Students' metarepresentational competence with matrix notation and Dirac notation in quantum mechanics

This article shares analysis regarding quantum mechanics students' metarepresentational competence (MRC) that is expressed as they engaged in solving an expectation value problem, which involves linear algebra concepts. The particular characteristic of MRC that is the focus of this analysis is students' critiquing and comparing the adequacy of representations, specifically matrix notation and Dirac notation, and judging their suitability for various tasks. With data of students' work during semistructured individual interviews, components of students' MRC were analyzed and categorized according to value-based preferences, problem-based preferences, and purpose and utility awareness. Detail is provided on two students who serve as paradigmatic examples of students' power and flexibility within different notation systems, and detail of a third student is given as a point of contrast. In addition to adapting MRC as a helpful construct for characterizing student understanding at the intersection of undergraduate mathematics and physics, we aim to demonstrate how students' rich understanding of linear algebra and quantum mechanics includes and is aided by their understanding and flexible use of different notational systems. For example, the problem-based preference aspects of MRC highlight that any particular problem-solving approach is itself intrinsically tied to a notational system. We suggest that any instruction with the goal of helping students develop a deep understanding of quantum mechanics and linear algebra should provide opportunities for students to use and improve their MRC.