The Amazing Story of Quantum Mechanics

by Kakalios, James

  Figure 34: Sketch of the band of quantum states from the highest energy occupied levels in a solid and the band formed from the next highest energy available quantum states. In an insulator (a) the lower band is analogous to a completely filled orchestra in an auditorium, where there is an energy gap separating the electrons in the lower band from the band of empty states (the balcony). The second figure (b) shows a situation where the lower orchestra is only half-filled and the electrons have ready access to empty seats—which describes a metal.

  In contrast, in metals the ground-state electrons are localized in “momentum space,” and the orchestra that can seat two trillion trillion electrons is occupied by only one trillion trillion electrons. There are therefore many empty seats in the half-filled orchestra, as sketched in Figure 34b, and it is easy for the electrons to move from seat to seat when carrying an electrical current.

  To construct a “glow-in-the-dark” nonmetallic solid, we need a filled orchestra, an empty balcony, and a “mezzanine” of seats, also unoccupied, just below the balcony (sketched in Figure 35). Let’s assume, for the sake of argument, that blue light is required to promote an electron from the orchestra to the balcony, but the mezzanine can be filled using lower-energy green light. The energy separation between the balcony and the mezzanine is in the infrared portion of the electromagnetic spectrum. These mezzanine seats may arise from a different element that is incorporated into the solid.

  Figure 35: Sketch of the band structure of a fluorescing solid, represented by a filled orchestra, an empty balcony at a high energy, and an unoccupied mezzanine level at a slightly lower energy than the bottom of the balcony. When the solid is illuminated with white light, electrons are easily promoted from the orchestra to the balcony, and photons are emitted when the electrons fall back into the lower level. Occasionally an electron will wind up in the mezzanine level, from which the transition rate to the orchestra is low. When the light exposure is stopped, these charges trapped in the mezzanine will eventually drop back into empty spots in the orchestra, emitting slightly lower energy photons in the process. In this way the material will give off light after being illuminated—that is, it will glow in the dark.

  Now assume that the transition rate from the orchestra to the balcony is high. This means that it is easy to promote the electron up from the filled lower auditorium to the balcony, and once up in these states, the electron quickly falls back to the orchestra. The mezzanine is different—it has a very low transition probability, so that it is very hard to promote an electron from the orchestra into these levels. Once in the mezzanine, the electron has a very low probability of dropping back to the lowest energy state—it will thus sit in this state for a long time before dropping down.

  Now, what will happen if we shine white light on this solid? White light is comprised of all visible colors in equal intensities. Due to the discrete nature of the quantized energy levels, the atom will ignore all colors except for the blue and the green (let’s not worry about the finite energy width of the orchestra and balcony for now). The blue light will be readily absorbed, as the transition rate for the orchestra to the balcony is high. Of course—easy come, easy go—and the electron in the balcony also has a high probability of dropping back down to the orchestra (in either its original seat or an empty seat created when another electron was promoted into the balcony), emitting a blue light photon as it does so (Figure 35a). For the most part this cycle continues—orchestra absorbs blue light, promoting electron to balcony; electron then releases another blue photon when falling back to the lower energy state. Occasionally, if we do this enough times, a seat in the mezzanine level becomes occupied, either by an electron being directly promoted from the orchestra to this level (just because the probability is low doesn’t mean it won’t happen if we try enough times) or possibly from the electron in the balcony dropping down into the lower-energy mezzanine instead of falling back to the orchestra (Figure 35b). We would not notice the infrared light emitted when the electron went from the balcony to the mezzanine unless we had specific detectors sensitive to this portion of the spectrum (alternatively, the electron can emit thermal energy as it moves from the balcony to the mezzanine). Once in the mezzanine, the electron will stay there until (1) an infrared photon excites it back into the balcony (not likely, as there is very little infrared light of the necessary energy in the white light source I am using); or (2) the electron drops back to an empty seat in the orchestra, emitting a green-light photon in the process (which can happen but has a low transition probability).

  So, as we expose this solid to white light, blue light is absorbed and we get blue light back, but eventually the solid ends up with electrons sitting in the mezzanine, leaving unoccupied seats in the orchestra. Now the light is turned off. All the electrons that are still up in the balcony rapidly drop down into the empty orchestra seats, and then as time goes on, the electrons in the mezzanine seats also fall back to the orchestra (Figure 35c), emitting photons as they do, even if the solid is now in a completely darkened room, glowing in the dark! Eventually, as the number of electrons in the mezzanine decreases, the light emitted by the solid becomes dimmer and dimmer, until it is recharged with another prolonged exposure to white light. From such simple quantum mechanical phenomena are totally awesome toys made.

  Doc Savage’s invisible writing must employ an “ink” for which the separation between the orchestra and the balcony is in the far ultraviolet portion of the spectrum, while the spacing between the mezzanine level and the filled orchestra corresponds to blue light. Doc used the “black-light” lamp that emits ultraviolet light to promote electrons to the balcony, which then subsequently charge up the mezzanine. From the fact, as described in the pulp adventure, that the blue writing rapidly fades, we can assume that the electrons do not stay in the mezzanine level for more than a few seconds. The intensity of ultraviolet light in the Planck spectrum for sunlight is apparently too weak to charge up these states, which is why Doc needed to use the “black-light” lamp.

