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
INTERFERENCE
Now that we know that wave functions can assign negative amplitudes to possible outcomes of observations, we can return to the question of why we ever need to talk about wave functions and superpositions in the first place, rather than just assigning probabilities to different outcomes directly. The reason is interference, and those negative numbers are crucial in understanding the way interference comes about—we can add two (nonvanishing) amplitudes together and get zero, which we couldn’t do if amplitudes were never negative.
To see how this works, let’s complicate our model of feline dynamics just a bit. Imagine that we see Miss Kitty leave the upstairs bedroom. From our previous observations of her wanderings through the house, we know quite a bit about how this quantum cat operates. We know that, once she settles in downstairs, she will inevitably end up either on the sofa or under the table, nowhere else. (That is, her final state is a wave function describing a superposition of being on the sofa and being under the table.) But let’s say we also know that she has two possible routes to take from the upstairs bed to whatever downstairs resting place she chooses: She will either stop by the food dish to eat, or stop by the scratching post to sharpen her claws. In the real world all of these possibilities are adequately described by classical mechanics, but in our idealized thought-experiment world we imagine that quantum effects play an important role.
Now let’s see what we actually observe. We’ll do the experiment two separate ways. First, when we see Miss Kitty start downstairs, we very quietly sneak behind her to see which route she takes, via either the food bowl or the scratching post. She actually has a wave function describing a superposition of both possibilities, but when we make an observation we always find a definite result. We’re as quiet as possible, so we don’t disturb her; if you like, we can imagine that we’ve placed spy cameras or laser sensors. The technology used to figure out whether she goes to the bowl or the scratching post is completely irrelevant; what matters is that we’ve observed it.
We find that we observe her visiting the bowl exactly half the time and the scratching post exactly half the time. (We assume that she visits one or the other, but never both, just to keep things as simple as we can.) Any one particular observation doesn’t reveal the wave function, of course; it can tell us only that we saw her stop at the post or at the bowl that particular time. But imagine that we do this experiment a very large number of times, so that we can get a reliable idea of what the probabilities are.
But we don’t stop at that. We next let her continue on to either the sofa or the table, and after she’s had time to settle down we look again to see which place she ends up. Again, we do the experiment enough times that we can figure out the probabilities. What we now find is that it didn’t matter whether she stopped at the scratching post or at her food bowl; in both cases, we o
bserve her ending up on the sofa exactly half the time, and under the table exactly half the time, completely independently of whether she first visited the bowl or the scratching post. Apparently the intermediate step along the way didn’t matter very much; no matter which alternative we observed along the way, the final wave function assigns equal probability to the sofa and the table.
Next comes the fun part. This time, we simply choose not to observe Miss Kitty’s intermediate step along her journey; we don’t keep track of whether she stops at the scratching post or the food bowl. We just wait until she’s settled on the sofa or under the table, and we look at where she is, reconstructing the final probabilities assigned by the wave functions. What do we expect to find?
In a world governed by classical mechanics, we know what we should see. When we did our spying on her, we were careful that our observations shouldn’t have affected how Miss Kitty behaved, and half the time we found her on the sofa and half the time on the table no matter what route she took. Clearly, even if we don’t observe what she does along the way, it shouldn’t matter—in either case we end up with equal probabilities for the final step, so even if we don’t observe the intermediate stage we should still end up with equal probabilities.
But we don’t. That’s not what we see, in this idealized thought-experiment world where our cat is a truly quantum object. What we see when we choose not to observe whether she goes via the food bowl or the scratching post is that she ends up on the sofa 100 percent of the time! We never find her under the table—the final wave function assigns an amplitude of zero to that possible outcome. Apparently, if all this is to be believed, the very presence of our spy cameras changed her wave function in some dramatic way. The possibilities are summarized in the table.
This isn’t just a thought experiment; it’s been done. Not with real cats, who are unmistakably macroscopic and well described by the classical limit, but with individual photons, in what is known as the “double slit experiment.” A photon passes through two possible slits, and if we don’t watch which slit it goes through, we get one final wave function, but if we do, we get a completely different one, no matter how unobtrusive our measurements were.
Here’s how to explain what is going on. Let’s imagine that we do observe whether Miss Kitty stops by the bowl or the scratching post, and we see her stop by the scratching post. After she does so, she evolves into a superposition of being on the sofa and being under the table, with equal probability. In particular, due to details in Miss Kitty’s initial condition and certain aspects of quantum feline dynamics, the final wave function assigns equal positive amplitudes to the sofa possibility and the table possibility. Now let’s consider the other intermediate step, that we see her stop by the food bowl. In that case, the final wave function assigns a negative amplitude to the table, and a positive one to the sofa—equal but opposite numbers, so that the probabilities turn out precisely the same.198
But if we don’t observe her at the scratching-post/food-bowl juncture, then (by the lights of our thought experiment) she is in a superposition of the two possibilities at this intermediate step. In that case, the rules of quantum mechanics instruct us to add the two possible contributions to the final wave function—one from the route where she stopped by the scratching post, and one from the food bowl. In either case, the amplitudes for ending up on the sofa were positive numbers, so they reinforce each other. But the amplitudes for ending up under the table were opposite for the two intermediate cases—so when we add them together, they precisely cancel. Individually, Miss Kitty’s two possible intermediate paths left us with a nonzero probability that she would end up under the table, but when both paths were allowed (because we didn’t observe which one she took), the two amplitudes interfered.
