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Changing basis is a common task when solving quantum mechanical problems. As part of a research project investigating student understanding of basis and change of basis in quantum mechanics, we developed a tutorial to support students in learning about basis in the context of spin-1/2 systems. We have since created an interactive online version of the basis tutorial as part of a freely available suite of online quantum tutorials called ACE Physics (acephysics.net). The ACE Physics tutorials include dynamic guidance elements and, unlike other tutorials, are intended for use outside the classroom without instructor facilitation. After extensive study in an instructor-supported environment, we assigned the ACE Physics basis tutorial as homework in two semesters of upper-division quantum mechanics, and we report on the effectiveness of the activity based on pre-/post-testing and comparison of student exam performance with a similar semester that did not include the activity. We find that the tutorial produces sufficient learning gains to justify continued assignment as a homework problem in our classes.

We investigate student comfort with the material in an upper-division spins-first quantum mechanics course. Pre-lecture surveys probing students' comfort were administered weekly, in which students assigned themselves a "discomfort level" on a scale of 0--10 and provided a written explanation for their choice. The weekly class-wide average discomfort level was effectively constant over the semester, suggesting that the class found no single unit especially jarring nor especially easy. Student written responses were coded according to their reported source of discomfort---math, math-physics connection, physics, and notation. The relative prevalence of these categories varied significantly over the semester, indicating that students find that different units present different challenges, and also that some of these challenges fade in importance as the semester progresses. Semi-structured interviews with students in a similar quantum mechanics course at a different institution provided additional context and insight into these results., Comment: 4 pages, 3 figures, to be published in 2020 Physics Education Research Conference Proceedings

We investigate upper-division student difficulties with direct integration in multiple contexts involving the calculation of a potential from a continuous distribution (e.g., mass, charge, or current). Integration is a tool that has been historically studied at several different points in the curriculum including introductory and upper-division levels. We build off of these prior studies and contribute additional data around student difficulties with multi-variable integration at two new points in the curriculum: middle-division classical mechanics, and upper-division magnetostatics. To facilitate comparisons across prior studies as well as the current work, we utilize an analytical framework that focuses on how students activate, construct, execute, and reflect on mathematical tools during physics problem solving (i.e., the ACER framework). Using a mixed-methods approach involving coded exam solutions and student problem-solving interviews, we identify and compare students' difficulties in these two different context and relate them to what has been found previously in other levels and contexts. We find that some of the observed student difficulties were persistent accross all three contexts (e.g., identifying integration as the appropriate tool, and expressing the difference vector), while other difficulties seemed to fade as students advanced through the curriculum (e.g., expressing differential line, area, and volume elements). We also identified new difficulties that appear in different contexts (e.g., interpreting and expressing the current density)., Comment: 13 pages, 2 figures, submitted to Phys. Rev. PER