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CHAPTER 3
Schrödinger’s Dog:
The Copenhagen Interpretation
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I’m in the kitchen, getting a glass of water, when Emmy trots in, tail wagging. “You should give me a treat,” she says.
“I should? Why should I give you a treat?”
“Because I’m a very good dog, and I deserve a treat!”
“I’m not going to give you a treat for no reason,” I say, “but I’ll tell you what I’ll do.” I reach into the treat jar, then hold out both fists. “Guess where the treat is, and you can have it.”
Immediately, her nose starts working.
“No sniffing, either.” I put my hands behind my back. “Just guess which hand has the treat.”
“Ummm … Okay. Both.”
“That’s not one of the choices.”
“But it’s the right answer,” she says, pouting. “It’s like that cat in the box.”
“What cat in what box?”
“You know, the one in the box. With the thing. It’s dead and alive at the same time. In the box.”
“Schrödinger’s cat?”
“Yes! That’s the one!” She wags her tail excitedly. “I like that experiment. You should do that.”
“For one thing, it’s just a thought experiment to show the absurdity of quantum predictions. Nobody ever did it for real. For another, I doubt that people would appreciate it if we started killing cats.”
“I don’t care about the killing. I just like the idea of putting cats in boxes. Cats belong in boxes.”
“I’ll pass that on to the scientific community. But what does this have to do with your treat?”
“Well, the treat could be in your left hand, and it could be in your right hand. I don’t know which it’s in, and you won’t let me sniff to see where it is, so that means that the treat is in a superposition state of both left and right hands. Until I measure which hand it’s in, the answer is that it’s in both hands at the same time.”
“That’s an interesting argument. It doesn’t apply here, though.”
“Yes it does. It’s basic quantum mechanics.”
“Well, yes, it’s true that unmeasured objects exist in superposition states as a general matter,” I say, “but those superposition states are extremely fragile. Any disturbance at all—absorbing or emitting even a single photon—will cause them to collapse into classical states with a definite value.”
“People have seen them, though.”
“Yes, there have been lots of ‘cat state’ experiments done, but the largest superposition anybody has managed to make involved something like a billion electrons.* That’s nowhere near the size of a dog treat, which would contain something like 1022 atoms.”
“Oh.”
“And on top of that, even in the most extreme variant of the Copenhagen interpretation, the wavefunction is collapsed by the act of observation by a conscious observer. Now, you can argue about who counts as an observer—”
“Not a cat, that’s for certain. Cats are stupid.”
“—but by any reasonable standard, I count as an observer. I know which hand the treat is in. So you’re dealing with a classical probability distribution, in which the treat is in either one hand or the other, not a quantum superposition in which the treat is in both hands at the same time.”
“Oh. Okay.” She looks disappointed.
“So, guess which hand the treat is in.”
“Ummm … I still say both.”
“Why is that?”
“Because I am an excellent dog, and I deserve two treats!”
“Well, yes. Also, I’m a sucker.” I give her both of the treats.
“Great! Treats!” she says, crunching happily.
One of the most vexing things about studying quantum mechanics is how stubbornly classical the world is. Quantum physics features all sorts of marvelous things—particles behaving like waves, objects being in two places at the same time, cats that are both alive and dead—and yet, we don’t see any of those things in the world around us. When we look at an everyday object, we see it in a definite classical state—with some particular position, velocity, energy, and so on—and not in any of the strange combinations of states allowed by quantum mechanics. Particles and waves look completely different, dogs can only pass on one side or the other of an obstacle, and cats are stubbornly, irritatingly alive and not happy about being sniffed by strange dogs.
We directly observe the stranger features of quantum mechanics only with a great deal of work, in carefully controlled conditions. Quantum states turn out to be remarkably fragile and easily destroyed, and the reason for this fragility is not immediately obvious. In fact, determining why quantum rules don’t seem to apply in the macroscopic world of everyday dogs and cats is a surprisingly difficult problem. Exactly what happens in the transition from the microscopic to the macroscopic has troubled many of the best physicists of the last hundred years, and there’s still no clear answer.
In this chapter, we’ll lay out the basic principles that are central to understanding quantum physics: wavefunctions, allowed states, probability, and measurement. We’ll introduce a key example system, and talk about a simple experiment that demonstrates all of the essential features of quantum physics. We’ll talk about the essential randomness of quantum measurement, and the philosophical problems raised by this randomness, which are disturbing enough that even some of the founders of quantum physics gave up on it entirely.
