Investigating and improving student understanding of the probability distributions for measuring physical observables in quantum mechanics

Abstract

A solid grasp of the probability distributions for measuring physical observables is central to connecting the quantum formalism to measurements. However, students often struggle with the probability distributions of measurement outcomes for an observable and have difficulty expressing this concept in different representations. Here we first describe the difficulties that upper-level undergraduate and PhD students have with the probability distributions for measuring physical observables in quantum mechanics. We then discuss how student difficulties found in written surveys and individual interviews were used as a guide in the development of a quantum interactive learning tutorial (QuILT) to help students develop a good grasp of the probability distributions of measurement outcomes for physical observables. The QuILT strives to help students become proficient in expressing the probability distributions for the measurement of physical observables in Dirac notation and in the position representation and be able to convert from Dirac notation to position representation and vice versa. We describe the development and evaluation of the QuILT and findings about the effectiveness of the QuILT from in-class evaluations.

QMwebJS—An Open Source Software Tool to Visualize and Share Time-Evolving Three-Dimensional Wavefunctions

“…While assessing the student's overall learning efficiency with this software would have great value, such a cohort study is beyond the scope of the present work. Nonetheless, several studies are available that establish some foundations for how such a case study could be carried out (see References [7,24,25] for cases that make use of questionnaires to investigate the impact of different methodologies in the context of quantum physics education).…”

Exploring student preferences when calculating expectation values using a computational features framework

“…They also noted that students would often write the Dirac expression hψjÂjψi as a template for matrix or wave function notation. In other cases, students have been shown to have difficulty using and connecting the different notations [10,11].…”

Improving student understanding of corrections to the energy spectrum of the hydrogen atom for the Zeeman effect

“…There have been a number of prior research studies aimed at investigating student reasoning in QM [23][24][25][26][27][28][29][30][31][32][33][34] and using the findings as resources for improving student understanding [35][36][37][38][39][40][41][42][43]. Guided by research studies conducted to identify student difficulties with QM and findings of cognitive research, we have been developing a set of research-based learning tools including the Quantum Interactive Learning Tutorials (QuILTs), which strive to help students develop a solid grasp of QM [44][45][46][47][48][49][50][51][52][53][54][55][56]. However, there has been relatively little research that focuses on student understanding of advanced topics in quantum mechanics, e.g., degenerate perturbation theory (DPT) [57][58][59].…”