How to Teach Physics to Your Dog

by CHAD ORZEL

  “People have seen them, though.”

  “Sure, 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 sure. Cats are dumb.”

  “—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, yeah. Also, I’m a sap.” I give her both of the treats.

  “Ooooo! 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 wave-function 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 sure 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 bunny 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.”

  “Oooh! 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.”

  “Awww . . . 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 leash up and down, which makes a vertically polarized wave in the leash, or by shaking the leash from side to side, which makes a horizontally polarized wave.

  Like a shaken leash, 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 vertica
l 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?”

  “Sure, 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?”

  “Yeah. 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.”