WHAT DOES A WAVEFUNCTION MEAN? INTERPRETATION OF
QUANTUM MECHANICS
Most of the philosophical problems with quantum mechanics center around the “interpretation” of the theory. This is a problem unique to quantum mechanics, as classical physics doesn’t require interpretation. In classical physics, you predict the position, velocity, and acceleration of some object, and you know exactly what those quantities mean and how to measure them. There’s an immediate and intuitive connection between the theory and the reality that we observe.
Quantum mechanics, on the other hand, is not nearly so obvious. We have the mathematical equations that govern the theory and allow us to calculate wavefunctions and predict their behavior, but just what those wavefunctions mean is not immediately clear. We need an “interpretation,” an extra layer of explanation, to connect the wavefunctions we calculate to the properties we measure in experiments.
The central elements of quantum mechanics can be presented in many different ways—as many different ways as there are books on the subject—but in the end, they all rest on four basic principles. You can think of these as the core principles of the theory, the basic rules that you have to accept in order to make any progress.*
CENTRAL PRINCIPLES OF QUANTUM MECHANICS
1. Wavefunctions: Every object in the universe is described by a quantum wavefunction.
2. Allowed states: A quantum object can only be observed in one of a limited number of allowed states.
3. Probability: The wavefunction of an object determines the probability of being found in each of the allowed states.
4. Measurement: Measuring the state of an object absolutely determines the state of that object.
The first principle is the idea of wavefunctions. Every object or system of objects in the universe is described by a wavefunction, a mathematical function that has some value at every point in space. It doesn’t matter what you’re describing—an electron, a dog treat, a cat in a box—it has a wavefunction, and that wavefunction has some value no matter where you look. The value could be positive, or negative, or zero, or even an imaginary number (like the square root of -1), but it has a value everywhere.
A mathematical formula called the Schrödinger equation (after the Austrian physicist and noted cad* Erwin Schrödinger, who discovered it) governs the behavior of wavefunctions. Given some basic information about the object of interest, you can use the Schrödinger equation to calculate the wavefunction for that object and determine how that wavefunction will change over time, similar to the way you can use Newton’s laws to predict the future position of a dog given her current position and velocity. The wavefunction, in turn, determines all the observable properties of the object.
The second principle is the idea of allowed states. In quantum theory, an object will only ever be observed in certain states. This principle puts the “quantum” in “quantum mechanics”—the energy in a beam of light comes as a stream of photons, and each photon is one quantum of light that can’t be split. You can have one photon, or two, or three, but never one and a half or pi.
Similarly, an electron orbiting the nucleus of an atom can only be found in certain very specific states.* Each of these states has a particular energy, and the electron will always be found with one of those energies, never in an in-between state. Electrons can move between those states by absorbing or emitting light of a particular frequency—the red light of a neon lamp, for example, is due to a transition between two states in neon atoms—but they make those jumps instantaneously, without passing through the intermediate energies. This is the origin of the term “quantum leap” for a dramatic change between two conditions—the actual energy jump is very small, but the change in the state happens in no time at all.
The third principle is the idea of probability. The wavefunction of an object determines the probabilities of the different allowed states. If you’re interested in the position of a dog, say, the wavefunction will tell you that there’s a very good probability of finding the dog in the living room, a lower probability of finding her in the closed bedroom, and an extremely low probability of finding her on one of the moons of Jupiter. If you’re interested in the energy of that same dog, the wavefunction will tell you that there’s a very good probability of finding her sleeping, a good probability of finding her leaping about and barking, and almost no chance of finding her calmly doing calculus problems.
Philosophical problems start to creep in at this point, because the one thing the wavefunction won’t give you is certainty. Quantum theory allows you to calculate only probabilities, not absolute outcomes. You can say that there’s some probability of finding the dog in the living room and some probability of finding the dog in the kitchen, but you can’t say for sure where she will be until you look. If you repeat the same measurement under the same conditions—asking “Where is the dog?” at four o’clock in the afternoon—you’ll get different results on different days, but when you put all the results together, you’ll see that they match the probability predicted from the wavefunction. You can’t say in advance what will happen for any individual measurement, only what will happen over many repeated experiments.
Quantum randomness is a tremendously disturbing idea for people raised on classical physics, where if you know the starting conditions of your experiment well enough, you can predict the outcome with absolute certainty: you know that the dog will be in the kitchen, and looking just confirms what you already knew. Quantum mechanics doesn’t work that way, though: identically prepared experiments can give completely different results, and all you can predict are probabilities. This randomness is the philosophical issue that led Einstein to make a variety of comments that have been rendered as “God does not play dice with the universe.”*
• • •
“Physicists are silly.”
