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. (shrink)

Common eigenfunctions of nontrivial projectors of x and p are constructed.

The arguments presented by Gibbins in his Note are based on a sharp distinction between the product Δx·Δp, which refers to the ranges of position and momentum of an individual system, and the uncertainty principle ΔX·ΔP ≥ ħ/2, which expresses a statistical relation for an ensemble of systems. A critical role in Gibbins’ reasoning is played by the theorem T which states that the restriction of the dynamical variable of position x of an individual system to a finite range Δx (...) excludes the possibility of restricting the canonically conjugate dynamical variable of momentum p to a finite range Δp. (shrink)

Although there is a complete consensus among working physicists with respect to the practical and operational meanings of quantum states, and also a rather loosely formulated general philosophic view called the Copenhagen interpretation, a great deal of confusion and divergence of opinions exist as to the details of the measurement process and its effects upon quantum states. This paper reviews the current expositions of the measurement problem, limiting itself for lack of space primarily to the writings of physicists; it calls (...) attention to inconsistencies and proposes resolutions. Except for a summary of the properties of statistical matrices which are needed in Part II, the first part is non-mathematical and deals largely with two kinds of probability, reducible and irreducible probabilities, which need to be distinguished for a proper understanding of the measurement act. (shrink)