*probability*that the photon will be emitted within any particular time interval can be calculated exactly. Quantum mechanics is an unambiguous and quantitative theory in the sense that probabilities can be calculated precisely. It is indefinite in that one can calculate only the chance that something will happen, never what will, in fact, happen for a particular atom or system. To check the calculated probability of photon emission against experiment, a single atom will never do. One must study a great many atoms and infer the probability for a single one by observing the average behavior of the whole collection. Similarly, one could never prove with a single toss of a coin that the probability of “heads” is exactly one half. Verification would require a great many tosses.

The decay of an unstable particle affords a simple and direct test of the working of probability at the fundamental level. If a great many pions are created under identical conditions at the target point in a particle accelerator, those that move away at a certain speed in a certain direction may be tracked in a detector—the multiwire proportional chamber, for example, or the older bubble chamber, both of which enable one to “see” the tracks. Some of the pions decay into muons and neutrinos after moving a very short distance; some, after moving a greater distance; a few, after moving very much farther. There will be some average distance and correspondingly some average time for the occurrence of the decay process. If this experiment is repeated time after time, each time with a very large number of pions, the *average *lifetime of the pions will be exactly the same for every group. This average is a perfectly definite quantity, a measure of the decay probability of the pion, and it can be measured to arbitrarily high precision if a large enough group of pions is used. Nevertheless, the length of time that any single pion will live is indeterminate. It might die long before most of its fellow pions, or it might outlive them all.

The idea that the fundamental processes of nature are governed by laws of probability should have hit the world of science like a bombshell. Oddly enough, it did not. It seems to have infiltrated science gradually over the first quarter of the twentieth century. Only after the quantum theory had become fully developed in about 1926 did physicists and philosophers sit up and take note of the fact that a revolution had occurred in our interpretation of natural laws.^{1}

As early as 1899, Ernest Rutherford and others studying the newly discovered phenomenon of natural radioactivity noticed that the decay of radioactive atoms seemed to follow a law of probability. Yet Rutherford and his fellow physicists did not shout from the housetops that the fundamental laws of nature must be laws of probability. Why was this? The answer is very simple. They did not recognize that they were dealing with *fundamental *laws. Probability in science was, after all, nothing new. What was new, but not yet recognized, was that for the first time probability was appearing in simple elementary phenomena in nature.

Classical physics was built solidly on the idea that nature follows exact deterministic or causal laws. If enough is known about a particle or a light wave or any system at one time, its future behavior is, according to pre-twentieth-century physics, exactly calculable. There is no question about where the Earth or Moon will be at some future date. A bridge can be built or an electromagnet designed with solid assurance that it will not collapse or fail to work because of some unpredictable fluctuation in the laws of mechanics or electromagnetism. Indeed, the exact determinism of physical laws had a powerful effect on philosophy in the nineteenth century; one popular view was that the universe can be regarded as a giant mechanism—the “world-machine”—inevitably and relentlessly unrolling history according to a predetermined plan.

Nevertheless we are all well aware of probability in everyday life and do not need to go to the fundamental-particle world to find it. As life-insurance actuaries and gamblers know, life and death and roulette wheels are governed by laws of probability. So is the behavior of bulk matter as it follows the second law of thermodynamics. The probability of the macroscopic world is a probability of ignorance (lack of detailed information); the probability of the submicroscopic world is a fundamental probability of nature. One contributor to macroscopic probability is ignorance of initial conditions. According to quantum mechanics it is impossible, in principle, as well as in fact, to calculate the exact moment when an atomic event occurs, no matter how precisely the initial conditions are known. One can know every possible thing there is to know about a pion, and still not be able to predict when it will decay.

We can now understand why Rutherford must not have been unduly surprised to discover a law of probability in radioactive decay. He assumed that he was dealing with a probability of ignorance. The interior of the atom was, as far as he knew, a complicated place with an elaborate structure, and the apparently random nature of the decay process could be attributed to unknown differences in the internal states of different atoms. Yet even before quantum mechanics was developed as a fully acceptable theory (in 1925) there were hints that the probability of the atomic world might be of a more fundamental kind. Rutherford himself discovered (with Frederick Soddy, in 1902) that radioactivity represented a sudden catastrophic change in an atom, and was not the result of a gradual process of change. This, in itself, made a radioactive transmutation event appear to be rather fundamental. Einstein’s photon theory of light in 1905 and Bohr’s theory of the hydrogen atom in 1913 also contained hints of a new fundamental role of probability.

