The Amazing Story of Quantum Mechanics (27 page)

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 possible, 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!
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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.
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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.

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.

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.

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 steel 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.”