Quantum Mechanics and Path Integrals [antikvár]

PROBABILITY IN QUANTUM MECHANICS1 From about the beginning of the twentieth century experimental physics amassed an impressive array of strange phenomena which demonstrated the inadequacy of classical physics. The attempts to discover a theoret- ical structure for the new phenomena led at first to a confusion in which it appeared that light, and electrons, behaved sometimes like waves and sometimes like particles. This apparent inconsistency was completely resolved in 1926 and 1927 in the theory called quantum mechanics. The new theory asserts that there are experiments for which the exact out- come is fundamentally unpredictable and that in these cases one has to be satisfied with computing probabilities of various outcomes. But far more fundamental was the discovery that in nature the laws of com- bining probabilities were not those of the classical probability theory of Laplace. The quantum-mechanical laws of the physical world approach very closely the laws of Laplace as the size of the objects involved in the experiments increases. Therefore, the laws of probabilities which are conventionally applied are quite satisfactory in analyzing the behavior of the roulette wheel but not the behavior of a single electron or a single photon of light. A Conceptual Experiment. The concept of probability is not altered in quantum mechanics. When we say the probability of a certain outcome of an experiment is p, we mean the conventional thing, i.e., that if the experiment is repeated many times, one expects that the fraction of those which give the outcome in question is roughly p. We shall not be at all concerned with analyzing or defining this concept in more detail; for no departure from the concept used in classical statistics is required. What is changed, and changed radically, is the method of calculating probabilities. The effect of this change is greatest when dealing with objects of atomic dimensions. For this reason we shall illustrate the laws of quantum mechanics by describing the results to be expected in some conceptual experiments dealing with a single electron. Our imaginary experiment is illustrated in Fig. 1-1. At A we have a source of electrons S. The electrons at S all have the same energy 1 Much of the material appearing in this chapter was originally presented as a lecture by R.P. Feynman and published as "The Concept of Probability in Quan- tum Mechanics" in the Second Berkeley Symposium on Mathematical Statistics and robabihty, University of California Press, Berkeley, Calif., pp. 533-541, 195]