Comparative analysis of spins-first and wave functions-first students’ understanding of expressions in quantum mechanics

[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-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>1</a:mn><a:mo>/</a:mo><a:mn>2</a:mn></a:mrow></a:math> 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.