Warped Passages - Lisa Randall [60]
An analogy might help elucidate Planck’s logic. You’ve probably eaten dinner with people who protest when it is time to order dessert. They’re afraid of eating too much fattening food, so they rarely order their own tasty treats. If the waiter promises that the desserts are small, they might order one. But they quail at the usual large, quantized portions of cake or ice cream or pudding.
There are two types of such people. Ike belongs to the first category. He has true discipline, and really doesn’t eat dessert. When a dessert is too big, Ike simply refrains from eating it. I’m more like the second type of person—Athena is also one—who thinks that the desserts are too big, and therefore doesn’t order any for herself, but, unlike Ike, has no compunction about taking bites from the desserts on everyone else’s plate. So even when Athena refuses to order her own portion, she still ends up eating quite a lot. If Athena were eating dinner with a large number of people, and hence could pick off a large number of plates, she would suffer from an unfortunate “calorie catastrophe.”
According to the classical theory, a blackbody is more like Athena. It would emit small amounts of light at any frequency, and theorists using classical reasoning would therefore predict an “ultraviolet catastrophe.” To avoid this predicament, Planck suggested that a blackbody was analogous to the truly abstemious type. Like Ike, who never eats a fraction of a dessert, a blackbody behaves according to Planck’s quantization rule and emits light of a given frequency only in quantized energy units, equal to the constant h times the frequency f. If the frequency were high, the quantum of energy would be simply too big for light to be emitted at that frequency. A blackbody would therefore emit most of its radiation at low frequencies, and high frequencies would be automatically cut out. In quantum theory, a blackbody doesn’t emit a substantial amount of high-frequency radiation and therefore emits far less radiation than is predicted by the classical theory.
When an object emits radiation, we call the radiation pattern—that is, how much energy the object emits at each frequency at a given temperature—its spectrum10 (see Figure 40). The spectra of certain objects such as stars can approximate that of a blackbody. Such blackbody spectra have been measured at many different temperatures, and they all agree with Planck’s assumption. Figure 40 shows that the emission is all at lower frequency; at high frequency, emission shuts off.
Figure 40. The blackbody spectrum of the cosmic microwave background of the universe. A blackbody spectrum gives the amount of light that is emitted at all frequencies when the temperature of the radiating object is fixed. Notice that the spectrum cuts off at high frequency.
One of the great achievements of experimental cosmology since the 1980s has been the increasingly accurate measurement of the blackbody spectrum that the radiation in our universe produces. Originally, the universe was a hot, dense fireball containing high-temperature radiation, but since then the universe has expanded and the radiation has cooled tremendously. That is because as the universe expanded, the wavelengths of the radiation did too. And longer wavelength corresponds to lower frequency, which corresponds to lower energy, which also corresponds to lower temperature. The universe now contains radiation that looks as if it has been produced by a blackbody with a temperature of only 2.7 degrees above absolute zero—considerably cooler than when it started.
Satellites have recently measured the spectrum of this cosmic microwave background radiation (which is what Figure 40 shows). It looks almost precisely like the spectrum of a blackbody with a temperature of 2.7 degrees K. The measurements tell us that deviations are smaller than one part in ten thousand. In fact, this relic radiation is the most accurately measured blackbody spectrum to date.
When asked in 1931 how he had come up with his outrageous