Mathematical reasoning in undergraduate quantum mechanics: eigenvalue equations and probability expressions

The ability to relate physical concepts and phenomena to expressions in multiple mathematical representations is a crucial outcome of physics instruction, particularly in upper-division quantum mechanics (QM). Students are required to work with various symbolic notations, some of which they may not have previously encountered. Developing the ability to generate and translate expressions in these notations is also essential. We have been investigating the ways that students reason about expressions commonly used in upper-division QM courses, particularly eigenvalue and equations and expressions associated with probability, as well as how their ability to generate and interpret expressions is impacted by multiple notations. Most of the work has taken place in an upper-division, "spins-first" QM course. One focus of research has been student interpretations of eigenequations, both in mathematics and QM contexts, particularly for position as the observable quantity in the transition from discrete to continuous quantities. Various frameworks were applied to the data, primarily symbolic forms and symbolic blending as well as mathematical sensemaking. We have also explored student-generated probability expressions in Dirac and wave function notations, employing symbolic forms to analyze how participants interpret and reason about these expressions. This analysis revealed multiple symbolic forms in both Dirac and wave function notations. Finally, in order to explore how students conceptualize expressions in Dirac and wavefunction notations in different instructional paradigms, a related study compared students' conceptual connections between these expressions in spins-first and wave functions-first courses. Network analysis and community detection techniques were used to compare the level of conceptual similarity between expressions as viewed by the students in both curricula. Excerpts of these studies, their findings and implications for instruction will be discussed.