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American Journal of Physics

Quantum measurement theory and the Stern–Gerlach experiment

M. Hannout, S. Hoyt, A. Kryowonos, A. Widom

Published May 1, 1998• 15 citations
Publisher

Abstract

The Stern–Gerlach experiment was one of the early illustrative examples of the Bohr view of quantum mechanics. In the Bohr view, a measurement is an interaction between a quantum object and an apparatus which acts in a classical fashion. There are still open problems with this concept, many of which enter into the discussion of the original Stern–Gerlach measurement.

DOI
10.1119/1.18876

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The Stern–Gerlach experiment was one of the early illustrative examples of the Bohr view of quantum mechanics. In the Bohr view, a measurement is an interaction between a quantum object and an apparatus which acts in a classical fashion. There are still open problems with this concept, many of which enter into the discussion of the original Stern–Gerlach measurement.
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.
Following the approach to quantum measurement, proposed in an earlier paper, in which the apparatus is treated as a purely classical instrument to be described by the laws of classical mechanics we examine some simple experiments for the purpose of illustration. The Stern-Gerlach experiment is examined in detail, within our approach, and we see explicitly how the relevant information is transferred to the apparatus variables. We also examine a magnetic interferometer arrangement and a crossed-field arrangement. Consideration of these experiments leads us to the inclusion of an extra procedural step in our approach. This extra step is necessary so that an observer may measure the classical apparatus variables at will after the interaction is completed.
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.
The Stern–Gerlach (SG) experiment (1922) was one of the earliest experiments for which the result could not be explained using classical physics by any means. Hence it demonstrated unambiguously the importance of quantum effects. The SG experiment can be explained using the quantum theory for the spin-½ particle, which is a quantum system that can be represented simply by 2 × 2 matrices. We will use the spin-½ particle to introduce some of the basic concepts of quantum mechanics, such as state space, operators, measurement, the uncertainty principle, and the evolution equation. Furthermore, the theory for the spin-½ particle can be readily generalized to finite dimensional quantum systems, i.e., systems that can be represented by finite dimensional matrices.
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.
We develop further a new approach to the quantum measurement process. In this approach, which we proposed in a recent paper, the apparatus is treated as a classical system; however, the classical apparatus is directly coupled to the quantum system. A principle of integrity, which requires that the observables of the classical apparatus retain their classical integrity, is introduced. We examine the constraints which this principle places upon the coupling between the apparatus and the quantum system. To illustrate our approach we use a model loosely based on the Stern-Gerlach experiment. For this model we exhibit a coupling which satisfies the principle of integrity.