  The energy separation between the balcony level and what we have termed mezzanine states, and how long electrons will remain in these states in the dark, depends on the particular elements that one introduces into the solid to produce these long-lived states. One does not need to use ultraviolet or visible light to promote electrons into these levels—any source of energy that can excite electrons from the orchestra to the balcony states can work.

  Back in the 1950s, the hands of some alarm clocks were painted with radium, and the continuous emission of alpha particles would provide the energy necessary to keep the balcony in the phosphor material occupied, thereby enabling the hands to glow in the dark. When the radium emits an alpha particle, the nucleus converts into radon, which is also radioactive. Eventually the materials for glow-in-the-dark alarm clocks were replaced with less toxic substances. Nevertheless, radioactive materials, and their ability to emit sources of energy at a uniform rate, are hard to give up. Smoke alarms use a radioactive isotope to create a beam of particles, and an alarm is triggered when this beam is obscured from its detector by smoke or haze. Certain wristwatches with glow-in-the-dark faces have replaced radium as the radioactive element that excites the phosphorescent material with high-energy electrons from the decay of tritium as the source of external energy. Most diners are likely relieved that Fiestaware dishes no longer employ uranium oxide in their bright orange-red glaze, as they did back in the 1930s. The shine on modern Fiestaware dinner plates may be not quite as bright, but it is much safer.

  CHAPTER FIFTEEN

  Death Rays and DVDs

  The popularity of the Buck Rogers newspaper strip led to a similarly successful radio serial program, and in 1934 a competing strip featuring the adventures of Flash Gordon was introduced. By the mid-1930s the demand for Buck Rogers- and Flash Gordon-inspired toy ray guns was so high that the Daisy Manufacturing Company, which had the license to create stamped-metal versions of Buck’s XZ-31 Rocket Pistol, ran out of both stee
l and cardboard boxes. Given the association of ray guns with the future conquest of space, perhaps it is not surprising that in 1960, when the development of the laser was announced, the first thing the public wanted to know was whether science had at last delivered the long-anticipated “death ray.”

  A patent for a laser, capable of projecting a high-intensity beam of visible light, designed by Charles H. Townes and Arthur L. Schawlow at Bell Labs, was filed in 1958, and in 1960 Theodore H. Maiman at Hughes Research Laboratory in California successfully constructed the first working device. At his press conference in 1960, Maiman was peppered with journalists’ questions about whether he had in fact invented a death ray. When speaking to the public, scientists from Bell Labs were instructed by management to deflect any questions concerning using the laser as a lethal weapon and took pains to avoid saying anything that might be misconstrued or misquoted. Yeah, good luck with that. In 1961, the report in the Detroit News of a lecture by a Bell Labs scientist involved in their laser program prominently featured “Death Ray” as the invention’s first potential application. Four years after Maiman’s announcement, in 1964’s MGM film Goldfinger James Bond is threatened with a slow, painful death while strapped to a table. The circular buzz saw of the 1959 novel was replaced in the movie with a high-power industrial laser, its beam slowly moving along the length of the table on a trajectory intended to bisect Agent 007.

  The physics of the laser is essentially that of a glow-in-the-dark solid. Depending on their chemical composition and material properties, lasers can emit not just green light, but red, green, blue, ultraviolet, or infrared photons. The two big differences between lasers and glow-in-the-dark solids is that in lasers, the mezzanine levels are nearly completely occupied with electrons, and, more important, when the electrons in the mezzanine level drop down to the ground state, they all do so at the same time.

  How can one ensure that all the electrons residing in the laser levels will choose to drop down to the ground state, emitting photons, simultaneously? Consider the auditorium analogy for a solid, shown in Figure 35.60 I use essentially the same argument as for the glow-in-the-dark situation from the last chapter. Electrons from the filled orchestra level are promoted up to the balcony by, for example, the absorption of light, or an electrical current. The electrons excited up into the balcony leave behind empty seats in the orchestra. The transition rate is high for electrons to go from the orchestra to the balcony, and it is similarly easy for these electrons to drop back down into the orchestra, emitting light as they do so. Occasionally, an electron will not fall from the balcony to the orchestra, but into a mezzanine seat instead. The transition rate into or out of these mezzanine levels is very low, so once the electron is in one of these quantum states it will stay there for quite some time. If electrons can be excited up to the balcony, and from there to the mezzanine, faster than they spontaneously drop down from the mezzanine level back to the orchestra, then we can obtain a situation where we have nearly as many electrons in the mezzanine level as in the orchestra.

  We are now ready for some laser action, as shown in Figure 36. There are two ways that an electron in the mezzanine band can return to an empty seat in the orchestra—it can fall or it can be pushed. The transition rate for an electron to spontaneously fall from the mezzanine to the orchestra can be, for some materials, up to a hundred million times slower than for the electron to move from the balcony to the orchestra. This was why we needed to go through the balcony levels in order to populate this intermediate energy band. What could push an electron down to the orchestra? Light.