“Why do you say that?”
“Well, what’s disturbing about randomness? I never know the outcome of anything for certain before it happens, and I’m fine.”
“Well, you’re a dog, not a physicist. But you do make a good point—any responsible practical treatment of classical physics has to include some element of probability in its predictions, just because you can never account for all the little perturbations that might affect the outcome of an experiment.”
“Like that butterfly in Brazil, causing all this weather.”
“Exactly. That’s the usual metaphor: a butterfly flaps its wings in the Amazon, and a week later, there’s a storm in Schenectady. It’s the classic example of chaos theory, which shows that probability is unavoidable even in classical physics, because you can never account for every single butterfly that might affect the weather.”
“Stupid chaos butterflies.”
“The thing is, quantum probability is a different game altogether. The probabilities we end up with in classical physics are a practical limitation. If, by some miracle, you really could keep track of every butterfly in the world, then you would be able to predict the weather with certainty, at least for a while. Quantum physics doesn’t allow that.”
“You mean the butterflies are covered by the uncertainty principle, so you don’t know where they are?”
“Only partly—it’s deeper than that. In quantum physics, even if you perform the same experiment twice under identical conditions—down to the very last butterfly wing-flap—you still won’t be able to predict the exact outcome of the second experiment, only the probability of getting various outcomes. Two identical experiments can and will give you different results.”
“Oh. You know what? That is pretty disturbing. Maybe you’re not so silly, after all.”
“Thanks for the vote of confidence.”
The final principle of quantum theory is the idea of measurement. In quantum mechanics, measurement is an active process. The act of measuring something creates the reality that we observe.*
To give a concrete example, let’s imagine that you have a dog treat in one of two boxes. The boxes are sealed, soundproof (so you can’t hear the treat rattling), and airtight (so you can’t sniff it out): you can’t tell which box the treat is in without opening one of the boxes.
If we want to describe this as a quantum mechanical object, we need to write down a wavefunction with two parts, one part describing the probability of finding the treat in the box on the left, and the other describing the probability of finding the treat in the box on the right. We can do this by adding together the wavefunctions for the treat being in the left-hand box only and the right-hand box only, just as we did in the preceding chapter (page 42) when we made a wave packet by adding together rabbit states.
Now, imagine that you open one of the boxes, and find the treat, then close the box back up. You still have one treat and two boxes, but you’ve measured the position of the treat. What does the wavefunction look like?
The wavefunction now has only one part—the piece describing a treat in the left-hand box—because we know exactly where the treat is. If you found it in the left-hand box, the next time you open that box, there’s a 100% chance that it will be there, and there’s a 0% chance of finding the treat in the right-hand box. The other part that was there before you opened the box, giving the probability of being in the right-hand box, is gone, due to the measurement you made.
Now throw away those boxes, take two new boxes prepared in the same manner as the first pair, and you’ll have a two-part wavefunction again. The result of opening the first box won’t necessarily be the same, though. You might very well find the treat in the right-hand box this time. If you do, and keep closing and reopening that set of boxes, you’ll always find the treat in the right-hand box. Again, you go from a two-part wavefunction to a one-part wavefunction.
So, what’s the big deal? After all, that’s just how probabilities work, right? In the first experiment, the treat was in the left-hand box all along, but you just didn’t know it, and in the second experiment, the treat was in the right-hand box. The state of the treat didn’t change, but your knowledge about the state of the treat did.
Quantum probabilities don’t work that way. When we have a two-part wavefunction (a “superposition state”), it doesn’t mean that the object is in one of the two states, it means that the object is in both states at the same time. The dog treat isn’t in the left-hand box all along, it’s simultaneously in both left and right boxes until after you open the box, and find it in one or the other.
“That’s pretty strange. Why should we believe it?”
“Well, we can demonstrate the weird features of quantum mechanics with an experiment called a quantum eraser.”
“Great! I like that! Let’s erase some cats!”
“It doesn’t work on macroscopic objects. It uses polarized light, which I have to explain first.”
“Oh dear … Why can’t we just erase stuff?”
“I’ll keep it as short as I can, but this is important stuff. Polarized light is the best system around for giving concrete examples of quantum effects. We’ll need it for this chapter, and also chapters 7 and 8.”
“Oh, all right. As long as I can erase stuff later.” “We’ll see what we can do.”
