Elegant Universe _2_ - Greene.pdf

Elegant Universe _2_ - Greene.pdf - 13M n g i i 7 >2 r3 Z)l...

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Unformatted text preview: 13M“ ._ . . n g; i i «7; >2 r3. Z)l The Elegant Universe gresses, but it is not the only way. In fact, we have already seen this: The search for a new theory of gravity was initiated, not by an experimental refutation of Newton’s theory, but rather by the conflict of Newtonian gravity with another theory—special relativity. It was only after the dis- 1 covery of general relativity as a competing theory of gravity that experi- mental flaws in Newton's theory were identified by seeking out tiny but measurable ways in which the two theories differ. Thus, internal theoret- ical inconsistencies can play as pivotal a role in driving progress as do ex- perimental data. For the last half century, physics has been faced with still another theoretical conflict whose severity is on par with that between special rel— ativity and Newtonian gravity. General relativity appears to be fundamen- tally incompatible with another extremely well-tested theory: quantum mechanics. Regarding the material covered in this chapter, the conflict prevents physicists from understanding what really happens to space, time, and matter when crushed together fully at the moment of the big bang or at the central point of a black hole. But more generally, the con- flict alerts us to a fundamental deficiency in our conception of nature. The resolution of this conflict has eluded attempts by some of the greatest the- oretical physicists, giving it a well-deserved reputation as the central prob- lem of modern theoretical physics. Understanding the conflict requires familiarity with some basic features of quantum theory, to which we now turn. 84 W Chapter 4 Microscopic Weirdness bit worn out from their trans-solar-system expedition, George and Gracie return to earth and head over to the H-iBar for some post- space-sojouming refreshments. George orders the (filial—papaya juice on the rocks for himself and a vodka tonic for Graci I and kicks back in his chair, hands clasped behind his head, to enjoy a freshly lit cigar. Just as he prepares to inhale, though, he is stunned to find that the cigar has vanished from between his teeth. Thinking that the cigar must somehow have slipped from his mouth, George sits forward expecting to find it burning a hole in his shirt or trousers. But it is not there. The cigar is not to be found. Gracie, roused by George’s frantic movement, glances over and spots the cigar lying on the counter directly behind George’s chair. “Strange,” George says, “how in the heck could it have fallen over there? It’s as if it went right through my head—but my tongue isn’t burned and I don’t seem to have any new holes.” Gracie examines George and reluc- tantly confirms that his tongue and head appear to be perfectly normal. As the drinks have just arrived, George and Gracie shrug} their shoulders and chalk up the fallen cigar to one of life’s little mysteries. But the weirdness at the H-Bar continues. ‘ George looks into his papaya juice and notices that the ice cubes are incessantly rattling around—bouncing off of each other and the sides of the glass like overcharged automobiles in a bumperfcar arena. And this time he is not alone. Gracie holds up her glass, which is about half the size 85 i g, k I“ ~71 L: graze; The Elegant Universe of George’s, and both of them see that her ice cubes are bouncing around even more frantically. They can hardly make out the individual cubes as they all blur together into an icy mass. But none of this compares to what happens next. As George and Gracie stare at her rattling drink with wide- eyed wonderment, they see a single ice cube pass through the side of her glass and drop down to the bar. They grab the glass and see that it is fully intact; somehow the ice cube went right through the solid glass without causing any damage. “Must be post-space—walk hallucinations,” says George. They each fight off the frenzy of careening ice cubes to down their drinks in one go, and head home to recover. Little do George and Gracie realize that in their haste to leave, they mistook a decorative door painted on a wall of the bar for the real thing. The patrons of the H—Bar, though, are well accustomed to people passing through walls and hardly take note of George and Gracie’s abrupt departura A century ago, while Conrad and Freud were illuminating the heart and the soul of darkness, the German physicist Max Planck shed the first ray of light on quantum mechanics, a conceptual framework that proclaims, among other things, that the H-Bar experiences of George and Gracie—— when scaled down to the microscopic realm—need not be attributed to clouded faculties. Such unfamiliar and bizarre happenings are typical of how our universe, on extremely small scales, actually behaves. The Quantum Framework Quantum mechanics is a conceptual framework for understanding the microscopic properties