Date of Award

Summer 8-18-2023

Level of Access Assigned by Author

Open-Access Dissertation

Degree Name

Doctor of Philosophy (PhD)




John R. Thompson

Second Committee Member

Saima Farooq

Third Committee Member

Michael D. Mason

Additional Committee Members

MacKenzie R. Stetzer

Thomas E. Stone


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 and an online survey was administered, all to students enrolled in upper-division quantum mechanics courses. The interviews were conducted at a single institution with a “spins-first” instructional approach, while the surveys were administered at ten institutions, including both “spins-first” and “wave functions-first” courses. The interviews and surveys focused on expressions for probabilities and their constituent terms. Analysis of student interviews used the symbolic forms framework to determine the ways that participants interpret and reason about these expressions. Survey responses were analyzed using network analysis techniques to determine the ways that students conceptualized these expressions and both whether and how they related analogous expressions between notations. Survey responses were also used to compare students’ understanding of these expressions and their relations between the two curricula studied.

Multiple symbolic forms—internalized connections between symbolic templates and their conceptual interpretations—were identified in both Dirac and wave function notation, suggesting that students develop an understanding of expressions for probability both in terms of their constituent pieces and as larger composite expressions. Network analysis techniques determined the relative strength of conceptual connections between expressions in different notations and were also used to understand the relative weight of different conceptualizations of expressions that share multiple possible interpretations. Comparative analysis between instructional approaches showed similarity in their conceptions of common expressions within quantum mechanics but also highlighted differences, such as a general preference for and better understanding of expressions in the notation taught first in their respective courses.

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