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arXiv:1911.00546

Stern-Gerlach: conceptually clean or acceptably vague?

Shaw, Robert

Published Aug 22, 2019• 0 citations
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Abstract

This paper develops a number of quantum mechanical characterisations of Stern-Gerlach. It discusses areas of vagueness in their formulation. Philosophers criticise quantum mechanics for unacceptable vagueness in connection with the measurement problem. The quantum formulation problems identified by this paper go beyond the locus of philosophical criticism. It concludes with an open question, are some areas of vagueness in quantum mechanics more acceptable philosophically than others and, if so, why?

DOI
10.48550/arxiv.1911.00546

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This paper develops a number of quantum mechanical characterisations of Stern-Gerlach. It discusses areas of vagueness in their formulation. Philosophers criticise quantum mechanics for unacceptable vagueness in connection with the measurement problem. The quantum formulation problems identified by this paper go beyond the locus of philosophical criticism. It concludes with an open question, are some areas of vagueness in quantum mechanics more acceptable philosophically than others and, if so, why?
While the Stern–Gerlach experiment is an old and familiar problem, no analysis of this experiment is presented in the pedagogical literature using modern quantum mechanical techniques. Expositions of the Stern–Gerlach experiment are usually based on ‘‘old quantum theory,’’ i.e., semiclassical Bohr–Sommerfeld quantum mechanics. The experiment is also a popular one in discussions of the postulate of quantum measurement, which asserts that the process of measuring an observable forces the state vector of the system into an eigenvector of that observable, and the value measured will be the eigenvalue of that eigenvector. The most enduring of the philosophical arguments associated with quantum mechanics ultimately revolves around the mechanism of quantum measurement. Since the postulate of quantum measurement is expressed in terms well outside the realm of the old quantum mechanics, the traditional analysis of the Stern–Gerlach experiment cannot reveal any information about the relationship between dynamics and measurement. Therefore, a modern analysis of the Stern–Gerlach experiment must be made if this experiment is to reveal any information about the relationship between dynamics and the mechanism of quantum measurement. This paper presents such an analysis, and develops some of the implications for the theory of quantum measurement.
A joint paper by Albert Einstein and \nPaul Ehrenfest, published just weeks \nafter the results of the Stern-Gerlach \nexperiment became known, shows \nwith remarkable clarity and prescience \nthe unsurmountable difficulties \nthat the experiment posed for \nany classical interpretation. With a \nfocus on the measurement process, \nrather than on the underlying theoretical \nalternatives for this experimentum \ncrucis, the authors almost \nanticipate what would later be recognized \nas a central conceptual difficulty \nof quantum mechanics, i.e., the \nquantum measurement problem.
We critically examine the quantum-mechanical modelling of a measurement process using the Stern–Gerlach (SG) setup in the most general context, probing in particular for nonideal situations, the subtleties involved in the connection between the notion of 'distinguishability' of apparatus states defined in terms of the inner product and the spatial separation among the wave packets emerging from the SG setup. The quantitative studies highlighting some of the unexplored features of this relationship are presented in terms of an appropriately defined measure for the spatial separation between the emerging wave packets. It is also indicated how the effects arising from such departures from the idealness can be empirically tested for different values of the relevant parameters.
There has been recent interest in formulating theories of non-representational indeterminacy. The aim of this paper is to clarify the relevance of quantum mechanics to this project. Quantum-mechanical examples of vague objects have been offered by various authors, displaying indeterminate identity, in the face of the famous Evans argument that such an idea is incoherent. It has also been suggested that the quantum-mechanical treatment of state-dependent properties exhibits metaphysical indeterminacy. In both cases it is important to consider the details of the metaphysical account and the way in which the quantum phenomenon is captured within it. Indeed if we adopt a familiar way of thinking about indeterminacy and apply it in a natural way to quantum mechanics, we run into illuminating difficulties and see that the case is far less straightforward than might be hoped.