of the universe. And just as special relativity and general relativity require dramatic changes in our worldview when things are moving very quickly or when they are very massive, quantum me- chanics reveals that the universe has equally if not more startling proper— ties when examined on atomic and subatomic distance scales. In 1965, Richard Feynman, one of the greatest practitioners of quantum mechan- ics, wrote, There was a time when the newspapers said that only twelve men un- derstood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did because 86 Microscopic Weirdness l he was the only guy who caught on, before he wrot ‘his paper. But after people read the paper a lot of people understood t e theory of relativ— ity in one way or other, certainly more than twelve? On the other hand I think I can safely say that nobody understands quantum mechanics.‘ Although Feynman expressed this view more than three decades ago, it applies equally well today. What he meant is that although the special and general theories of relativity require a drastic revision of previous ways of seeing the world, when one fully accepts the basic principles underly- ing them, the new and unfamiliar implications for space and time follow directly from careful logical reasoning. If you ponder the descriptions of Einstein’s work in the preceding two chapters with adequate intensity, you will—if even for just a moment—recognize the inevitability of the conclusions we have drawn. Quantum mechanics is !different. By 1928 or so, many of the mathematical formulas and rules of 3quantum mechanics had been put in place and, ever since, it has been used to make the most precise and successful numerical predictions in thle history of science. But in a real sense those who use quantum mechanics find themselves fol- lowing rules and formulas laid down by the “founding fathers" of the theory—calculational procedures that are straightforward to carry out—— without really understanding why the procedures we or what they really mean. Unlike relativity, few if any people ever grasp at a “soulful” level. In What are we to make of this? Does it mean that on a microscopic level the universe operates in ways so obscure and unfamiliar that the human mind, evolved over eons to cope with phenomena on familiar everyday scales, is unable to fully grasp “what really goes on”? Or, might it be that through historical accident physicists have constructed an extremely awk- ward formulation of quantum mechanics that, although quantitatively successful, obfuscates the true nature of reality? No one knows. Maybe some time in the future some clever person will see clear to a new for- mulation that will fully reveal the “whys” and the “whats" of quantum mechanics. And then again, maybe not. The only thing we know with cer- tainty is that quantum mechanics absolutely and unequivocally shows us that a number of basic concepts essential to our understanding of the fa- miliar everyday world fail to have any meaning when our focus narrows to ‘the microscopic realm. As a result, we must significantly modify both our uantum mechanics 87 a 1' i The Elegant Universe language and our reasoning when attempting to understand and explain . the universe on atomic and subatomic scales. In the following sections we will develop the basics of this language and describe a number of the remarkable surprises it entails. If along the way quantum mechanics seems to you to be altogether bizarre or even ludi— crous, you should bear in mind two things. First, beyond the fact that it is a mathematically coherent theory, the only reason we believe in quantum mechanics is because it yields predictions that have been verified to as- tounding accuracy. If someone can tell you volumes of intimate details of your childhood in excruciating detail, it’s hard not to believe their claim of being your long—lost sibling. Second, you are not alone in having this re- action to quantum mechanics. It is a View held to a greater or lesser extent by some of the most revered physicists of all time. Einstein refused to ac- cept quantum mechanics fully. And even Niels Bohr, one of the central pi- oneers of quantum theory and one of its strongest proponents, once remarked that if yoii do not get dizzy sometimes when you think about quantum mechanics, then you have not really understood it. It’s Too Hot in the Kitchen The road to quantum mechanics began with a puzzling problem. Imagine that your oven at home is perfectly insulated, that you set it to some tem— perature, say 400 degrees Fahrenheit, and you give it enough time to heat up. Even if you had sucked all the air from the oven before turning it on, by heating its walls you generate waves of radiation in its interior. This is the same kind of radiation—heat and light in the form of electromagnetic waves—that is emitted by the surface of the sun, or a glowing—hot iron poker. Here’s the problem. Electromagnetic waves carry energy—life on earth, for example, relies crucially on solar energy transmitted