Figure 57: Alternative evolutions for Miss Kitty’s wave function. At the top, we observe her stop at the scratching post, after which she could go to either the table or the sofa, both with positive amplitude. In the middle, we observe her go to the food bowl, after which she could go to either the table or the sofa, but this time the table has a negative amplitude (still a positive probability). At the bottom, we don’t observe her intermediate journey, so we add the amplitudes from the two possibilities. We end up with zero amplitude for the table (since the positive and negative contributions cancel), and positive amplitude for the sofa.
That’s why the wave function needs to include negative numbers, and that’s how we know that the wave function is “real,” not just a bookkeeping device to keep track of probabilities. We have an explicit case where the individual probabilities would have been positive, but the final wave function received contributions from two intermediate steps, which ended up canceling each other.
Let’s catch our breath to appreciate how profound this is, from our jaundiced classically trained perspective. For any particular instantiation of the experiment, we are tempted to ask: Did Miss Kitty stop by the food bowl or the scratching post? The only acceptable answer is: no. She didn’t do either one. She was in a superposition of both possibilities, which we know because both possibilities ended up giving crucial contributions to the amplitude of the final answer.
Real cats are messy macroscopic objects consisting of very large numbers of molecules, and their wave functions tend to be very sharply concentrated around something resembling our classical notion of a “position in space.” But at the microscopic level, all this talk of wave functions and superpositions and interference becomes brazenly demonstrable. Quantum mechanics strikes us as weird, but it’s the way Nature works.
COLLAPSE OF THE WAVE FUNCTION
The thing about this discussion that tends to rub people the wrong way—and with good reason—is the crucial role played by observation. When we observed what the cat was doing at the scratching-post/food-bowl juncture, we got one answer for the final state; when we didn’t make any such observation, we got a very different answer. That’s not how physics is supposed to work; the world is supposed to evolve according to the laws of Nature, whether we are observing it or not. What counts as an “observation,” anyway? What if we set up surveillance cameras along the way but then never looked at the tapes? Would that count as an observation? (Yes, it would.) And what precisely happened when we did make our observation?
This is an important collection of questions, and the answers are not completely clear. There is no consensus within the physics community about what really constitutes an observation (or “measurement”) in quantum mechanics, nor about what happens when an observation occurs. This is the “measurement problem” and is the primary focus of people who spend their time thinking about interpretations of quantum mechanics. There are many such interpretations on the market, and we’re going to discuss two: the more or less standard picture, known as the “Copenhagen interpretation,” and a view that seems (to me) to be a bit more respectable and likely to conform to reality, which goes under the forbidding name the “many-worlds interpretation.” Let’s look at Copenhagen first.199
The Copenhagen interpretation is so named because Niels Bohr, who in many ways was the godfather of quantum mechanics, helped to develop it from his institute in Copenhagen in the 1920s. The actual history of this perspective is complicated, and certainly involves a great deal of input from Werner Heisenberg, another quantum pioneer. But the history is less crucial to our present purposes than the status of the Copenhagen view as what is enshrined in textbooks as the standard picture. Every physicist learns this first, and then gets to contemplate other alternatives (or choose not to, as the case may be).
The Copenhagen interpretation of quantum mechanics is as easy to state as it is hard to swallow: when a quantum system is subjected to a measurement, its wave function collapses. That is, the wave function goes instantaneously from describing a superposition of various possible observational outcomes to a completely different wave function, one that assigns 100 percent probability to the outcome that was actually measured, and 0 percent to anything else. That kin
d of wave function, concentrated entirely on a single possible observational outcome, is known as an “eigenstate.” Once the system is in that eigenstate, you can keep making the same kind of observation, and you’ll keep getting the same answer (unless something kicks the system out of the eigenstate into another superposition). We can’t say with certainty which eigenstate the system will fall into when an observation is made; it’s an inherently stochastic process, and the best we can do is assign a probability to different outcomes.
We can apply this idea to the story of Miss Kitty. According to the Copenhagen interpretation, our choice to observe whether she stopped by the food bowl or the scratching post had a dramatic effect on her wave function, no matter how sneaky we were about it. When we didn’t look, she was in a superposition of the two possibilities, with equal amplitude; when she then moved on to the sofa or the table, we added up the contributions from each of the intermediate steps, and found there was interference. But if we chose to observe her along the way, we collapsed her wave function. If we saw her stop at the scratching post, once that observation was made she was in a state that was no longer a superposition—she was 100 percent scratching post, 0 percent food bowl. Likewise if we saw her stop at the food bowl, with the amplitudes reversed. In either case, there was nothing left to interfere with, and her wave function evolved into a state that gave her equal probabilities to end up on the sofa or under the table.200
There is good news and bad news about this story. The good news is that it fits the data. If we imagine that wave functions collapse every time we make an observation—no matter how unobtrusive our observational strategy may be—and that they end up in eigenstates that assign 100 percent probability to the outcome we observed, we successfully account for all of the various quantum phenomena known to physicists.
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 30