SUPERPOSITION AND POLARIZATION: AN EXAMPLE SYSTEM
We can show both the existence of superposition states and the effects of measurement using the polarization of light. Polarized photons are extremely useful for testing the predictions of quantum mechanics, and will show up again and again in coming chapters, so we need to take a little time to discuss polarization of light, which comes from the idea of light as a wave.
A wave, such as a beam of light, is defined by five properties. We have already talked about four of these: the wavelength (distance between crests in the wave pattern), frequency (how many times the wave oscillates per second at a given point), amplitude (the distance between the top of a crest and the bottom of a trough), and the direction in which the wave moves. The fifth is the polarization, which is basically the direction along which the wave oscillates. An impatient dog owner out for a walk can attempt to get his dog’s attention by shaking the lead up and down, which makes a vertically polarized wave in the lead, or by shaking the lead from side to side, which makes a horizontally polarized wave.
Like a shaken lead, a classical light wave has a direction of oscillation associated with it. The oscillation is always at right angles to the direction of motion, but can point in any direction around that (that is, left, right, up, or down, relative to the direction the light is moving). Physicists typically represent the polarization state of a beam of light by an arrow pointing along the direction of oscillation—a vertically polarized beam of light is represented by an arrow pointing up, and a horizontally polarized beam of light is represented by an arrow pointing to the right, as seen in the figure below.
Left: vertical polarization, represented as an up arrow. Middle: horizontal polarization, represented as a right arrow. Right: polarization between vertical and horizontal, which can be thought of as a sum of horizontal and vertical components.
• • •
“Wait, what are these pictures, again?”
“Imagine that you’re right behind the beam of light, and looking down the direction of motion. The arrow indicates the direction of the oscillation of the wave. An up arrow means that you’ll see the wave moving up and down; a right arrow means that you’ll see it moving side to side.”
“So … an up arrow is like chasing a bunny that bounds up and down, while a right arrow is like chasing a squirrel that zigzags back and forth?”
“Okay, that works.”
“Are up and to the right the only options?”
“You can have arrows in other directions, too. An arrow to the left also indicates a side-to-side oscillation, but it’s out of phase with the arrow to the right.”
“So, a right arrow is a squirrel that zigs to the right first, and a left arrow is a squirrel that zags to the left first?”
“Yes. If you insist on examples involving prey animals.” “I like prey animals!”
• • •
The polarization of a wave can be horizontal or vertical, but also any angle in between. We can think of the in-between angles as being made up of a horizontal part and a vertical part, as shown in the figure above. Each of these components is less intense (that is, it has a smaller amplitude, indicated in the figure by the length of the arrow) than the total wave, but they add together to give the same final intensity at some angle. You can think of this addition as a combination of steps, just like the way that we can get from one point to another by either taking three steps east followed by four steps north, or by taking five steps in a direction about 37° east of due north.
“So, an in-between angle is like a bunny that’s zigzagging left and right, while also hopping up and down?”
“Yes, that’s right.”
“Or a squirrel that’s jumping up and down while it zigzags left and right?”
“I think that’s about enough prey examples for now.”
“You’re no fun.”
Thinking of in-between polarizations as a sum of horizontal and vertical components is a useful trick because it makes it easy to see what happens when light encounters a polarizing filter. Polarizing filters are devices that will allow light polarized at a particular angle—vertical, say—to pass through unimpeded, while light polarized at an angle 90° away—horizontal—will be completely absorbed. You can understand the effect by imagining a dog on a lead that passes through a picket fence. If you shake the lead up and down, the wave will pass right through, but side-to-side shaking will be blocked by the boards of the fence.
When light at an angle between vertical and horizontal strikes a vertically oriented polarizing filter, only the vertical component of the light will pass through. This lowers the intensity of the light on the other side, by an amount that depends on the angle. For small angles, most of the light makes it through—at an angle of 30°, the beam on the far side is three-fourths as bright as the initial beam—while for larger angles, most of the beam is blocked—at 60° from vertical, the beam on the far side is only one-fourth as bright as the initial beam. At an angle of 45° midway between horizontal and vertical, exactly half of the light will pass through the filter.
The light on the far side of the filter is polarized at the angle of the filter, no matter what angle it started at. For this reason, polarizing filters are commonly called polarizers: light passing through a vertically oriented polarizing filter will emerge as vertically polarized light, whether it started with vertical polarization or at some other angle. The overall amount of light will be different, but the polarization will be the same. All of the light passing through a vertically oriented filter will pass through a second vertical filter, and all of it will be blocked by a horizontally oriented filter.
“What is all this good for, anyway?”