from the sun to the earth by electromagnetic waves. At the beginning of the twentieth century, physicists calculated the total energy carried by all of the electromagnetic radiation inside an oven at a chosen temperature. Using well-established calculational procedures they came up with a ridiculous answer: For any chosen temperature, the total energy in the oven is infinite. It was clear to everyone that this was nonsense—a hot oven can em- 88 Microscopic Weirdness l 1 body significant energy but surely not an infinite amount. To understand the resolution proposed by Planck it is worth understanding the problem in a bit more detail. It turns out that when Maxwell’s electromagnetic theory is applied to the radiation in an oven it shows that the waves gen— erated by the hot walls must have a whole number of peaks and troughs that fit perfectly between opposite surfaces. Some examples are shown in Figure 4.1. Physicists use three terms to describe these waves: wave— length, frequency, and amplitude. The wavelength is he distance between successive peaks or successive troughs of the wavesias illustrated in Fig- ure 4.2. More peaks and troughs mean a shorter wav length, as they must all be crammed in between the fixed walls of the oven. The frequency refers to the number of up—and-down cycles of oscillation that a wave completes every second. It turns out that the frequency is determined by . the wavelength and vice versa: longer wavelengths imply lower frequency; ' shorter wavelengths imply higher frequency. To see why, think of what ._ happens when you produce waves by shaking a long r0pe that is tied down “ at one end. To generate a long wavelength, you leisurely shake your end up and down. The frequency of the waves matches the number of cycles per second your arm goes through and is consequently fairly low. But to generate short wavelengths you shake your end more frantically—more frequently, so to speak—and this yields a higher-frequency wave. Finally, physicists use the term amplitude—to describe the inaximum height or depth of a wave, as also illustrated in Figure 4.2. ‘ In case you find electromagnetic waves a bit abstract, another good i Figure 4.1 Maxwells theory tells us that the radiation waives in an oven have a whole number of crests and troughs—they fill out complete wave—cycles. 89 The Elegant Universe wavelength amplitude Figure 4.2 The wavelength is the distance between successive peaks or troughs of a wave. The amplitude is the maximal height or depth of the wave. analogy to keep in mind are the waves that are produced by plucking a vi- olin string. Different wave frequencies correspond to different musical notes: the higher the frequency, the higher the note. The amplitude of a wave on a violin string is determined by how hard you pluck it. A harder pluck means that you~put more energy into the wave disturbance; more en- ergy therefore corresponds to a larger amplitude. You can hear this, as the resulting tone is louder. Similarly, less energy corresponds to a smaller amplitude and a lower volume of sound. By making use of nineteenth-century thermodynamics, physicists were able to determine how much energy the hot walls of the oven would pump into electromagnetic waves of each allowed wavelength—how hard the walls would, in effect, “pluck" each wave. The result they found is simple to state: Each of the allowed waves—regardless of its wavelength—carries the same amount of energy (with the precise amount determined by the temperature of the oven). In other words, all of the possible wave patterns within the oven are on completely equal footing when it comes to the amount of energy they embody. At first this seems like an interesting, albeit innocuous, result. It isn’t. It spells the downfall of what has come to be known as classical physics. The reason is this: Even though requiring that all waves have a whole number of peaks and troughs rules out an enormous variety of conceivable wave patterns in the oven, there are still an infinite number that are possible—those with ever more peaks and troughs. Since each wave pat- tern carries the same amount of energy, an infinite number of them trans- lates into an infinite amount of energy. At the turn of the century, there was a gargantuan fly in the theoretical ointment. 