Abstract:  Quantum mechanics (QM) is a particularly challenging subject for upper-level undergraduate and graduate students in physics. There have been a number of research studies aimed at investigating student reasoning in QM and improving student understanding of QM. However, there have been relatively few investigations into student difficulties with fundamental concepts involving degenerate perturbation theory (DPT) or a system of identical particles. Through researching students' understanding and reasoning about DPT or a system of identical particles, I have found many common student difficulties that may hinder their development of a consistent and coherent knowledge structure pertaining to this topic. These common student difficulties were then used as a guide to help develop a research-based learning tool, i.e., a Quantum Interactive Learning Tutorial (QuILT). I describe the methodology for investigating these student difficulties along with the development and evaluation of the corresponding research-based QuILT that strives to help students develop a functional understanding of the fundamental concepts involved in constructing the many-particle stationary state wavefunction for a system of identical particles.

The QuILT incorporates guided inquiry-based learning sequences which consist of a set of questions, each building upon the previous question(s) that require the students to take a stand and actively engage in the learning process. The QuILT also includes hypothetical student conversations in which the students must analyze each hypothetical student's statement to determine whether they are correct and explain why they agree or disagree with each student. Many of the common student difficulties were used as a guide when constructing these hypothetical conversations and inquiry-based sequences with the goal being that students would identify an inconsistency in their reasoning and then use the provided support to reconcile these inconsistencies. For example, there are a number of hypothetical student conversations in which one or more students make statements reflecting these common difficulties and provide incorrect reasoning mirroring those given by actual students. Other students in these hypothetical conversations disagree with their incorrect reasoning and provide correct reasoning and often note an issue with the incorrect statement(s). As the students work through the QuILT, they must consider each student's argument and reflect upon their own reasoning in order to determine which student(s) are correct. Similarly, the guided inquiry-based sequences often include excerpts that strive to present the students with a contradiction between the answer to the questions in the sequence and their prior knowledge that they must then reconcile. Checkpoints are provided at the end of each section that allow the students to go back and reconcile any remaining difference between the correct reasoning and their own reasoning before moving on the next section.