How to Teach Physics to Your Dog (8 page)

Zero-point energy is one of the most counterintuitive ideas in quantum physics, telling us that nothing can ever be perfectly at rest. It means that there is always some energy present in any system, no matter how hard you try to extract all the energy. Even empty space has zero-point energy, which leads to some surprising consequences, including the spontaneous emission of photons from atoms and tiny forces (called “Casimir forces”) between metal plates in a vacuum. The zero-point energy of empty space can even produce short-lived pairs of “virtual” particles, as we’ll see in
chapter 9
.

Zero-point energy is probably the most dramatic manifestation of the uncertainty principle. Its existence is a direct consequence of the quantum nature of all the particles making up our universe.

“So, the point of all this is that position and momentum do not have definite values?”

“Yes, that’s it exactly.”

“Are those the only things this happens with?”

“No, there are lots of different uncertainty relationships between pairs of physical quantities. There’s uncertainty in angular momentum, for example—you can’t know both the direction of a rotating object and how fast it’s spinning. There’s also an uncertainty relationship between the number of photons in a light beam and the phase of the wave associated with the beam. Uncertainty relationships are all over the place in quantum physics.”

“So, basically, nothing is defined in an absolute sense? Isn’t that kind of . . . postmodern?”

“It’s not all that bad—it’s not like different experimenters get to make up their own results. The uncertainty due to quantum effects is generally very small, so for all practical purposes, we can treat macroscopic objects as if they have definite properties. But at the microscopic level, there really isn’t any single defined value for any of these quantities.”

“But you talked earlier about measuring bunnies in definite positions. How does that work, if they don’t have definite positions?”

“That’s a very good question, and takes us into a whole new area of weird stuff, in the next chapter.”

*
To give you an idea of the breadth of subjects in which this shows up, in June 2008, Google turned up citations of the Heisenberg uncertainty principle in (among others) an article from the
Vermont Free Press
about traffic cameras, a
Toronto Star
article citing the influence of YouTube on underground artists, and a blog article about the Phoenix Suns of the NBA. Incidentally, all of these articles also use the uncertainty principle incorrectly—by the end of this chapter, you should hopefully understand it better than any of them.

*
This is why scientists use electron microscopes to look at very small things: electron microscopes use electrons instead of visible light, and electrons have much shorter wavelengths than visible light.

*
“State” in physics refers to a particular collection of properties—position, momentum, energy, etc. A bunny in the yard with a given momentum is said to be in a momentum state. A second bunny in the same yard with the same momentum is in the same state; a second bunny in the same yard with a different momentum is in a different state. A third bunny with the same momentum as the first, but in a different yard, is in a third state, and so on.

*
The symbol “Δ” is a capital Greek letter delta, as any fraternity dog can tell you. In science, it’s used to indicate a change in something or a difference between two things. Δx is the uncertainty in the position, or the amount of difference you can expect between the position you eventually measure and the most likely position from the wavefunction. When you say “I buried a nice bone sixteen steps from the big oak tree, give or take a step,” the “sixteen steps” is the most likely position of the bone, and the “give or take a step” is Δx.

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?”

“Yeah! 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, yeah, 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.”

“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 10
²²
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.

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.