9O m. was“ a. i mt.) “1 em MK Microscopic Weirdness t Making Lumps at the Turn of the Century f In 1900 Planck made an inspired guess that allowed a way out of this puz- f zle and would earn him the 1918 Nobel Prize in physics.2 To get a feel for f his resolution, imagine that you and a huge crowd of pebple——“infinite” in ' number—are crammed into a large, cold warehouse rurl by a miserly land- lord. There is a fancy digital thermostat on the wall that: controls the tem- perature but you are shocked when you discover the} charges that the landlord levies for heat. If the thermostat is set to 50 degrees Fahrenheit everyone must give the landlord $50. If it is set to 55 degrees everyone must pay $55, and so on. You realize that since you are sharing the ware- house vvith an infinite number of companions, the landl ‘rd will earn an in— finite amount of money if you turn on the heat at all. But on closer reading of the landlord’s rules of payrn nt you see a loop- hole. Because the landlord is a very busy man he does? not want to give change, especially not to an infinite number of individual tenants. So he i _ 1 works on an honor system. Those who can pay exactly what they owe, do so. Otherwise, they pay only as much as they can Without requiring change. And so, wanting to involve everyone but wanting to avoid the ex- orbitant charges for hear, you compel your comrades to organize the wealth of the group in the following manner: One person carries all of the pennies, one person carries all of the nickels, one came1 all of the dimes, one carries all of the quarters, and so on through dollar bills, five-dollar bills, ten-dollar bills, twenties, fifties, hundreds, thousands, and ever larger (and unfamiliar) denominations. You brazenly set the thermostat to 80 degrees and await the landlord’s arrival. When he does come, the person carrying pennies goes to pay first and turns over 8,000. The person carry- ing nickels then turns over 1,600 of them, the person carfying dimes turns over 800, the person with quarters turns over 320, the person with dollars gives the landlord 80, the person with five-dollar bills turns over 16, the person with ten-dollar bills gives him 8, the person with ’enties gives him 4, and the person with fifties hands over one (since ilfifw-dollar bills would exceed the necessary payment, thereby requiri g change). But everyone else carries only a denomination—a minimal “lulf'np” of money—— that exceeds the required payment. Therefore they cannot pay the land- 91 The Elegant Universe lord and hence rather than getting the infinite amount of money he ex- pected, the landlord leaves with the paltry sum of $690. Planck made use of a very similar strategy to reduce the ridiculous re- sult of infinite energy in an oven to one that is finite. Here’s how. Planck boldly guessed that the energy carried by an electromagnetic wave in the oven, like money, comes in lumps. The energy can be one times some fundamental “energy denomination,” or two times it, or three times it, and so forth—but that’s it. Just as you can’t have one-third of a penny or two and a half quarters, Planck declared that when it comes to energy, no fractions are allowed. Now, our monetary denominations are deter— mined by the United States Treasury. Seeking a more fundamental expla- nation, Planck suggested that the energy denomination of a wave—the minimal lump of energy that it can have—is determined by its frequency. Specifically, he posited that the minimum energy a wave can have is proportional to its frequency: larger frequency (shorter wavelength) implies larger minimum energy;~smaller frequency (longer wavelength) implies smaller minimum energy. Roughly speaking, just as gentle ocean waves are long and luxurious while harsh ones are short and choppy, long- wavelength radiation is intrinsically less energetic than short—wavelength radiation. Here's the punch line: Planck’s calculations showed that this lumpiness of the allowed energy in each wave cured the previous ridiculous result of infinite total energy. It’s not hard to see why. When an oven is heated to some chosen temperature, the calculations based on nineteenth-century thermodynamics predicted the common energy that each and every wave would supposedly contribute to the total. But like those comrades who cannot contribute the common amount of money they each owe the land- lord because the monetary denomination they carry is too large, if the minimum energy a particular wave can carry exceeds the energy it is sup- posed to contribute, it can’t contribute and instead lies dormant. Since, ac- cording to Planck, the minimum energy a wave can carry is proportional to its frequency, as we examine waves in the oven of ever larger frequency (shorter wavelength), sooner or later the minimum energy they can carry is bigger than the expected energy contribution. Like the comrades in the warehouse entrusted with denominations larger than fifty-dollar bills, these waves with ever—larger frequencies cannot contribute the amount of 92 Microscopic Weirdness energy demanded by nineteenth-century physics. And so, fiust as only a finite number of comrades are able to contribute to the total heat payment—~leading to a finite amount of total money—only a finite num- ber of waves are able to contribute to the oven 5 total energy-again lead- ing to a finite amount of total energy. Be it energy or money, the lumpiness of the fundamental units—and the ever increasing size of these lumps as we go to higher frequencies or to larger monetary denominations— changes an infinite answer to one that is finite.3 l By eliminating the manifest nonsense of an infinite result, Planck had taken an impor...
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