“Other than helping demonstrate quantum physics? Plenty. Light polarization is an extremely useful thing. Digital displays on watches, mobile phones, and televisions use a polarizer in front of a light source to vary the amount of light that gets through. And polarizing filters are also used to make sunglasses.”
“Sunglasses?”
“Yes, those sunglasses that I wear when I take you for walks are actually polarizing filters. The light from the sun is unpolarized—it’s as likely to be horizontal as vertical—but when light reflects off a surface, it tends to become slightly polarized. Light reflecting off the road out in front of us when we’re walking has more horizontal polarization than vertical, so by wearing vertical polarizers as sunglasses, I can block most of that light.”
“What’s the point of that? Doesn’t it make it harder to see?”
“Actually, it reduces the glare off the road, and makes it easier to see things up ahead.”
“Things … Like bunnies in the road?”
“For example, yes.”
“Can I have some polarized sunglasses so I can see bunnies?”
“The ones I have won’t fit on your ears, but we’ll look into it. Later. First, I have to talk about quantum measurement with polarized light.”
“Oh, okay. Quantum physics. Right.”
How does all this apply to light as a particle, though? We spent a good chunk of chapter 1 describing how a beam of light is both a stream of photons and a smooth wave. The last few pages have been discussing polarization in classical terms. How do we handle light polarization in quantum physics?
When we’re dealing with classical light waves, it’s easy to understand how part of a wave can pass through the filter. When we talk about light in terms of photons, though, the filter is an all-or-nothing proposition. Any given photon either makes it through, or gets absorbed by the filter. There are no “parts” of photons.
We handle the interaction between photons and polarizing filters by saying that each photon has a probability of passing through the filter that is equal to the fraction of the total wave that makes it through in the classical model. If a beam of light with a polarization at 60° from vertical encounters a vertical polarizing filter, the beam on the far side will be one-fourth as bright, meaning that it has one-fourth as many photons. That means that each individual photon has only one chance in four of making it through the polarizing filter.
Each photon making it through the filter will also have its polarization determined by the filter. Only one photon in four may make it through a vertically oriented polarizing filter, but every one of those photons will pass through a second vertical filter, and none of them will pass through a horizontal filter. “Vertical” and “horizontal” are then the allowed states of the single photon’s polarization—when we measure the polarization using a filter, we will find the photon in one of those two states (either passing through the vertical filter, or being absorbed by it), and not anywhere in between.
Polarized photons thus provide an excellent system for looking at the core principles of quantum mechanics. Each individual photon can be described in terms of a wavefunction, with two parts corresponding to the two allowed states, horizontal and vertical polarization. That wavefunction gives you the probability of the photon passing through a polarizing filter, and after you make a measurement of the polarization with a filter, the photons are in only one of the allowed states. A single photon passing through a polarizing filter demonstrates all the essential features of quantum physics. As a result, polarized photons have been used in many experiments demonstrating quantum phenomena.
“So, let me get this straight. A photon at an angle between horizontal and vertical is in a superposition state? And sending it through a polarizing filter is the same as measuring it?”
“Yes. You get all the features of quantum superposition and measurement—wavefunctions, allowed states, probability, and measurement—using single polarized photons.”
“But I thought you said all this stuff worked the same way when you talked about light as a classical wave?”
“Well, yes. The end result is the same as the classical polarized wave description.”
“What’s the big deal, then? I mean, your big example of quantum weirdness is something that just reproduces classical results?”
“Well, no. I mean, that’s not my big example. The big example of quantum weirdness is in the next section.”
“Oh. Well, carry on, then.”
(UN)MEASURING A PHOTON: THE QUANTUM ERASER
One of the best demonstrations of the weirdness of quantum superpositions is an experiment called a quantum eraser. The quantum eraser encapsulates everything that’s strange about single-particle quantum physics in a single experiment: particle-wave duality, superposition states, and the active nature of measurement. If you can understand the quantum eraser, you’ve understood the essential elements of quantum physics.
Many different variants of quantum-eraser experiments have been done over the years,* but the simplest starts with a variant of Young’s double-slit experiment (page 18). If we send a beam of photons at a pair of narrow slits, we will see an interference pattern on the far side of the slits, built up out of single photons detected at particular points (as shown in the figure on the next page). We can see the pattern only because light passes through both slits at the same time. If we block one slit, the interference pattern will disappear, and we’ll see only a broad scattering of photons due to the light passing through the unblocked slit.
