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arXiv:1601.05555 · WORLD SCIENTIFIC (EUROPE) eBooks

A Top-down versus a Bottom-up Hidden-variables Description of the Stern–Gerlach Experiment

M. Arsenijević, J. Jeknić-Dugić, M. Dugić

Published Nov 14, 2016• 2 citations
arXivHTMLPDF

Abstract

We employ the Stern-Gerlach experiment to highlight the basics of a minimalist, non-interpretational top-down approach to quantum foundations. Certain benefits of the here highlighted "quantum structural studies" are detected and discussed. While the top-down approach can be described without making any reference to the fundamental structure of a closed system, the hidden variables theory ?a la Bohm proves to be more subtle than it is typically regarded.

DOI
10.1142/9781786341419_0015

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We employ the Stern-Gerlach experiment to highlight the basics of a minimalist, non-interpretational top-down approach to quantum foundations. Certain benefits of the here highlighted "quantum structural studies" are detected and discussed. While the top-down approach can be described without making any reference to the fundamental structure of a closed system, the hidden variables theory ?a la Bohm proves to be more subtle than it is typically regarded.
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.
We discuss whether the Stern-Gerlach experiment accepts hidden-variables theories. We discuss that the existence of two spin-1/2 pure states ∣↑⟩|\uparrow\rangle∣↑⟩ and ∣↓⟩|\downarrow\rangle∣↓⟩ rules out the existence of probability space of specific quantum measurement. If we detect ∣↑⟩|\uparrow\rangle∣↑⟩, then measurement outcome is +1+1+1. If we detect ∣↓⟩|\downarrow\rangle∣↓⟩, then measurement outcome is −1-1−1. This hidden-variables theory does not accept the transition probability ∣⟨↑∣↓⟩∣2=0|\langle\uparrow|\downarrow\rangle|^2=0∣⟨↑∣↓⟩∣2=0. Therefore we have to give up the hidden-variables theory. This implies the Stern-Gerlach experiment cannot accept the specific hidden-variables theory. And we study whether quantum phase factor accepts hidden-variables theories. We use the transition probability for two spin-1/2 pure states (∣↑⟩+∣↓⟩)/2(|\uparrow\rangle+|\downarrow\rangle)/\sqrt{2}(∣↑⟩+∣↓⟩)/2​ and (∣↑⟩+eiθ∣↓⟩)/2(|\uparrow\rangle+e^{i \theta}|\downarrow\rangle)/\sqrt{2}(∣↑⟩+eiθ∣↓⟩)/2​. It is cos⁡2(θ/2)\cos^2(\theta/2)cos2(θ/2). We discuss that the phase factor does not accept another specific hidden-variables theory. We explore the phase factor is indeed a quantum effect, not classical. Our research gives a new insight to the quantum information processing which relies on quantum phase factor, such as Deutsch's algorithm.