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THE INTERPRETATION OF QUANTUM MECHANICS. III. MATHEMATICAL FORMALISM

A.B. Datzeff

Published Jan 1, 1961• 0 citations
OpenAlex

Abstract

As a summary of the ideas developed in the previous works, the mathematical formalism of the prerelativistic quantum mechanics is given. It is shown that the relations of uncertainty as well as the attempts for their interpretation have always a statistical sense and from them one cannot draw a conclusion for the behavior of a separate particle. (auth)

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Cambridge University Press eBooks•Jan 31, 2013•1 citation
As a summary of the ideas developed in the previous works, the mathematical formalism of the prerelativistic quantum mechanics is given. It is shown that the relations of uncertainty as well as the attempts for their interpretation have always a statistical sense and from them one cannot draw a conclusion for the behavior of a separate particle. (auth)
A survey is made of the hitherto proposed answers to the following crucial questions arising in the physical interpretation of the mathematical formalism of wave mechanics: (I) What is the meaning of the dynamical variables and of their eigenvalues? (II) What is the meaning of the wave function? (III) What is the nature and origin of Heisenberg's uncertainty relations? (IV) What is the nature of the systems dealt with by wave mechanics? Taking into account the interpretations that have been proposed in the last few years, a surprisingly large number of answers is found. The present situation in quantum mechanics is compared with similar crises in the past history of physics. The mere multiplicity of consistent interpretations of quantum mechanics is regarded as a warning against the dogmatic adherence to any one of them, to the exclusion of new possibilities.
This paper deals with the problem of the interpretation of quantum mechanics, the nature of physical theory, and the future of physics. First, the question of why there are so many different interpretations of quantum mechanics proposed and accepted by outstanding quantum experts is treated. This requires a discussion of the relation between physics and philosophy and drawing a distinction between a physical formalism and a physical explanation. The Copenhagen interpretation is considered, criticized, and found to be unsatisfactory in some respects. Finally, an interpretation that contains ideas from previous interpretations but which may itself be new, is proposed. Hopefully, this is superior to the previous interpretations. It has, I believe, the advantage of logical clarity. In the process of doing this, I introduce a theory of levels of physical theory, according to which microscopic physical theory, as it develops in successive stages, becomes more and more adaptable to explaining—or at least taking cognizance of—fundamental biological or ultimately psychological phenomena.
In completing the story of spin, we have run far ahead of the continued development of the understanding of the matrix, operator and wave mechanical approaches to quantum mechanics. The reconciliation of these approaches was described in Chap. 15, but there remained the issue of the interpretation of the wavefunction and the deeper implications of the theory. The understanding came gradually with Born's interpretation of the wavefunction, Ehrenfest's demonstration of the equivalence of the classical and quantum pictures and Heisenberg's enunciation of the uncertainty principle. These led to what became known as the Copenhagen interpretation of quantum mechanics. At the same time, the formal mathematical foundations of the different approaches to quantum phenomena were set on a secure foundation thanks to the efforts of Hilbert and many others. These developments resulted in what may be referred to as the completion of quantum mechanics, in the sense that it laid the foundations for all the future development of physics at the atomic and subatomic level – some of these achievements are summarised in Chap. 18. Schrödinger's interpretation (1926) Schrödinger regarded wave mechanics as superior to the matrix mechanical approach to quantum physics, not only because it was based upon the well-known eigenfunction techniques of classical physics, but also because it was much more visualisable. His first attempt at interpreting the wavefunction appeared in the final Sect. 7 of the fourth part of his great series of papers (Schrödinger, 1926f) and was entitled On the physical significance of the field scalar.