Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)

Chapter 2. Size Is Absolute

THE FUNDAMENTAL NATURE OF SIZE is central to understanding the differences between the aspects of the everyday world that fit into our intuitive view of nature and the world of quantum phenomena, which is also manifested all around us. We have a good feel for the motion of baseballs, but we mainly gloss over our lack of knowledge of what gives things different colors or why the heating element in an electric stove gets hot and glows red. The motion of baseballs can be described with classical mechanics, but color and electrical heating are quantum phenomena. The differences between classical and quantum phenomena depend on the definition of size.

The quantum mechanical concept of size is the correct view, and it is different from our familiar notion of size. Our common concept of size is central to classical mechanics. The failure to treat size properly, and all of the associated consequences of that failure, is ultimately responsible for the inability of classical mechanics to properly describe and explain the behavior of the basic constituents of matter. A quantum mechanical description of matter is at the heart of technological fields as diverse as microelectronics and the computer design of pharmaceuticals.

SIZE IS RELATIVE IN EVERYDAY LIFE

In classical mechanics, size is relative. In quantum mechanics, size is absolute. What does relative versus absolute size mean, and why does it matter?

In classical mechanics and in everyday life, we determine whether something is big or small by comparing it to something else. Figure 2.1 shows two rocks. Looking at them, we would say that the rock on the left is bigger than the rock on the right. However, because there is nothing else to compare them to, we can’t tell if they are what we might commonly call a big rock and a small rock. Figure 2.2 shows the rock on the left again, but this time there is something to compare it to. The size of the rock is clear because we have the size of a human hand as a reference. Because we know how big a typical hand is, we get a good feel for how big the rock is relative to the hand. Once we have the something against which to make a size comparison, we can say that the rock is relatively small, but not tiny. If I were to describe the rock over the phone, I could say it is somewhat bigger than the palm of your hand, and the person I am talking to would have a good idea of how big the rock is. In the absence of something of known size for comparison, there is no way to make a size determination.

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FIGURE 2.1. Two rocks.

Figure 2.1 demonstrates how much we rely on comparing one thing to another to determine size. In Figure 2.1, the two rocks are on a white background, with no other features for reference. Their proximity immediately leads us to compare them and to decide that the rock on the left is larger than the rock on the right. Figure 2.3 shows the rock on the right in its natural setting. Now we can see that it is actually a very large rock. The hand on the rock gives a very good reference from which to judge its size. Like the rock in the hand, the rock with the hand on top provides us with a scale that permits a relative determination of size. It is clear from these simple illustrations that under normal circumstances, we take size to be relative. We know how big something is by comparing it to something else.

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FIGURE 2.2. The rock from Figure 2.1 in a hand.

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FIGURE 2.3. The other rock from Figure 2.1, but now in a context from which its size can be judged.

OBSERVATION METHOD CAN MATTER

Why does the definition of size, relative versus absolute, matter? To observe something, we must interact with it. This is true in both classical and quantum mechanics.

Figure 2.4 illustrates the observation of a rose. In a totally dark room, we cannot see the rose. In Figure 2.4, however, light emanating from the light bulb falls on the rose. Some of the light is absorbed, and some of it bounces off. (Which colors are absorbed, and therefore, which colors bounce off to make the leaves look green and the petals look red, is a strictly quantum mechanical phenomenon that will be discussed in Chapter 8.) A portion of the light that bounces off is detected by the eye and processed by the brain to observe the rose. The observer is interacting with the rose through the light that bounces off of it.

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FIGURE 2.4. The light bulb illuminates the rose. The light that bounces off the rose enters the eye, enabling us to see the rose.

Once we recognize that we must interact with an object to observe it, we are in a position to define big and small. The definition of what is big and what is small is identical in classical mechanics and quantum mechanics. If the disturbance to an object caused by an observation, which is another way of saying a measurement, is negligible, then the object is big. If the disturbance is nonnegligible, the object is small. In classical mechanics, we make the following assumption.

Assume: When making an observation, it is always possible to find a way to make a negligible disturbance.

