Structural features of algebraic quantum notations

Abstract

[This paper is part of the Focused Collection on Upper Division Physics Courses.] The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation. The varied features of these representations affect how computations are performed. For example, identifying probabilities of measurement outcomes for a state described in Dirac notation may involve identifying expansion coefficients by inspection, but if the state is described as a function, identifying those expansion coefficients often involves performing integrals. In this study, we focus on three notational systems: Dirac notation, algebraic wavefunction notation, and matrix notation. These quantum notations must include information about basis states and their associated complex probability amplitudes. In this theory paper, we identify four structural features of quantum notations, which we term individuation, degree of externalization, compactness, and symbolic support for computational rules. We illustrate how student reasoning interacts with these structural features with episodes from interviews with advanced undergraduate physics majors reasoning about a superposition state of an infinite square well system. We find evidence of the students coordinating different notations through the use of Dirac notation, using an expression in Dirac notation to guide their work in another notation. These uses are supported by the high degree of individuation, compactness, and symbolic support for computation and the moderate degree of externalization provided by Dirac notation.

Case study of a successful learner’s epistemological framings of quantum mechanics

“…Brookes and Etkina [101] analyzed the use of metaphors in QM (e.g., a "potential well") and hypothesized that a student could be "distracted by an overly literal interpretation." Gire and Price [102] showed evidence of student reasoning interacting with features of Dirac, wave function, and matrix notations. Modir and Sayer [61] examined students' group problem solving on a free particle problem and identify a mathematical resource called "pulling out" that aids them in sense making.…”

The use of ACER to develop and analyze student responses to expectation value problems

“…We look at the representations chosen through the lens of structural aspects of quantum notations [4]. In our questions, the representations and the methods used to solve for the expectation value are linked.…”

“…For example, it is possible to solve for the expectation value using matrix equations, Dirac notation, or integral expressions. In different contexts, one method may be preferred, but research shows that students tend to be more comfortable using a single notation and use it in most contexts [4]. Our study is designed to probe how students solve for expectation values of angular momentum operators and which representation they choose to use.…”

Sense-making with Inscriptions in Quantum Mechanics

“…Students who construct the diagram unprompted tend to do better at integrating the diagram with conceptual reasoning. Others, for example Gire et al [9] and Parnafes [10], have shown that different physical features within representations may serve different roles in student reasoning, providing support for computation or for conceptual ideas such as "fast." The picture that emerges from these studies is that representation use by physics learners is contextual and tied closely with their conceptual reasoning.…”