7 PROBABILITY AND RANDOMNESS

0 4.7 PROBABILITY AND RANDOMNESS

Any single measurement of the position or momentum of a particle can be made FIGURE 4.26

with as much precision as our experimental skill permits. How then does the The smaller the initial wavelike behavior of a particle become observable? How does the uncertainty in wave packet, the more quickly it

position or momentum affect our experiment?

grows. Suppose we prepare an atom by attaching an electron to a nucleus. (For this example we regard the nucleus as being fixed in space.) Some time after preparing our atom, we measure the position of the electron. We then repeat the procedure, preparing the atom in an identical way, and find that a remeasurement of the position of the electron yields a value different from that found in our first measurement. In fact, each time we repeat the measurement, we may obtain

a different outcome. If we repeat the measurement a large number of times, we find ourselves led to a conclusion that runs counter to a basic notion of classical physics—systems that are prepared in identical ways do not show identical subsequent behavior. What hope do we then have of constructing a mathematical theory that has any usefulness at all in predicting the outcome of a measurement, if that outcome is completely random?

The solution to this dilemma lies in the consideration of the probability of obtaining any given result from an experiment whose possible results are subject to the laws of statistics. We cannot predict the outcome of a single flip of a coin or roll of the dice, because any single result is as likely as any other single result. We can, however, predict the distribution of a large number of individual measurements. For example, on a single flip of a coin, we cannot predict whether the outcome will be “heads” or “tails”; the two are equally likely. If we make a large number of trials, we expect that approximately 50% will turn up “heads” and 50% will yield “tails”; even though we cannot predict the result of any single toss of the coin, we can predict reasonably well the result of a large number of tosses.

Our study of systems governed by the laws of quantum physics leads us to a similar situation. We cannot predict the outcome of any single measurement of the position of the electron in the atom we prepared, but if we do a large number of measurements, we ought to find a statistical distribution of results. We cannot develop a mathematical theory that predicts the result of a single measurement, but we do have a mathematical theory that predicts the statistical behavior of a system (or of a large number of identical systems). The quantum theory provides this mathematical procedure, which enables us to calculate the average or probable outcome of measurements and the distribution of individual outcomes about the average. This is not such a disadvantage as it may seem, for in the realm of quantum physics, we seldom do measurements with, for example, a single atom. If we were studying the emission of light by a radiant system or the properties of a solid or the scattering of nuclear particles, we would be dealing with a large number of atoms, and so our concept of statistical averages is very useful.

In fact, such concepts are not as far removed from our daily lives as we might

4.7 | Probability and Randomness 127

a 50% chance of rain tomorrow? Will it rain 50% of the time, or over 50% of the city? The proper interpretation of the forecast is that the existing set of atmospheric conditions will, in a large number of similar cases, result in rain in about half the cases. A surgeon who asserts that a patient has a 50% chance of surviving an operation means exactly the same thing—experience with a large number of similar cases suggests recovery in about half.

Quantum mechanics uses similar language. For example, if we say that the electron in a hydrogen atom has a 50% probability of circulating in a clockwise direction, we mean that in observing a large collection of similarly prepared atoms we find 50% to be circulating clockwise. Of course a single measurement shows either clockwise or counterclockwise circulation. (Similarly, it either rains or it doesn’t; the patient either lives or dies.)

Of course, one could argue that the flip of a coin or the roll of the dice is not

a random process, but that the apparently random nature of the outcome simply reflects our lack of knowledge of the state of the system. For example, if we knew exactly how the dice were thrown (magnitude and direction of initial velocity, initial orientation, rotational speed) and precisely what the laws are that govern their bouncing on the table, we should be able to predict exactly how they would land. (Similarly, if we knew a great deal more about atmospheric physics or physiology, we could predict with certainty whether or not it will rain tomorrow or an individual patient will survive.) When we instead analyze the outcomes in terms of probabilities, we are really admitting our inability to do the analysis exactly. There is a school of thought that asserts that the same situation exists in quantum physics. According to this interpretation, we could predict exactly the behavior of the electron in our atom if only we knew the nature of a set of so-called “hidden variables” that determine its motion. However, experimental evidence disagrees with this theory, and so we must conclude that the random behavior of

a system governed by the laws of quantum physics is a fundamental aspect of nature and not a result of our limited knowledge of the properties of the system.