If you perform the correct experiment, then the disturbance that accompanies the measurement is negligible. Therefore, you can observe a system without changing it. However, if you do the wrong experiment in trying to study a system, you make a nonnegligible disturbance, and the object is small. A nonnegligible disturbance changes the system in some way and, it is desirable, if possible, to make a measurement that doesn’t change the thing you are trying to measure. Classical theory assumes that you can reduce the size of the disturbance to be as small as desired. No matter what is under observation, it is possible to find an experimental method that will cause a negligible disturbance. This assumed ability to find an experiment that produces a negligible disturbance implies that size is only relative. The size of an object depends on the object and on your experimental technique. There is nothing inherent. Any object can be considered to be big by observing it with the correct method, a method that causes a negligible disturbance.

Suppose you decide to examine the wall of a room in which you are sitting by throwing many billiard balls at it. In your experiment, you will observe where the balls land after they bounce off of the wall. You start throwing balls and, pretty soon, plaster is flying all over the place. Holes appear in the wall, and the balls you throw later on don’t seem to bounce off the same way the earlier balls did. This may not be surprising because of the gaping holes that your measurement method is making in the wall. You decide that this is not a very good experiment for observing the wall. You start over again after having a good painter restore the wall to its original state. This time you decide to shine light on the wall and observe the light that bounces off of the wall. You find that this method works very well. You can get a very detailed look at the wall. You observe the wall with light for an extended period of time, and the properties you observe do not change.

BIG OR SMALL—IT’S THE SIZE OF THE DISTURBANCE

When the wall was observed with billiard balls, it was small because the observation made a nonnegligible disturbance. When the wall was observed with light, it was big. The observation made a negligible disturbance. In these experiments, which can be well described with classical mechanics, the wall’s size was relative. Do the poor experiment (observation with billiard balls), and the wall is small. Do a good experiment (observation with light), and the wall is big.

In classical mechanics, there is nothing intrinsic about size. Find the right experiment, and any object is big. It is up to the experimental scientist to design or develop the right experiment. Nothing intrinsic in classical mechanics theory prevents a good experiment from being performed. A good experiment is one that produces a negligible disturbance during the measurement. In other words, a good experiment does not change the object that is being observed, and, therefore, the observation is made on a big object.

CAUSALITY FOR BIG OBJECTS

The importance of being able to make any object big is that it can be observed without changing it. Observing an object without changing it is intimately related to the concept of causality in classical mechanics. Causality can be defined and applied in many ways. One statement of causality is that equal causes have equal effects. This implies that the characteristics of a system are caused by earlier events according to the laws of physics. In other words, if you know in complete detail the previous history of a system, you will know its current state and how it will progress. This idea of causality led Pierre-Simon, Marquis de Laplace (1749-1827), one of the most renowned physicists and mathematicians, to declare that if the current state of the world were known with complete precision, the state of the world could be computed for any time in the future. Of course, we cannot know the state of the world with total accuracy, but for many systems, classical mechanics permits a very accurate prediction of future events based on accurate knowledge of the current state of a system. The prediction of the trajectory of a shell in precision artillery and the prediction of solar eclipses are examples of how well causality in classical mechanics works.

As a simple but very important example, consider the trajectory of a free particle, such as a rock flying through space. A free particle is an object that has no forces acting on it, that is, no air resistance, no gravity, etc. Physicists love discussing free particles because they are the simplest of all systems. However, it is necessary to point out that a free particle never really exists in nature. Even a rock in intergalactic space has weak gravity influencing it, weak light shining on it, and occasionally bumps into a hydrogen atom out there among the galaxies. Nonetheless, a free particle is useful to discuss and can almost be realized in a laboratory. So our free particle is a hypothetical true free particle despite its impossibility.

The free particle was set in motion some time ago with a momentum p, and at the time we will call zero, t = 0, it is at location x. x is the particle’s position along the horizontal axis. The trajectory of the rock is shown in Figure 2.5 beginning at t = 0. The momentum is p = mV where m is the mass of the object and V is its velocity. The mass on earth is just its normal weight. If the rock is on the moon, it has the same mass, but it would have one-sixth the weight because of the weaker pull of gravity on the moon.

