Comparing student understanding of quantum mechanics notations and expressions across curricula

The ability to relate physical concepts and phenomena to multiple mathematical representations—and to move fluidly between these representations—is a critical outcome expected of physics instruction. In upper-division quantum mechanics, students must work with multiple symbolic notations, including some that they have not previously encountered. Thus, being able to translate expressions between these notations is an important skill to develop, and the extent to which students are able to relate these expressions is crucial to understand. To investigate student understanding of the relationships between expressions used in these various notations, a survey was distributed to students in upper-division quantum mechanics courses at multiple institutions over two years. These courses included those structured as "spins-first" and "wave functions-first." Network analysis techniques were used to investigate student understanding of common expressions used in these courses, in both Dirac and wave function notation. Comparing results between students in spins-first and wave function-first courses suggests that the former appear to conceptualize Dirac expressions as more vector-like than the latter, consistent with common instructional emphasis.