Student Understanding of Eigenvalue Equations in Quantum Mechanics: Symbolic Forms Analysis

Prior research on the use of mathematics in physics, primarily at the introductory level, has demonstrated that students think about mathematical tools, operations, and structures differently in mathematics and physics problem solving. Quantum mechanics is flush with opportunities to observe physics students use of more advanced mathematics. Eigentheory is central to the mathematization of quantum mechanics. As part of an effort to examine students’ mathematical sensemaking in a “spins-first quantum” mechanics course, students at two institutions were asked to construct an eigenvalue equation for a one-dimensional position operator on an in class written assignment. Sherin’s symbolic forms framework guided the deconstruction of student responses into symbol templates and conceptual schemata. Analysis yielded three symbolic forms for an eigenvalue equation, all sharing a single symbol template but with unique conceptual schemata, as well as an unproductive application of Sherin’s parts-of-a-whole form. Interview data complements a subset of these findings. Our results corroborate prior literature on a construction task rather than a comparison or deconstruction task, and with a continuous variable after instruction on discrete systems.