The interference pattern that we see indicates that the photons are in a superposition state: the wavefunction describing each photon has two parts, one for the photon passing through the left slit, and the other for the photon passing through the right slit. Each photon has passed through both slits, at the same time, and the interference between those two components is what produces the pattern we see. Interference patterns always turn up when you have a two-part wavefunction. When we block one slit, we only have a one-part wavefunction, destroying the superposition, and there is no interference pattern.
• • •
“Wait, I thought the interference was between two different photons—one that went through the left slit, and one that went through the right slit?”
Interference pattern built up from single photons. Left to right, 1/30 second, 1 second, 100 seconds. Images by Lyman Page at Princeton University, reprinted with permission.
“That’s an easy thing to think, since we usually send in lots of photons at the same time. We can show that that’s not the case, though, by sending light at the slits one photon at a time.”
“How does a single photon give you an interference pattern?”
“It doesn’t. Each individual photon shows up as a single spot, at a particular position on the screen, and where any individual photon turns up is random.”
“There’s the probability thing again.”
“Exactly. The individual photons are random, but if you repeat the experiment over and over again, and keep track of all the photons, you’ll see them add up to form an interference pattern. There are some places where you’re very likely to find a photon, and other places where there’s absolutely no chance of finding one. The overall pattern is determined by the probability distribution you get from the wavefunction for each individual photon interfering with itself.”
“So it’s one particle, but it goes through both slits, and then ends up at one place on the other side?”
“Exactly.”
“That’s just weird.”
“That’s quantum physics.”
Instead of blocking one slit, though, let’s imagine covering the two slits with two different polarizing filters, one vertical and one horizontal. We put a filter on the left slit that will pass only horizontally polarized light, and we put a filter on the right slit that will pass only vertically polarized light. If we send in light polarized at an angle of 45º to the vertical, it has a 50% chance of going through a horizontal polarizing filter, and a 50% chance of going through a vertical polarizing filter, so we get some light through each slit.
This arrangement of filters gives us a way of measuring which slit the light went through. If we put a vertical polarizer in front of our detector, we will only see light that went through the right-hand slit, and if we put a horizontal polarizer in front of our detector, we will only see light that went through the left-hand slit. The polarizer in front of the detector lets us tell which slit the photon went through, just as if we had put a detector right next to the slit and measured the position directly.
What happens when we do this? When we look at the light with the filters over the two slits, we don’t see any sign of an interference pattern. When we measure the polarization of the light, we measure which slit the light went through, and that takes us from a two-part wavefunction, which produces an interference pattern, to a one-part wavefunction, which does not. The act of measuring which slit the photon went through destroys the component of the wavefunction describing the photon going through the other slit, just as the act of opening one of the boxes destroyed the component for the treat being in the other box.
We don’t even need to put a polarizing filter on the detector—by putting the polarizers over the slits, we have “tagged” each photon, and the mere fact that we can measure which slit it went through is enough to destroy the pattern. In the treats-in-boxes example, this is like someone writing “Treat” on the outside of the box containing the treat—we no longer need to open the box to destroy the superposition.
The disappearing pattern is pretty weird in its own right, but things get weirder: we can undo the measurement after the fact by using a 45º polarizer instead of a horizontal or vertical polarizer to look at the light after the slits. If we do this, we see an interference pattern again! A 45º polarizer will pass either horizontal or vertical polarization, each with a 50% probability, which means any light we detect after the polarizer could have gone through either of the two slits, or even both at once. The third filter “erases” the information we had gained by tagging the photon, like somebody removing the label from the box containing the treat. Inserting the extra polarizer makes it as if we had never made the measurement at all. The second part of the wavefunction isn’t destroyed after all, and we can see interference.
The quantum-eraser experiment encapsulates everything that is strange about the core principles of quantum mechanics. The appearance of the interference pattern shows the superposition of quantum states, as each photon goes through both slits at the same time, and the disappearance and reappearance of the pattern when we add polarizing filters shows the active nature of quantum measurement. Just the fact that it is possible to measure which slit the particle went through is enough to completely change the results of the experiment.
WHAT YOU SEE IS ALL THERE IS:
THE COPENHAGEN INTERPRETATION
These four ideas—wavefunctions, allowed states, probability, and measurement—are the central elements of quantum theory. The interference patterns we see in experiments with photons* confirm that quantum particles really do occupy multiple states at the same time. The disappearance and reappearance of the pattern in the quantum-eraser experiment confirms that measurement is an active process and determines what happens in subsequent experiments.
