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American Journal of Physics

A modern analysis of the Stern–Gerlach experiment

Daniel E. Platt

Published Apr 1, 1992• 48 citations

Abstract

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.

DOI: 10.1119/1.17136

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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.

This popular undergraduate quantum mechanics textbook is now available in a more affordable printing from Cambridge University Press. Unlike many other books on quantum mechanics, this text begins by examining experimental quantum phenomena such as the Stern-Gerlach experiment and spin measurements, using them as the basis for developing the theoretical principles of quantum mechanics. Dirac notation is developed from the outset, offering an intuitive and powerful mathematical toolset for calculation, and familiarizing students with this important notational system. This non-traditional approach is designed to deepen students' conceptual understanding of the subject, and has been extensively class tested. Suitable for undergraduate physics students, worked examples are included throughout and end of chapter problems act to reinforce and extend important concepts. Additional activities for students are provided online, including interactive simulations of Stern-Gerlach experiments, and a fully worked solutions manual is available for instructors.

The Stern-Gerlach experiment holds key role for students, particularly those studying quantum mechanics, modern physics, atomic physics, and physics education.It is one of the simplest experiments that demonstrates that the microscopic world does not behave accordingly withdynamics of classical physics, thusthe Stern-Gerlach experiment allows students to follow a logical progression: Classical expectation → Experimental observation → Failure of classical explanation → Quantum hypothesis → Spin states → Mathematical formulation → Quantum measurement.

Stern-Gerlach experiment is a paradigm of measurement theory in quantum mechanics. Notwithstanding several analysis given in literature, no clear understanding of the apparent collapse has been given so far. Indeed, one can imagine a Stern-Gerlach device where all environmental effects are removed and ask the question on how the measurement goes on. In this letter we will prove that a the Stern-Gerlach device behaves as a true measurement apparatus, as expected by the Copenaghen interpretation, by the Ehrenfest theorem. In this way we recover, by other means, a limit on the Stern-Gerlach device, given by Bohm, based on scrambling of phases due to a large oscillation frequency.

This popular undergraduate quantum mechanics textbook is now available in a more affordable printing from Cambridge University Press. Unlike many other books on quantum mechanics, this text begins by examining experimental quantum phenomena such as the Stern-Gerlach experiment and spin measurements, using them as the basis for developing the theoretical principles of quantum mechanics. Dirac notation is developed from the outset, offering an intuitive and powerful mathematical toolset for calculation, and familiarizing students with this important notational system. This non-traditional approach is designed to deepen students' conceptual understanding of the subject, and has been extensively class tested. Suitable for undergraduate physics students, worked examples are included throughout and end of chapter problems act to reinforce and extend important concepts. Additional activities for students are provided online, including interactive simulations of Stern-Gerlach experiments, and a fully worked solutions manual is available for instructors.

We show that it is possible to explain the quantum measurement process within the framework of quantum mechanics without any additional postulates. The key concept of the theory is decoherence, which appears as an inherent characteristic of quantum mechanics and results from the uncertainty relation. In contrast to environment-induced decoherence, this decoherence exists prior to a measurement being made. To clarify our idea, we examine three elemental experiments: a Stern-Gerlach-like experiment, the Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) experiment, and the double-slit experiment. By considering the first experiment, we explain how the uncertainty relation between position and momentum introduces decoherence prior to measurement. Consideration of the EPR-Bohm experiment leads us to conclude that the correlation of the EPR pair is not a consequence of what is known as the collapse of the wave function. Our theory of decoherence can also be applied to experiments with continuous eigenvalues, such as the double-slit experiment. Consideration of the double-slit experiment leads us to understand how pure quantum mechanics describes the fact that quanta behave as interfering particles.