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European Journal of Physics

Models for complementarity

A Juhasz, George Marx

Published Jul 1, 1983• 0 citations
Publisher

Abstract

Understanding complementarity is a challenge for college students with modest mathematical experience. In the authors' paper noncommuting operators have been introduced in a three-dimensional 'Hilbert space'. Later an electronic model of the quantum-mechanical measurement has been described.
DOI
10.1088/0143-0807/4/3/005

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In any course of quantum mechanics it is necessary to explain the behaviour of incompatible dynamical variables when measurements are made of them and also the noncommutation of their corresponding operators. At the outset this presents the student with considerable conceptual difficulties since these ideas are quite outside his everyday experience. The authors present a simple model, an electronic device designed to simulate some of the features of incompatible observables, which can play a useful role in aiding the student to digest the mathematics of noncommuting operators and their eigenfunctions as they appear in an introductory course.
The Stern–Gerlach experiment can play an important role in teaching the formalism of quantum mechanics. In the context of a finite-dimensional Hilbert space students can learn how to prepare a specific quantum state starting from an arbitrary state, issues related to the time evolution of the wave function and quantum measurement. The Stern–Gerlach experiment can also be used to teach the distinction between the physical space where the experiment is performed and the Hilbert space where the state of the system lies, and how information about the state of the system in the Hilbert space can be exploited to interpret the possible outcomes of the experiment in physical space. Students can learn the advantages of choosing an appropriate basis to make predictions about the outcomes of experiments with different arrangements of Stern–Gerlach devices. The latter can also help students understand that an ensemble of identically prepared systems is not the same as a mixture. We discuss student difficulties with the Stern–Gerlach experiment based on written tests and interviews with advanced undergraduate and graduate students in quantum mechanics courses. We also discuss preliminary data which suggest that the Quantum Interactive Learning Tutorial on the Stern–Gerlach experiment is helpful in improving student understanding of these concepts.
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