A very qualitative way to think about momentum is that it is a measure of the force that an object could exert on another object if they collided. Imagine that a small boy weighing 50 pounds runs into you going 20 miles per hour. He will probably knock you down. Now imagine that a 200-pound man runs into you going 5 miles per hour. He will probably also knock you down. The small boy is light and moving fast. The man is heavy and moving slow. Both have the same momentum, 1000 lb-miles/hour. (lb is the unit for pound.) In some sense, both would have the same impact when they collide with you. Of course, this example should not be taken too literally. The boy might hit you in the legs while the man would hit you in the chest. But in a situation where these types of differences did not occur, either would have essentially the same effect when running into you.

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FIGURE 2.5. A free particle in the form of a rock is shown moving along a trajectory.

Momentum is a vector because the velocity is a vector. A vector has a magnitude and a direction. The velocity is the speed and the direction. Driving north at 60 mph is not the same as driving south at 60 mph. The speed is the same, but the direction is different. The momentum has a magnitude mV and a direction because the velocity has a direction. In Figure 2.5, the motion is from left to right across the page.

At t = 0 we observe (make measurements of) the rock’s position and momentum. Once we know x and p at t = 0, we can predict the trajectory of the rock at all later times. For a free particle, predicting the trajectory is very simple. Because there are no forces acting on the particle, no air resistance to slow it down or gravity to pull it down to earth, the particle will continue in a straight line indefinitely. At some later time called tʹ (t prime), t = tʹ, the rock will have moved a distance d = Vtʹ. The distance is the velocity multiplied by how long the particle has been traveling. Since we started at time equal to zero, t = 0, then tʹ is how long the particle has been moving—for example, one second. So at time tʹ we know exactly where to look for the rock. We can make an observation to see if the particle is where we think it should be and, sure enough, there it is, as shown in Figure 2.5. We can predict where it will be at a later time, and observe that it is in fact there. This is shown on the right side of Figure 2.5. We have predicted where the particle will be, and when we make an observation, it is there. The rock is traveling with a well-defined trajectory, and the principle of causality is obeyed.

NONNEGLIGIBLE DISTURBANCES MATTER

Now consider Figure 2.6. The rock is prepared identically to the situation shown in Figure 2.5. At t = 0, it has position x and momentum p. Again it is observed at t = tʹ.

Its position is as predicted from the values of x and p at t = 0. However, some time after t = tʹ, a bird flies into a rock. (You will have to forgive my drawing of the bird. This is the best I can do on a computer with a mouse.) In the jargon of physics, we might refer to this as a bird-rock scattering event. The bird hitting the rock makes a nonnegligible disturbance. Therefore, it is not surprising that a measurement of the position and momentum made some time after the scattering event will not coincide with the predictions made based on the trajectory determined at t = 0. According to the precepts of classical mechanics, if we knew everything about the bird, the rock, and how they interact (collide with each other), we could make a prediction of what would happen after the bird-rock scattering event. We could test our predictions by observation. Observation is possible in classical mechanics because we can find a method for observation that makes a negligible disturbance of the system. That is, we can always find a way to make the system big. But the important point is that following a nonnegligible disturbance, it is not surprising that predictions are not fulfilled, as they were based on the known trajectory that existed prior to the disturbance.

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FIGURE 2.6. A free particle in the form of a rock is moving along a trajectory. At time t = 0, it has position x and momentum p. At a later time, t = tʹ, it has moved to a new position where it is observed, and its future position is predicted. However, some time later, a bird flies into the rock. The prediction made at tʹ is no longer valid.

THERE IS ALWAYS A DISTURBANCE

Quantum theory is fundamentally different from classical mechanics in the way it treats size and experimental observation; the difference makes size absolute. Dirac succinctly put forward the assumption that makes size absolute.

Assume: There is a limit to the fineness of our powers of observation and the smallness of the accompanying disturbance, a limit that is inherent in the nature of things and can never be surpassed by improved technique or increased skill on the part of the observer.

This statement is a wild departure from classical thinking. It says that whenever you observe a system (make a measurement), there is always a disturbance; it may be small, but it is always there. The size of this disturbance is part of nature. No improvements in instrumentation or new methods of observation can make this minimum disturbance vanish or become smaller.

