp magazine title that bore his name, and his first story, The Man of Bronze, was published in March 1933, written by Lester Dent. Before the year was out, Doc Savage would be one of the top-selling pulps on the newsstand. Dent would go on to write 160 more full-length Doc Savage novels over the next sixteen years, at a pace of nearly one a month.56 Even at the pay rate of a penny a word, his writing income enabled Dent and his wife to live a life of personal adventure and travel that would inform his fictional tales. Doc Savage and his team would often travel the high seas in one of Doc’s yachts or his personal submarine, battling modern-day pirates or exploring an island where dinosaurs still walked the Earth. Meanwhile, Dent and his wife lived for several years on a forty-foot schooner, traveling along the eastern seaboard, fishing and diving for buried treasure in the Caribbean by day and writing pulp adventures by night. Dent was a licensed pilot and radio operator, climbed mountains, prospected for gold in Death Valley, was a vast storehouse of obscure information, and was elected a member of the Explorers Club.
Dent’s most famous literary creation would serve as the inspiration for Superman and Batman (Doc would retreat to an arctic sanctuary to develop new inventions that he called his Fortress of Solitude, and he carried many of his crime-fighting gadgets in a utility vest), James Bond and the Man from U.N.C.L.E. (Doc’s tie and jacket buttons hid the chemical ingredients of thermite and his car could produce a smokescreen to blind pursuers), and Marvel Comics’ Fantastic Four (the comic-book superhero foursome also lived in a skyscraper headquarters, and the friendly bickering between two of Doc’s teammates presaged the relationship between the Thing and the Human Torch), and even Star Trek’s Mr. Spock (Doc could incapacitate foes by pinching certain nerves in their neck).
Doc’s gadgets were similarly ahead of his time. In 1934 Doc employs a version of radar, long before its debut in World War II. (According to Dent, a reference to radar in a 1943 Savage novel was censored by the military immediately prior to publication, requiring him to scramble to come up with an alternative plot device).57 Doc Savage employed shark repellant and colored dyes to mark a pilot’s location when forced to eject over the ocean a good ten years before the navy would adopt these innovations. He invented a small tracking device that, when affixed to an automobile, would transmit a radio signal, enabling the car’s position to be monitored from a remote location. And one of Doc’s inventions—ultraviolet writing—employed in his first pulp adventure makes use of the same quantum mechanical principles that underlie the laser.
In 1933’s The Man of Bronze, Doc and his team of adventurers search their quarters on the eighty-sixth floor for a message from Doc’s recently deceased father. Knowing that his father would often leave him missives using a form of invisible writing, Doc brings out a small metal box that resembles a magic lantern. Showing the interior of the mechanism to Long Tom, the group’s electrical expert, Doc tests his companion, asking him whether he recognizes the device. “Of course. [. . .] That is a lamp for making ultraviolet rays, or what is commonly called black light. The rays are invisible to the human eye, since . . . [their wavelengths] are shorter than ordinary light.” Long Tom then points out that while we may not see in the ultraviolet, many common substances, such as quinine and Vaseline, fluoresce when so illuminated. When they shine this ultraviolet light on a window in Doc’s office, sure enough, a message from his father is revealed in glowing blue letters, directing them to the hiding place where they would find important papers that would in turn send them on a perilous journey to the fictional Central American nation of Hidalgo. The mechanism by which Doc and his father, and in later pulp adventures Doc and his teammates, communicate through ultraviolet writing relies on the variation in transition rates for quantized levels.
We have seen that electrons bound in atoms are constrained to particular energy levels. A consequence of this discreteness is that the atoms can absorb or lose energy only when it enables transitions between these allowed energy states (we will neglect transitions of protons or neutrons within the nucleus, as these energy scales are in the gamma-ray range, and we are interested now in transitions in the visible portion of the electromagnetic spectrum). Any energy interacting with the atom, in the form of a light photon or a collision with another electron or atom, will not induce an electronic transition if the change in energy does not correspond to the difference between two energy levels.
In our analogy of students in a classroom, where the rows of seats represent allowed energy levels, students may be promoted from their original seats at the front of the room to empty seats near the rear of the lecture hall. However, students are not allowed to stand between rows and may change their seats only if the energy they absorb takes them exactly from one row to another (and if the seat they are moving to is unoccupied). When an atom relaxes from some high-energy state back to the ground state, it similarly may do so only by emitting a photon whose energy is equal to the difference between the starting and final energy levels. That is, only electronic transitions that satisfy the principle of conservation of energy are allowed. This accounts for the discrete-line spectrum, with only a very select number of wavelengths observed (see Figure 13 in Chapter 5) when an atom is placed in a high-temperature environment. Different elements will have their allowed quantum levels at different energies, so that the spacing between levels, and hence the frequency of the light emitted when the electron moves between states, will differ.
Just because an electron can make a jump between two quantized energy levels does not determine how fast or slow such a transition may be. For a collection of atoms, the light will be brighter for those transitions for which the probability of a jump is higher. Some lines will be present, but very faint, as the probability of a transition occurring at any given moment might be very low. One of the great successes of the quantum theory is that it actually makes predictions of the transition rates, that is, the probability per second that an atom with an electron in an excited state would drop down to a lower energy state, emitting a photon in the process. Thus, the quantum theory correctly predicts not only what wavelengths will be observed for a given atom, but even how bright the lines will be.
What determines these transition rates is fairly complicated and depends on details of the wave functions for the initial and final states. The important point is that quantum mechanics is able to account for the following: (1) the fact that electrons in atoms may have only certain energies, (2) the fact that only certain transitions between allowed states are pos
sible, and (3) the probability per second of a given transition occurring. That is, the theory can explain why only discrete lines rather than continuous spectra will be observed for the light emitted by an atom, as well as predicting the wavelengths of the line spectrum and the intensity of the lines, all in excellent agreement with experimental observations. We now know enough about how atoms interact with light to explain two of the most important inventions of the twentieth century: lasers and glow-in-the-dark action figures!58
Let’s first consider glow-in-the-dark materials. Each atom in the solid has a highest occupied energy level (as in Figure 31), and when a trillion trillion of these atoms are collected, all of these “seats” broaden into an auditorium of quantum states, as illustrated in Figure 34. In Chapter 12 we saw that, thanks to the Pauli exclusion principle, each seat is actually a “love seat” in which two electrons can sit, if they have opposite spins (one with + ħ/2 and the other with -ħ/2). The trillion trillion “seats” in this “ground-state auditorium” can therefore accommodate two trillion trillion electrons.
If the atoms in the solid form bonds by keeping their electrons in “boxes,” as in the case of the carbon-carbon bonds in diamond (Figure 32), then every love seat in the auditorium has two electrons, and the auditorium is completely filled (Figure 34a). The electron thus has to move to a higher energy (the next available empty quantum state) in order to find a vacant level. All of these higher energy states will also broaden into an “auditorium” of seats. Atoms that form solids similar to diamond can be considered to have an orchestra of seats, all of which are completely filled, and a higher-energy balcony with an equal number of seats, which are all empty.59 When a current flows in a solid in response to an applied voltage, the electrons gain kinetic energy, but this cannot happen if there are no unoccupied higher energy states accessible to the electrons. Consequently, only those electrons promoted to the balcony, by either heat or light, will be able to participate in an electrical current, moving along the newly available empty seats. Diamond is an electrical insulator because normally there are too few electrons in the balcony to provide an appreciable current.
The Amazing Story of Quantum Mechanics Page 18