  During the transition from the mezzanine to the orchestra, the electron’s wave function can be expressed as the overlap of the orchestra and mezzanine quantum states. During this process the electron’s average location may be considered to oscillate between its value for each state. An oscillating electric charge emits electromagnetic waves at the frequency of vibration. A formal quantum mechanical analysis of this process finds that the energy emitted is in a discrete packet of energy (that is, a photon) whose energy corresponds to the energy difference between the mezzanine and orchestra levels.61

  Once a photon is emitted, this quantum of the electromagnetic wave can induce oscillations in another electron up in the mezzanine level, making it easier for this second electron to jump down into the orchestra, emitting its own light quantum in the process. This second photon can stimulate another electron to make the transition, generating yet another photon with an energy given by the separation of the mezzanine and orchestra bands. In this way a cascade of falling electrons, each induced (pushed) by the oscillating electric field of a light quantum, results. One photon in therefore leads to potentially trillions of photons out, all with exactly the same energy, emitted all at the same time. As the photons are fast, as in speed-of-light fast, there is no noticeable time lag between the first electron falling from the mezzanine and the trillions of electrons stimulated by other photons. The device produces light amplification by stimulated emission of radiation and is called a “laser” for short.

  Figure 36: The auditorium model from the last chapter, only now the occupation of the mezzanine level is quite high. A single photon can stimulate an electron in the mezzanine to drop down to an empty seat in the orchestra, emitting a photon in the process. This photon can in turn induce another electron to make this transition, with the net effect that a very large number of electrons may be stimulated into dropping down to the lower energy band, all emitting identical energy photons. This procedure is the basic physical mechanism underlying the laser.

  Of course, if I want this stimulated emission of light to occur more than once, I have to continue to excite electrons up to the balcony level, so that I can maintain the population inversion of electrons in the mezzanine. Thus, it will take a great deal of energy to run the laser. The more photons that I want to be emitted per second, the more energy I have to expend maintaining the occupancy of the mezzanine level. A laser pointer used in a lecture presentation is relatively low intensity and can be run from two AA or AAA batteries, while the high-power versions used in industrial-laser cutting procedures require a thousand Watts of power, enough energy to run a standard household.

  Lasers make use of the fact that the emitted light is coherent (that is, all the light waves are in phase with one another, as in the constructive interference example from Chapter 2, Figure 4). The material that is being stimulated to emit photons is typically housed in a long cylinder, both ends of which are mirrored, with one end having a small hole for light to escape. Consequently all the walls of the auditorium reflect photons, and only those light quanta moving in exactly the right direction toward the single exit will depart the hall.62 Those photons that do not leave the chamber will bounce back and forth, inducing more transitions from the mezzanine to the lower level. The laser light thus forms a tightly focused beam, and as the photons are in phase, they will exhibit minimal spreading upon leaving the laser cavity. Laser light is therefore invisible unless you look directly at the aperture of the laser cylinder, unlike incandescent lightbulbs, from which the illumination spreads out uniformly in all directions. We can see light from an incandescent bulb regardless of where we are looking, but in a sense these photons’ energies are wasted, as light is hitting objects I don’t care about seeing. The laser beam can be seen only if it reflects off a surface. If there is no dust or particulates in the air to scatter the laser beam, the only way to see it is when it gets to where it is going. A tight, narrow laser beam, sent out from a lab on Earth, was measured to have broadened out to a width of only about two miles after traveling 240,000 miles to the moon.

  Thanks to the quantization of energy levels, when the electrons drop from the mezzanine to the lower-energy orchestra in response to the photon stimulation, they will all emit light of exactly the same energy. The light from a laser will thus be of a single frequency, that is, one color, with remarkably small variations. An efficient mechanism to generate red laser light is to
use a mixture of two gases, helium and neon. Both of these elements have completely filled outer quantum levels (as shown in Figure 31b) and are thus chemically inert—they do not lower their energy by forming any type of chemical bond. When an electron beam is passed through this gas mixture, the kinetic energy of the electron current can be transferred when it collides with a helium atom. An electron in the helium atom is excited from the ground state to an “excited state”—which we have been terming the balcony level. The spacing of their quantum levels is such that when the helium atom with its electron in the higher-energy state collides with a neon atom, it promotes an electron into a very long-lived excited state in the neon atom that acts as the mezzanine level. When light of the necessary frequency stimulates the neon atoms, they drop back to their ground state, emitting red photons.

  By using electrically charged (that is, ionized) argon gas instead of a helium-neon mixture, green light can be produced. Using semiconducting diodes (much more on this in the next chapter), one can dispense with the gases and construct a completely solid-state laser, capable of producing red, green, or even blue light. Red light has a lower energy, of 1.9 electron Volts, and longer wavelength (about 650 nanometers) compared to blue light’s photon energy of 2.6 electron Volts and a wavelength of 475 nanometers. The difference in wavelength may not seem like much, but it makes a big difference in your DVD player.