SIZE IS ABSOLUTE

Dirac’s statement has ramifications that are part of all formulations of quantum theory. His assumption immediately makes size absolute. An object is big in the absolute sense if the minimum disturbance that accompanies a measurement is negligible. An object is small in the absolute sense if the inherent minimum disturbance is not negligible. At the most fundamental level, classical mechanics is not set up to describe objects that are small in the absolute sense. In classical mechanics, any object can be made “big” by finding the right experiment to use in making an observation. In the development of classical mechanics, it was never envisioned that because of the inherent properties of nature, it was impossible to improve methodology to the point where an observation did not change a system. Therefore, classical mechanics is not set up to deal with objects that are small in an absolute sense. Its inability to treat objects that are absolutely small, such as electrons or atoms, is the reason that classical mechanics fails when it is applied to the description of such objects.

Figure 2.7 illustrates the nature of the problem. An electron is a particle that is small in the absolute sense. (Later we will discuss in detail the meaning of the word particle, which is not the same as the classical concept of particle.) At t = 0, it is moving along a trajectory. As with the rock, we want to see if it is actually doing what we think it is doing so that we can make subsequent predictions. We use the least invasive method to observe the electron; we let it interact with a single particle of light, a photon. (Below is a detailed discussion of the nature of light and what it means to have a particle of light.) Here is what makes this problem completely different from that illustrated in Figure 2.5. Because an electron is absolutely small, even observing it with a single particle of light causes a nonnegligible disturbance. The electron is changed by the observation. We cannot make subsequent predictions of what it will do once we observe it to see if it is doing what we think it is doing. Causality applies to undisturbed systems. The act of observing the electron disturbs it. You can predict what a system is doing as long as you don’t look to see if it is actually doing what you think it should be doing. Therefore, causality does not apply to systems that are absolutely small. The act of observation destroys causality. Indeterminacy, that is a certain type of indefiniteness, comes into the calculation of observables for absolutely small systems. A system is absolutely small if the minimum disturbance that accompanies a measurement is not negligible. An absolutely small system can’t be observed without changing it.

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FIGURE 2.7. At time, t = 0, an electron is moving along some trajectory. At time, t = tʹ, we observe it in a minimally invasive manner by letting it interact with a single particle of light, a photon. (Photons are discussed in detail later.) The electron-photon interaction causes a nonnegligible disturbance. It is not possible to make a causal prediction of what happens after the observation.

CAN’T CALCULATE THE FUTURE—ONLY PROBABILITIES

Unlike in classical mechanics, once an observation is made for a quantum system, it is not possible to say exactly what another observation will yield. This lack of exactitude is not like the bird hitting the rock in Figure 2.6. In the bird-rock case, it is possible, if difficult, in principle to predict the result of the next observation. We would need to know all of the properties of the bird and the rock, as well as the exact details of how the bird hit the rock (e.g., the velocities and masses of the bird and the rock and the angle at which they hit). In the electron-photon case, it is impossible to predict exactly what the results of the next observation will be. What quantum theory can do is predict the probability of obtaining a particular result. In the Schrödinger’s Cats example, when a box was opened, either a dead cat or a live cat was found. There was no way to predict which it would be. Opening the box (observing the cat) changed the cat from being in a type of mixed live-dead state into either a pure live state or a pure dead state. If many boxes were opened, the probability of finding a cat alive was 50%, but there was no way to predict what would happen when a particular box was opened (a single measurement). The cat problem is not physically realizable and, therefore, it is not a true quantum mechanical problem. A physically real problem that is akin to the cat problem is discussed in a later chapter. The cat problem was intended to introduce the idea that an observation can change a system and that only a probability could be ascertained from a series of experiments. For real systems that are absolutely small, quantum mechanics is the theory that permits the calculation and understanding of the distribution of probabilities that are obtained when measurements are made on many identically prepared systems. How quantum mechanical probability distributions arise and how to think about the nature of the disturbance that accompanies measurements on absolutely small systems are discussed in the following chapters.