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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.

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