Probability in the submicroscopic world shows itself in several ways. First, and most directly, it manifests itself through a randomness of microscopic events. If you are a prospector using a Geiger counter to search for uranium deposits, you can easily listen in on the randomness of radioactive decay. The Geiger counter should be of the common type arranged to give an audible click when a high-speed particle triggers the device. When you hold the Geiger counter close to a uranium-bearing rock, you will hear a sequence of individual clicks. It will be obvious that the clicks are not coming in a regular sequence like the ticks of a clock, but occur in an apparently random fashion. Indeed, a mathematical analysis would show that they are exactly random. The time at which a given click occurs is completely unrelated to the time elapsed since the previous click or to the time at which any other click occurs.

In doing an experiment like this, you feel in unusually close touch with the submicroscopic world. A single audible click means that somewhere among the countless billions of atoms in the rock, one nucleus has suddenly spontaneously ejected a particle at high speed and transmuted itself into a different nucleus. Very literally, a nuclear explosion has occurred and, in the private world of the nucleus, the time at which the explosion occurred has been governed exclusively by a law of probability. An identical neighboring nucleus may have long since exploded, or it may be destined to live yet a long time. (In this experiment, only some of the clicks come from radioactivity in the rock. Some come also from high-speed muons coursing through the Geiger counter from high in the sky, but the same principle of randomness applies to these “cosmic rays” as to the radioactive particle from the rock.)

Probability manifests itself in another way that is not so obvious to the eye or ear, but that is equally convincing to someone with a little mathematical training. This is through the exponential law of decay. Rutherford, in fact, discovered the role of probability in radioactivity in this way, for in 1899 he did not yet have any way to observe single transmutation events (it was several years later that his research student, Hans Geiger, invented the Geiger counter). Rutherford noticed that when the total intensity of the radioactivity was graphed as a function of time, a curve like that in the figure below resulted. This is called an exponential curve. Its most marked characteristic is that it falls vertically from any value whatever to half that value in a fixed horizontal distance. This meant in Rutherford’s experiment that a definite fixed time was needed for the radioactivity to diminish in intensity by half, regardless of the initial intensity. This fixed time is called the half life of the material, designated *t*_{½}.

The curve above is described mathematically by the function

*I = I*_{0}*e*^{–}^{t/τ},

where *τ *is the mean lifetime, or average lifetime, of the radioactive nuclei. When *t *= *τ*, the intensity is *I = I*_{0}*e*^{–1 }= 0.368*I*_{0}. When *t *= *t*_{½ }= 0.694*τ*, *I = I*_{0}*e*^{–0.694 }= 0.5*I*_{0}. In the figure, the initial intensity *I*_{0 }is set equal to one. In other words, the half life is about 69 percent of the mean life—for radioactive nuclei and unstable particles, but not for everything that lives and dies.^{2}

What Rutherford knew and what we here state without proof is that the exponential curve results from the action of a particular law of probability on the individual radioactive-decay events. For each single nucleus, the probability of decay per unit time is constant, and the half life represents a halfway point in probability. The chance that the nucleus will decay in less time is one half; the chance that it will decay after a longer time is one half. When this probabilistic law acts separately on a large collection of identical nuclei, the total rate of radioactive decay falls smoothly downward along an exponential curve. The same is true in the particle world. The mean lives of unstable particles are measured by studying the exponential curve of decay for each kind of particle. (The times themselves are usually not measured directly, but they can be inferred by measuring the speeds of the particles and the distances traveled.)

The span of known half lives from the shortest to the longest is unimaginably great. At one extreme are the super-short-lived particles, or resonances, with half lives of 10^{–20} s or less. The more “respectable” particles live from 10^{–10} s up to several minutes—except, of course, for the stable particles, which, as far as we know, live forever. Radioactive nuclei are known with half lives ranging from about 10^{–3} s up to more than 10^{15} years. Regardless of the half life, the decay of every kind of unstable particle or nucleus proceeds inflexibly along an exponential curve.

The radioactivity created by nuclear explosions consists of a mixture of many different radioactive species, so that in the days when bomb tests were conducted above ground, the fallout subsequent to the test did not follow a single simple exponential curve. Some of the radioactive nuclei decayed so quickly after the explosion—within seconds or minutes—that they contaminated only a local region and were not a public-health problem. Other species have such a long life—millions of years—that their rate of decay remains always very small. In between are the nuclei with half lives of a few years up to a few thousand years; these constitute the greatest threat from fallout. The often-discussed Co^{60} (cobalt 60) has a half life of 5.3 years, and Sr^{90} (strontium 90) a half life of 29 years. An isotope of carbon, C^{14}, which has been so useful for dating archaeological finds because of its half life of 5,700 years, was also formed in the atmosphere by bomb tests. This will greatly complicate, perhaps invalidate altogether, the C^{14} dating method for archaeologists of future millennia.

