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[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Instructors teaching upper-division quantum mechanics have had two primary options when it comes to textbook choice and thus curriculum sequence: starting with wave functions and the Schrödinger equation, referred to as “wave functions-first;” and starting with discrete spin-1/2 systems and Dirac notation, known as “spins-first” courses. Given the very different structures of these courses, particularly as it pertains to the notations and formalisms both emphasized and used, it begs the question as to whether and to what extent students in these different courses conceptualize symbolic expressions in Dirac and wave function notations differently. To investigate this, online surveys were administered to students in spins-first courses at six institutions and in wave functions-first courses at four institutions. As a follow-up to a prior study focused on the results from the spins-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. Conceptual interpretations of individual expressions in both Dirac and wave function notations were also directly compared between the two populations. The primary difference observed between the two populations appears to lie in the way they interpret Dirac bras and kets: spins-first students were found to more strongly connect these expressions to vectorlike interpretations, while wave functions-first students were found to interpret them as more wave functionlike. This suggests that the choice of text and/or curricular style should be informed by the interpretation that best matches the goals of the instructor. 

[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] 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, developing the ability to generate and translate expressions in these notations is of great importance, and the extent to which students can relate these expressions to physical quantities and phenomena is crucial to understand. To investigate student understanding of the expressions used in these notations and the ways they relate, clinical think-aloud interviews were conducted with students enrolled in an upper-division quantum mechanics course. Analysis of these interviews used the symbolic forms framework to determine the ways that participants interpret and reason about these expressions. Multiple symbolic forms—internalized connections between symbolic templates and their conceptual interpretations—were identified in both Dirac and wave function notations, suggesting that students develop an understanding of expressions for probability both in terms of their constituent pieces and as larger composite expressions. 

One expected outcome of physics instruction is for students to be capable of relating physical concepts to multiple mathematical representations. In quantum mechanics (QM), students are asked to work across multiple symbolic notations, including some they have not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, a survey was developed and distributed to students at six different institutions. All of the courses studied were structured as “spins-first,” in which the course begins with spin-1/2 systems and Dirac notation before transitioning to include continuous systems and wave function notation. Network analysis techniques such as community detection methods were used to investigate conceptual connections between commonly used expressions in upper-division QM courses. Our findings suggest that, for spins-first students, Dirac bras and kets share a stronger identity with vectorlike concepts than are associated with quantum state or wave function concepts. This work represents a novel way of using well-developed network analysis techniques and suggests such techniques could be used for other purposes as well. 

As part of an effort to examine student understanding of expressions for probability in an upper-division spins-first quantum mechanics (QM) context, clinical think-aloud interviews were conducted with students following relevant instruction. Students were given various tasks to showcase their conceptual understanding of the mathematics and physics underpinning these expressions. The symbolic forms framework was used as an analytical lens. Various symbol templates and conceptual schemata were identified, in Dirac and function notations, with multiple schemata paired with different templates. The overlapping linking suggests that defining strict template-schema pairs may not be feasible or productive for studying student interpretations of expressions for probability in upper-division QM courses. 

One important outcome of physics instruction is for students to be capable of relating physical concepts and phenomena to multiple mathematical representations. In quantum mechanics (QM), students are asked to work between multiple symbolic notations, some not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, many of which describe analogous physical concepts, a survey was distributed to students enrolled in upper-division QM courses at multiple institutions. Network analysis techniques were shown to be useful for gaining information about how students relate these expressions. Preliminary analysis suggests that students view Dirac bras and kets as more similar to generic vectors than to their physically analogous wave function counterparts, and that Dirac bras and kets serve as a bridge between vector and wave function expressions.