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Journal of Mathematical Physics

Compatibility of the uncorrelated joint distribution with the classical limit

Roderick I. Sutherland

Published Jun 1, 1983• 1 citation
Publisher

Abstract

It is shown that the uncorrelated joint distribution, when adopted for quantum mechanics, is not (despite initial appearances to the contrary) in conflict with the known correlations between position and momentum in the classical limit. It is therefore not excluded on these grounds as a possible joint distribution expression for quantum mechanics.

DOI
10.1063/1.525881

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•Mar 18, 2022•0 citations
It is shown that the uncorrelated joint distribution, when adopted for quantum mechanics, is not (despite initial appearances to the contrary) in conflict with the known correlations between position and momentum in the classical limit. It is therefore not excluded on these grounds as a possible joint distribution expression for quantum mechanics.
Present quantum theory, which is statistical in nature, does not predict joint probability distribution of position and momentum because they are noncommuting. We propose a deterministic quantum theory which predicts a joint probability distribution such that the separate probability distributions for position and momentum agree with usual quantum theory. Unlike the Wigner distribution the suggested distribution is positive definite. The theory predicts a correlation between position and momentum in individual events.
The fundamental problem considered is that of the existence of a joint probability distribution for momentum and position at a given instant. The philosophical interest of this problem is that for the potential energy functions (or Hamiltonians) corresponding to many simple experimental situations, the joint “distribution” derived by the methods of Wigner and Moyal is not a genuine probability distribution at all. The implications of these results for the interpretation of the Heisenberg uncertainty principle are analyzed. The final section consists of some observations concerning the axiomatic foundations of quantum mechanics.
A theoretical investigation is made into the difficulties encountered in constructing a deterministic model for quantum mechanics and into the restrictions that can be placed on the form of such a model.The various implications of the known 11 impossi bil ity proofs II are examined.A possible explanation for the non-locality required by Bell's proof is suggested in terms of backwards-in-time causality.The efficacy of the Kochen and Specker proof is brought into doubt by showing that there is a possible way of avoiding its implications in the only known physically realizable situation to which it applies.Also, a new thought experiment is put forward to show that a particle's predetermined momentum and energy values cannot satisfy the laws of momentum and energy conservation without conflicting with the predictions of quantum mechanics.Special attention is paid to a particular class of deterministic models which appear not to have been explicitly studied before.The models considered are those for which the individual outcomes of measurements are not dependent on hidden variables associated with the measuring apparatus and for which the hidden variables of a particle do not need to be randomized after each measurement.Specific models of this type are constructed and examined for the cases of both discrete and continuous spectra of eigenvalues.In developing these models a need is found for a joint distribution for position and momentum.In relation to this point, it is demonstrated that the quantum mechanical wave equations actually point to a particular joint distribution expression, namely the Margenau and Hill distribution.The schemes considered are found to have certain shortcomings and possible steps towards more satisfactory models are suggested.