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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.

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