I have discussed so far only the probability of time, which determines the characteristic decay pattern of unstable particles and nuclei. Probability also manifests itself in other ways in the submicroscopic world. There is the probability of “branching ratio.” A kaon may decay in various ways: into two pions or into a muon and a neutrino, among other possibilities. Which branch any particular kaon will choose to follow is completely indeterminate, but the probability for each branch is readily measurable (given enough kaons). There is also a probability of position and probability of angular deflection in scattering. A particularly fascinating aspect of probability at the fundamental level is the phenomenon of “tunneling.” If a particle is held on one side of a wall that, according to classical physics, is totally impenetrable, there is a chance that it will emerge on the other side. In certain transistors, electrons tunnel through potential-energy barriers. The alpha decay of nuclei can also be explained as a tunneling phenomenon. As with so many other aspects of quantum mechanics, tunneling is of significance only in the submicroscopic world. A man leaning idly against the wall of his hotel room need have no fear that he will suddenly find himself in the next room. Nor should a student in a dull lecture put any hope in the tunneling phenomenon as a way out.

Not every aspect of nature is uncertain and probabilistic. Many of the properties of stable systems—for example the spin of an electron or its mass—are precisely defined. Even where a law of probability is at work, the probability of an event may be so close to zero or to one that its nonoccurrence or occurrence can be regarded in practice as a certainty. The chance that the tunneling phenomenon will be experienced by a person can be said to be effectively zero. The chance that a proton will decay in a billion years is essentially zero (this has been measured). The chance that a pion will live for two hours is as good as zero. Because quantum-mechanical probabilities in the macroscopic world are always so close to zero or one, the deterministic laws of classical physics are completely adequate and accurate for describing large-scale phenomena.

Is the probability of the submicroscopic world really a fundamental probability of nature, or is it perhaps, after all, a probability of ignorance, arising out of a complicated deeper, as yet undiscovered, substructure of matter? The simplest answer that can be given is: “No one knows.” Most scientists regard it as not a very interesting question. Since nothing of a deeper substructure is known, it is not fruitful at this moment in history to discuss it. So far as we know now, the probability is indeed fundamental, but one need cling to this idea no more firmly than to any other in science.

Nevertheless, some of the greatest scientists of modern times did find the question interesting and have discussed it. Those arguing for the fundamental nature of the laws of probability have a bit better time of it, for they have all of the successes of quantum mechanics on their side. Those favoring the view that the probability of quantum mechanics is really a probability of ignorance can adduce at best philosophic, and not scientific, arguments. Einstein, for example, liked to remark that he did not believe in God playing dice, and in 1953 he wrote, “In my opinion it is deeply unsatisfying to base physics on such a theoretical outlook, since relinquishing the possibility of an objective description . . . cannot but cause one’s picture of the physical world to dissolve into a fog.”^{3}

The arguments in favor of the truly fundamental role of probability in nature are more subtle, being based on the theory of quantum mechanics. We shall give just one argument here. The manufacturers of baseballs try hard to make all of their balls identical. It is, of course, an impossible task. No two can ever be precisely alike down to the last microscopic detail, because each ball is a complicated structure with many constituents—more than 10^{25} if we count atoms. On the other hand, there is quite good evidence that any two electrons are, in fact, truly identical, and that relatively few parameters are required to specify an electron completely. In short, the electron seems to be a decidedly much more elementary structure than a baseball. This is not a trivial conclusion. If there were infinitely many layers of nature to uncover, the electron could just as well be about as complex as the baseball. Since the electron follows laws of probability, one is led to suspect that these laws are themselves of an elementary and fundamental kind, not merely a reflection of the fact that a complicated unknown structure resides within the electron.

Although arguments of this kind sound scientific, they are no more rigorous than Einstein’s statement of belief. We can only wait and see.

1^{} The *fundamental* role of probability in nature seems to have been first clearly emphasized by Niels Bohr, Hendrik Kramers, and John Slater in 1924. Their effort to build a new quantum theory failed, but success came the following year to Werner Heisenberg, and in 1926 Max Born gave to the new theory the probability interpretation that remains as a keystone of quantum mechanics to the present day.

2^{} In 2011, the average life expectancy of an American male was 77 years, but his half life was about four years greater. He would need to reach the age of 81 years in order to outlive half of his contemporaries. (The numbers for American women were greater by several years.)

3^{} A. Einstein, in Scientific Papers Presented to Max Born (New York: Hafner, 1953), p. 40. Original in German.