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**Unformatted text preview: **13M“ ._ . . n g;
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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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