Joint Probability Distributions in Quantum Mechanics

Abstract

The notion and methods of calculating probabilities in quantum mechanics is strikingly different from classical mechanics and standard probability theory. Quantum mechanics manipulates mysteriously, from a classical probabilistic point of view, complex functions and operators to nevertheless end up with observable quantities which are interpreted as if they were derived from a standard type of probability theory. Whether the strange methods of obtaining the predictions of the theory are essential, in the sense that given the type of results quantum mechanics predicts implies a quantum mechanical type of mathematical formalism, is not totally clear. Certainly many of the paradoxes seem to be based on the mode of calculating probabilities. For example, the paradox of the double slit experiment arises from the fact that the probability of the particle being at a certain point on the screen is not the same probability that it went through one slit or the other. Since standard probability theory arose out of simplistic considerations it may be that it is restricted and that its future development will generalize it to a degree where quantum mechanics will fit in naturally. Weiner (1963) made some attempts along these lines by showing some examples from ordinary probability theory that yield results similar in mathematical form as in quantum mechanics. Another approach to the understanding of the probabilistic methods is to attempt to formulate quantum mechanics as a traditional probability theory. We shall describe this attempt, and although we shall show that this is strictly speaking impossible, the impossibility will illuminate certain aspects of the theory. In particular we will emphasize the question of joint distributions of non-commuting variables and show the one-to-one relationship between so-called correspondence rules and the distribution functions. As we shall see, the question as to whether there can exist a consistent rule for obtaining quantum mechanical operators (a correspondence rule) from their classical counterpart and the question whether there exists a legitimate joint distribution function are one and the same.