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arXiv:quant-ph/0703183 · Journal of Physics A Mathematical and Theoretical

Aspects of nonideal Stern–Gerlach experiment and testable ramifications

Dipankar Home, A. K. Pan, Md. Manirul Ali, A. S. Majumdar

Published Oct 31, 2007• 32 citations

Abstract

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.

DOI: 10.1088/1751-8113/40/46/010

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The Stem-Gerlach (SG) experiments considered as a quantum measurement process measuring the spin states of particles from their spatial distribution. The wave packet collapse or quantum decoherence can he described as a dynamical evolution governed by the interaction between the space degrees of freedom and the spin degrees of freedom. The analytic solution of this exactly solvable model shows that the decaying process of coherence with time in the SG experiment obeys no longer the linear exponential law.

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.

The Stern-Gerlach experiment is the fundamental experiment in order to exhibit the quantization of spin and understand the measurement problem in quantum mechanics. However, although the Stern-Gerlach experiment plays an essential role in the teaching of quantum mechanics, no complete analysis of this experiment using Pauli spinors is presented in the pedagogical literature. This paper presents such an analysis and develops implications for the theory of quantum measurement. We first propose an analytic expression of both the wave function and the probability density in the Stern-Gerlach experiment. Our explicit solution is obtained via a complete integration of the Pauli equation over time and space. The probability density evolution describes a slipping of the wave packet into two separate packets due to the measurement device, but it cannot account for impacts. We therefore calculate the de Broglie-Bohm trajectories, which not only explain impacts naturally, but also accounts for the spin quantization following the magnetic field gradient. It is then possible to propose a clear explanation of measurement effects in the Stern-Gerlach experiment.

The theory of measurement is reasonably interpreted without any abrupt and noncausal change of a state, namely, with no “reduction of a wave packet”, in the framework of quantum mechanics. Though a process of measurement is causal in the usual quantum-mechanical sense, it is possible to utilize the concept of the “reduction of a wave packet” without any loss of compatibility. The Stern-Gerlach experiment is put to a careful study to show that the present interpretation is consistent with experiment.