We still have a problem, though, because there is no mathematical process for describing how to get from a probability to the result of a measurement. We use the Schrödinger equation to calculate the wavefunctions for the allowed states of a physical object, and we use the wavefunction to calculate the probability distribution, but we cannot use the probability distribution to predict the exact result of an individual measurement. Something mysterious happens in the process of making a measurement.
This “measurement problem” is the origin of the competing interpretations of quantum mechanics, and the point where physics is forced to become philosophy. All interpretations use the same methods to calculate probabilities for the outcomes of repeated measurements. They differ only in how they explain the step from a quantum superposition state, where the wavefunction consists of two (or more) states at the same time, to the classical result of a single measurement, where the object is found in one and only one state.
The first interpretation put forward for quantum theory was developed by Niels Bohr and coworkers at his institute in Denmark, and is thus known as the Copenhagen interpretation. The Copenhagen interpretation is a very ad hoc approach to the problem of measurement in quantum mechanics (which is in some ways typical of Bohr’s approach*).
The Copenhagen interpretation tries to avoid the problems of superposition and measurement by drawing a strict line between microscopic and macroscopic physics. Microscopic objects—photons, electrons, atoms, and molecules—are governed by the rules of quantum mechanics, but macroscopic objects—dogs, physicists, and measurement apparatus—are governed by classical physics. There’s an absolute separation between the two, and you will never see a macroscopic object behaving in a quantum manner.
Quantum measurement involves the interaction of a macroscopic measurement apparatus with a microscopic object, and that interaction changes the state of the microscopic object. The usual description is that the wavefunction “collapses” into a single state. This “collapse,” in the Copenhagen interpretation, is an actual change of the wavefunction from a spread-out quantum state with multiple possible measurement outcomes to a state with a single measured value.*
In the most extreme variants of the Copenhagen interpretation, the collapse requires not only a macroscopic measurement apparatus, but also a conscious observer to note the measurement. In this view, a tree that falls in the forest hasn’t really fallen until some person (or dog) comes along and observes it.
“So, wait, doesn’t that mean rejecting the entire idea of an objective physical reality?”
“In its most extreme forms, yes. Werner Heisenberg was probably the most radical of the Copenhagen crowd, and he insisted quite strongly that it was a mistake to talk about electrons having an independent reality. In Heisenberg’s view, the only things we can really talk about are the outcomes of specific measurements. He rejected the whole notion of talking about what the electrons were doing between measurements.”
“That’s … pretty radical. I don’t think I like that.”
“You’re not alone, believe me.”
The Copenhagen interpretation raises a great many problems, among them that there’s no obvious reason for the absolute distinction between microscopic and macroscopic physics. As we’ve already seen, while it gets more difficult to detect quantum behavior as objects get larger and more complicated, it is still possible to see wave behavior in rather large molecules. Macroscopic objects ought to be described by quantum wavefunctions and quantum rules.
Another problem is who or what counts as an “observer” for the purposes of collapsing the wavefunction. The requirement that somebody observe the outcome of a measurement before the measurement really “counts” seems to assign some sort of mystical quality to “consciousness,” and that idea makes many physicists uncomfortable.
Even the idea of the “collapse” itself is problematic. No mathematical formula exists to describe the collapse—you can use the Schrödinger equation to describe how a wavefunction changes between measurements, but there is no way to describe the “collapse” process. All you can do is choose a result, and start over with a new wavefunction after the measurement. Many physicists find this a little too magical for comfort.
The most famous illustrations of the problems with the Copenhagen interpretation are the infamous “Schrödinger’s cat” thought experiment, and the follow-up thought experiment of “Wigner’s friend.” Despite his role in creating quantum theory, Erwin Schrödinger, like Einstein, had deep philosophical problems with its interpretation, and became disillusioned with the entire field. Schrödinger’s cat, which is arguably more famous than his equation, is a diabolical thought experiment through which Schrödinger attempted to illustrate the absurdity of the Copenhagen interpretation. He imagined placing a cat in a sealed box with a radioactive atom that has a 50% chance of decaying within one hour, and a device that will release poison gas if the atom decays, killing the cat. What, he asked, is the state of the cat at the end of the hour?
As Schrödinger noted, according to the Copenhagen interpretation the wavefunction describing the cat would be equal parts “alive” and “dead.” This would last until the experimenter opens the box, at which point it would collapse into one of the two states.* This seems completely absurd, though—the idea of a cat that is both dead and alive at the same time is outlandish. And yet this is exactly what seems to happen with photons.
