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American Journal of Physics

The Spinning Nucleus in a Magnetic Field: A Source of Classroom Problems

William J. Archibald

Published Sep 1, 1952• 2 citations
Publisher

Abstract

The object of this paper is to call to the attention of those teaching quantum mechanics several of the problems arising from the behavior of nuclei possessing a magnetic moment in an external magnetic field. These problems are simple enough to serve as student exercises. They are presented both from the classical and quantum-mechanical points of view, and this is particularly instructive in connection with the problems considered because both techniques of calculation lead to the same answers, but the results in spite of formal similarity are to be interpreted quite differently. This provides the teacher with an excellent opportunity to discuss the conceptual differences of the old and new mechanics.

DOI
10.1119/1.1933238

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The object of this paper is to call to the attention of those teaching quantum mechanics several of the problems arising from the behavior of nuclei possessing a magnetic moment in an external magnetic field. These problems are simple enough to serve as student exercises. They are presented both from the classical and quantum-mechanical points of view, and this is particularly instructive in connection with the problems considered because both techniques of calculation lead to the same answers, but the results in spite of formal similarity are to be interpreted quite differently. This provides the teacher with an excellent opportunity to discuss the conceptual differences of the old and new mechanics.
The purpose of this tutorial is to derive the precessional characteristics of the magnetic moments of hydrogen nuclei in the presence of a constant external magnetic field using the Dirac bra-ket formulation of quantum mechanics (QM). This behaviour has many applications, most notably in nuclear magnetic resonance (NMR) and magnetic resonance (MR) imaging. Many NMR and MR imaging textbooks claim that the QM expectation value of the magnetic moment of a proton in a magnetic field reduces to the classical picture of a precessing magnetic dipole. This paper validates this conclusion by reducing the cumbersome QM integrals using Dirac notation and matrix algebra, and then comparing this result with the classical picture. It illustrates the connections between the quantum and classical pictures and demonstrates quantitatively how they differ and how they are similar. This tutorial targets students and researchers interested in the fundamental physics behind the MR phenomenon and assumes that the reader has a basic understanding of QM. This may be helpful for undergraduate QM students learning about spin angular momentum and Dirac notation.
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Recent years have seen increased interest towards quantum physics education research, and the topic has been studied from many perspectives both empirically and theoretically. The consensus of previous studies has been that the theory poses an exceptional challenge for learning and teaching, warranting more attention and further research. This dissertation aims to contribute to this discussion by asking what the underlying reasons for these challenges are and how they could potentially be alleviated. On one hand, the topic is approached from an interpretational point of view, considering the ontological foundations of the theory. It is argued that many of the fundamental difficulties in learning and teaching quantum mechanics stem from the ontological foundations of the theory. On the other hand, the historical development and the mathematical structure of the theory are discussed from an educational point of view. The dissertation consists of five articles that explore the above themes from different perspectives. The first study considers the development of one historical formulation of quantum mechanics, namely matrix mechanics. In general, the historical development of quantum theory has been complex and meandering, and its results may seem confusing to students. In our study, we discuss the use of matrix mechanics in quantum mechanics instruction to provide students with a clearer picture of the theory, its historical development, and the relations between its various formulations and used formalisms. The second and third studies add to the discussion on the roles of mathematics in physics and physics education. In quantum mechanics, the role of mathematics is especially pronounced, as the understanding of the theory heavily relies on its mathematical framework. Therefore, we argue that the focus of quantum mechanics instruction should be shifted away from calculational aspects to a more integrated approach that treats the conceptual and mathematical sides of the theory intertwined. In our studies, we provide an overview of the roles assigned to mathematics in previous physics education research literature and present our own layered characterisation of the roles, aiming at diversifying the discussion on the roles of mathematics within the physics education research community and in physics instruction. The fourth and fifth studies empirically examine the challenges of quantum mechanics instruction in the context of an introductory quantum mechanics course. In the fourth study, we explore the effect of initial mathematics knowledge and self-efficacy on student learning, while in the fifth study, we consider the effects of the instructional approach and different teaching strategies on student conceptual knowledge and self-efficacy. The specific focus is on the impact of implementing a new instructional approach, the so-called spin first approach, that emphasises the mathematical basis and conceptual understanding of the theory instead of repeated mathematical calculations. Our results show that the spin-first approach, combined with an active learning model, is able to foster a high level of student self-efficacy and conceptual learning. Quantum mechanics is one of the most successful theories in the history of physics. Nevertheless, learning and teaching its basic principles continue to pose significant challenges for both instructors and students. In this dissertation, we argue for an approach to teaching the theory that places greater emphasis on its mathematical foundations and treats physics and mathematics as intertwined disciplines.