From Eternity to Here (43 page)

But now we know better: Classical mechanics isn’t correct. In the early decades of the twentieth century, physicists trying to understand the behavior of matter on microscopic scales were gradually forced to the conclusion that the rules would have to be overturned and replaced with something else. That something else is
quantum mechanics
, arguably the greatest triumph of human intelligence and imagination in all of history. Quantum mechanics offers an image of the world that is radically different from that of classical mechanics, one that scientists never would have seriously contemplated if the experimental data had left them with any other choice. Today, quantum mechanics enjoys the status that classical mechanics held at the dawn of the twentieth century: It has passed a variety of empirical tests, and most researchers are convinced that the ultimate laws of physics are quantum mechanical in nature.

But despite its triumphs, quantum mechanics remains somewhat mysterious. Physicists are completely confident in how they
use
quantum mechanics—they can build theories, make predictions, and test against experiments, and there is never any ambiguity along the way. Nevertheless, we’re not completely sure we know what quantum mechanics really
is
. There is a respectable field of intellectual endeavor, occupying the time of a substantial number of talented scientists and philosophers, that goes under the name “interpretations of quantum mechanics.” A century ago, there was no such field as “interpretations of classical mechanics”—classical mechanics is perfectly straightforward to interpret. We’re still not sure what is the best way to think and talk about quantum mechanics.

This interpretational anxiety stems from the single basic difference between quantum mechanics and classical mechanics, which is both simple and world shattering in its implications:

According to quantum mechanics, what we can
observe
about the world is only a tiny subset of what actually
exists
.

Attempts at explaining this principle often water it down beyond recognition. “It’s like that friend of yours who has such a nice smile, except when you try to take his picture it always disappears.” Quantum mechanics is a lot more profound than that. In the classical world, it might be difficult to obtain a precise measurement of some quantity; we need to be very careful not to disturb the system we’re looking at. But there is nothing in classical physics that prevents us from being careful. In quantum mechanics, on the other hand, there is an unavoidable obstacle to making complete and nondisruptive observations of a physical system. It simply can’t be done, in general. What exactly happens when you try to observe something, and what actually counts as a “measurement”—those are the locus of the mystery. This is what is helpfully known as the “measurement problem,” much as having an automobile roll off a cliff and smash into pieces on the rocks hundreds of feet below might be known as “car trouble.” Successful physical theories aren’t supposed to have ambiguities like this; the very first thing we ask about them is that they be clearly defined. Quantum mechanics, despite all its undeniable successes, isn’t there yet.

None of which should be taken to mean that all hell has broken loose, or that the mysteries of quantum mechanics offer an excuse to believe whatever you want. In particular, quantum mechanics doesn’t mean you can change reality just by thinking about it, or that modern physics has rediscovered ancient Buddhist wisdom.
¹⁹⁵
There are still rules, and we know how the rules operate in the regimes of interest to our everyday lives. But we’d like to understand how the rules operate in every conceivable situation.

Most modern physicists deal with the problems of interpreting quantum mechanics through the age-old strategy of “denial.” They know how the rules operate in cases of interest, they can put quantum mechanics to work in specific circumstances and achieve amazing agreement with experiment, and they don’t want to be bothered with pesky questions about what it all means or whether the theory is perfectly well-defined. For our purposes in this book, that is often a pretty good strategy. The problem of the arrow of time was there for Boltzmann and his collaborators before quantum mechanics was ever invented; we can go very far talking about entropy and cosmology without worrying about the details of quantum mechanics.

At some point, however, we need to face the music. The arrow of time is, after all, a fundamental puzzle, and it’s possible that quantum mechanics will play a crucial role in resolving that puzzle. And there’s something else of more direct interest: That process of measurement, where all of the interpretational tangles of quantum mechanics are to be found, has the remarkable property that it is
irreversible
. Alone among all of the well-accepted laws of physics, quantum measurement is a process that defines an arrow of time: Once you do it, you can’t undo it. And that’s a mystery.

It’s very possible that this mysterious irreversibility is of precisely the same character as the mysterious irreversibility in thermodynamics, as codified in the Second Law: It’s a consequence of making approximations and throwing away information, even though the deep underlying processes are all individually reversible. I’ll be advocating that point of view in this chapter. But the subject remains controversial among the experts. The one sure thing is that we have to confront the measurement problem head-on if we’re interested in the arrow of time.

THE QUANTUM CAT

Thanks to the thought-experiment stylings of Erwin Schrödinger, it has become traditional in discussions of quantum mechanics to use cats as examples.
¹⁹⁶
Schrödinger’s cat was proposed to help illustrate the difficulties involved in the measurement problem, but we’re going start with the basic features of the theory before diving into the subtleties. And no animals will be harmed in our thought experiments.

Imagine your cat, Miss Kitty, has two favorite places in your house: on the sofa and under the dining room table. In the real world, there are an infinite number of positions in space that could specify the location of a physical object such as a cat; likewise, there are an infinite number of momenta, even if your cat tends not to move very fast. We’re going to be simplifying things dramatically, in order to get at the heart of quantum mechanics. So let’s imagine that we can completely specify the state of Miss Kitty—as it would be described in classical mechanics—by saying whether she is on the sofa or under the table. We’re throwing out any information about her speed, or any knowledge of exactly what part of the sofa she’s on, and we’re disregarding any possible positions that are not “sofa” or “table.” From the classical point of view, we are simplifying Miss Kitty down to a two-state system. (Two-state systems actually exist in the real world; for example, the spin of an electron or photon can either point up or point down. The quantum state of a two-state system is described by a “qubit.”)

