This is the second, mathematically more detailed part of a paper consisting of two articles, the first having appeared in the immediately preceding issue of this Journal. It shows that a measurement converts a pure case into a mixture with reducible probabilities. The measurement as such permits no inference whatever as to the state of the physical system subjected to measurement after the measurement has been performed. But because the probabilities after the act are classical and therefore reducible, it is (...) often possible to adjust them so that Von Neumann's projection postulate is true. Among the more specific features dealt with in Part II is the occurrence of negative joint probabilities for the measurement of non-commuting operators in certain (not all!) quantum states. The general conclusions reached are stated at the end of the article. (shrink)

Recent discussions in the physical literature, designed to clarify the logical position of modern physical theory, have brought to light an amazing divergence of fundamental attitudes which may well bewilder the careful student of physics as well as philosophy. Quantum mechanics, representing an abstract formalism, should be capable of having its logical structure analyzed with great precision like any other mathematical discipline. Its consequences in all problems to which its method can be applied are so unambiguous, consistent, and successful in (...) predicting physical experience as to disperse immediately all thoughts of possible discrepancies in its fundamental texture. Yet it must be said that even the founders of quantum theory are not in harmony in their various expositions of the bases of that theory. However, while this situation seems disquieting on the face of it, there is no cause for serious brow raising, for it is a fact that there exists agreement with regard to the central axioms of the theory, and that the ambiguities affect only their philosophical interpretation, a field in which differences of opinion may at present be honestly entertained. (shrink)

It is the purpose of this note to comment on some important problems which have been already vividly discussed by several authors. Besides the well known former discussions of Schrödinger and J. v. Neumann I should like to mention here especially H. Margenau's article, “Critical Points in Modern Physical Theory,” which strongly influenced my present discussion.