The Copenhagen interpretation also seems to be saying that physical reality does not exist until a measurement is made, which poses its own philosophical problems. Eugene Wigner brought this out by adding another layer to the cat experiment, imagining that the entire thing was conducted by a friend, and only reported to him later. Wigner asked when the wavefunction collapsed: When the friend opened the box, or later, when Wigner heard the result? Has a tree in a forest really fallen before your dog tells you that it’s on the ground?
None of the Copenhagen interpretation’s answers to these questions are very satisfying, philosophically. While quantum mechanics does an outstanding job of describing the behavior of microscopic objects and collections of objects, the world we see remains stubbornly, infuriatingly classical. Something mysterious happens in the transition from the weird world of simple quantum objects to the much larger world of everyday objects. The Copenhagen approach of insisting on an absolute division between microscopic and macroscopic strikes many physicists as simply dodging the question: it says what happens, but not why.
How best to handle the transition between quantum and classical remains a subject of active debate. Some future theory may lead to a detailed understanding of what, exactly, happens when we make a measurement of a quantum object. Until then, we’re stuck with one of the various interpretations of quantum mechanics.
“I don’t think I like this interpretation. It’s awfully solipsistic, isn’t it?”
“You’re not alone. There aren’t very many physicists these days who are really happy with the Copenhagen interpretation.”
“So, what interpretation do you like?”
“Me? I tend to go with the ‘shut up and calculate’ interpretation. The name is sometimes attributed to Richard Feynman,* but the idea is just to avoid thinking about it. Quantum mechanics gives us very good tools for calculating the results of experiments, and the question of what goes on during measurement is probably better left to philosophy.”
“I don’t think I like that one, either. It’s hard to work a calculator without opposable thumbs.”
“Well, there are all sorts of different interpretations—there’s the many-worlds interpretation, David Bohm’s non-local mechanics, and something called the ‘transactional interpretation.’ There are almost as many interpretations of quantum mechanics as there are people who have thought deeply about quantum mechanics.”
“I like the many-worlds interpretation. You should talk about that.”
“Good idea. That’s the next chapter.”
* Sort of like “Thou shalt not climb on the furniture” for dogs living with humans.
* Schrödinger was almost as notorious for his womanizing as for his contributions to physics. He came up with the equation that bears his name while on a ski holiday with one of his many girlfriends, and fathered daughters with three different women, none of them his wife (who, incidentally, knew about his affairs). His unconventional personal life cost him a position at Oxford after he left Germany in 1933, but he carried on living more or less openly with two women (one the wife of a colleague) for many years.
* As we discussed at the end of chapter 2.
* Einstein had many negative things to say about the probabilistic nature of quantum mechanics, but the origin of the usual formulation is a letter to Max Born in 1926, in which he wrote, “The theory delivers a lot, but hardly brings us closer to the secret of the Old One. I for one am convinced that He does not throw dice” (quoted in David Lindley’s Uncertainty, p. 137).
* Werner Heisenberg went so far as to say that the results of measurements were the only reality—that it made no sense to talk about where an electron was or what it was doing between measurements.
* The May 2007 issue of Scientific American even describes a quantum-eraser experiment that you can do at home, using a laser pointer, tinfoil, wire, and a few pieces of cheap polarizing film.
* And electrons, and atoms, and molecules …
* As we said in the last chapter (page 49), Bohr’s first great contribution to physics was a simple quantum model of hydrogen. It was a cobbled-together mix of quantum and classical ideas with no clear justification that happened to give the right result, and it’s unclear what led Bohr to put it forth. It did, however, point the way toward the modern quantum theory that we’re discussing in this book.
* The word “collapse” has come to be strongly associated with Copenhagen-type interpretations. There are other approaches to the problem of the projection of a multicomponent wavefunction onto a single measurement result that don’t involve a physical change in the wavefunction. We’ll look at the best-known example of these “no-collapse” interpretations in chapter 4.
* Or, as Terry Pratchett described it in his novel Lords and Ladies, applied to a particularly nasty cat: “Technically, a cat locked in a box may be alive or it may be dead. You never know until you look. In fact, the mere act of opening the box will determine the state of the cat, although in this case there were three determinate states the cat could be in: these being Alive, Dead, and Bloody Furious” (p. 226, Harper paperback).
* Feynman tends to get credit for anything clever said by a physicist in the latter half of the twentieth century. “Shut up and calculate” probably isn’t Feynman, though—its first appearance in print seems to be a David Mermin column in Physics Today (April 1989, p. 4), as he explains in the May 2004 issue (p. 10).