Here is the first major difference between quantum mechanics and classical mechanics: In quantum mechanics, there is
no such thing
as “the location of the cat.” In classical mechanics, it may happen that we don’t know where Miss Kitty is, so we may end up saying things like “I think there’s a 75 percent chance that she’s under the table.” But that’s a statement about our ignorance, not about the world; there really is a fact of the matter about where the cat is, whether we know it or not.

In quantum mechanics, there is no fact of the matter about where Miss Kitty (or anything else) is located. The space of states in quantum mechanics just doesn’t work that way. Instead, the states are specified by something called a
wave function
. And the wave function doesn’t say things like “the cat is on the sofa” or “the cat is under the table.” Rather, it says things like “if we were to look, there would be a 75 percent chance that we would find the cat under the table, and a 25 percent chance that we would find the cat on the sofa.”

This distinction between “incomplete knowledge” and “intrinsic quantum indeterminacy” is worth dwelling on. If the wave function tells us there is a 75 percent chance of observing the cat under the table and a 25 percent chance of observing her on the sofa, that does not mean there is a 75 percent chance that the cat
is
under the table and a 25 percent chance that she
is
on the sofa. There is no such thing as “where the cat is.” Her quantum state is described by a
superposition
of the two distinct possibilities we would have in classical mechanics. It’s not even that “they are both true at once”; it’s that there is no “true” place where the cat is. The wave function is the best description we have of the reality of the cat.

It’s clear why this is hard to accept at first blush. To put it bluntly, the world doesn’t look anything like that. We see cats and planets and even electrons in particular positions when we look at them, not in superpositions of different possibilities described by wave functions. But that’s the true magic of quantum mechanics: What we see is not what there is. The wave function really exists, but we don’t see it when we look; we see things as if they were in particular ordinary classical configurations.

None of which stops classical physics from being more than good enough to play basketball or put satellites in orbit. Quantum mechanics features a “classical limit” in which objects behave just as they would had Newton been right all along, and that limit includes all of our everyday experiences. For objects such as cats that are macroscopic in size, we never find them in superpositions of the form “75 percent here, 25 percent there”; it’s always “99.9999999 percent (or more) here, 0.0000001 percent (or much less) there.” Classical mechanics is an approximation to how the macroscopic world operates, but a very good one. The real world runs by the rules of quantum mechanics, but classical mechanics is more than good enough to get us through everyday life. It’s only when we start to consider atoms and elementary particles that the full consequences of quantum mechanics simply can’t be avoided.

HOW WAVE FUNCTIONS WORK

You might wonder how we know any of this is true. What is the difference, after all, between “there is a 75 percent chance of observing the cat under the table” and “there is a 75 percent chance that the cat is under the table”? It seems hard to imagine an experiment that could distinguish between those possibilities—the only way we would know where it is would be to look for it, after all. But there is a crucially important phenomenon that drives home the difference, known as
quantum interference
. To understand what that means, we have to bite the bullet and dig a little more deeply into how wave functions really work.

In classical mechanics, where the state of a particle is a specification of its position and its momentum, we can think of that state as specified by a collection of numbers. For one particle in ordinary three-dimensional space, there are six numbers: the position in each of the three directions, and the momentum in each of the three directions. In quantum mechanics the state is specified by a wave function, which can also be thought of as a collection of numbers. The job of these numbers is to tell us, for any observation or measurement we could imagine doing, what the probability is that we will get a certain result. So you might naturally think that the numbers we need are just the probabilities themselves: the chance Miss Kitty will be observed on the sofa, the chance she will be observed under the table, and so on.

As it turns out, that’s not how reality operates. Wave functions really are wavelike: A typical wave function oscillates through space and time, much like a wave on the surface of a pond. This isn’t so obvious in our simple example where there are only two possible observational outcomes—“on sofa” or “under table”—but it becomes more clear when we consider observations with continuous possible outcomes, like the position of a real cat in a real room. The wave function is like a wave on a pond, except it’s a wave on the space of all possible outcomes of an observation—for example, all possible positions in a room.

When we see a wave on a pond, the level of the water isn’t uniformly higher than what it would be if the pond were undisturbed; sometimes the water goes up, and sometimes it goes down. If we were to describe the wave mathematically, to every point on the pond we would associate an
amplitude
—the height by which the water was displaced—and that amplitude would sometimes be positive, sometimes be negative. Wave functions in quantum mechanics work the same way. To every possible outcome of an observation, the wave function assigns a number, which we call the amplitude, and which can be positive or negative. The complete wave function consists of a particular amplitude for every possible observational outcome; those are the numbers that specify the state in quantum mechanics, just as the positions and momenta specify the state in classical mechanics. There is an amplitude that Miss Kitty is under the table, and another one that she is on the sofa.

There’s only one problem with that setup: What we care about are probabilities, and the probability of something happening is never a negative number. So it can’t be true that the amplitude associated with a certain observational outcome is equal to the probability of obtaining that outcome—instead, there must be a way of calculating the probability if we know what the amplitude is. Happily, the calculation is very easy! To get the probability, you take the amplitude and you square it.

(probability of observing
X
) = (amplitude assigned to
X
)
²
.

So if Miss Kitty’s wave function assigns an amplitude of 0.5 to the possibility that we observe her on the sofa, the probability that we see her there is (0.5)
²
= 0.25, or 25 percent. But, crucially, the amplitude could also be -0.5, and we would get exactly the same answer: (-0.5)
²
= 0.25. This might seem like a pointless piece of redundancy—two different amplitudes corresponding to the same physical situation—but it turns out to play a key role in how states evolve in quantum mechanics.
¹⁹⁷