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Abstract Quantum information science is a rapidly growing interdisciplinary field that is attracting the attention of academics and industry experts alike. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of quantum information science, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a tutorial on the Bloch sphere, a useful visual tool for developing intuition about single quantum bits (qubits), which are the basic building block of any quantum computer. Students’ understanding was evaluated after they received traditional lecture-based instruction on the requisite topics, and again after engaging with the tutorial. We observe, analyze, and discuss their improvement in performance on concepts covered in the tutorial. 

Abstract Research-validated multiple-choice questions comprise an easy-to-implement instructional tool for scaffolding student learning and providing formative assessment of students’ knowledge. We present findings from the implementation of a research-validated multiple-choice question sequence on the basics of two-state quantum systems, including inner products, outer products, translation between Dirac notation and matrix representation in a particular basis, and change of basis. This study was conducted in an advanced undergraduate quantum mechanics course, in both online and in-person learning environments, across three years. For each cohort, students had their learning assessed after traditional lecture-based instruction in relevant concepts before engaging with the multiple-choice question sequence. Their performance was evaluated again afterward with a similar assessment and compared to their earlier performance. We analyze, compare, and discuss the trends observed in the three implementations. 

We describe a study focusing on students' and faculty members' reasoning about problems of differing cognitive complexity related to the double-slit experiment (DSE) with single particles. In the first phase of the study, students in advanced quantum mechanics courses were asked these questions in written form. Additionally, individual interviews were conducted with ten students in which they were asked follow-up questions to make their thought processes explicit on the challenging problems. Students did well on the straightforward problem, showing they had some knowledge of the DSE after traditional instruction, but they struggled on the more complex ones. Even if explicitly asked to do so in interviews, students were often uncomfortable performing calculations or making approximations and simplifications, instead preferring to stick with their gut feeling. In the second phase of the study, the problems were broken down into more pointed questions to investigate whether students had knowledge of relevant concepts, whether they would do calculations as part of their solution approach if explicitly asked, and whether they explicitly noted using their gut feeling. While the faculty members' responses suggest that they could seamlessly move between conceptual and quantitative reasoning, most students were unable to combine concepts represented by different equations to solve the problems quantitatively. We conclude with instructional implications. 

[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] We discuss how research on student difficulties was used as a guide to develop, validate, and evaluate a Quantum Interactive Learning Tutorial (QuILT) to help students learn how to determine the completely symmetric bosonic or completely antisymmetric fermionic wave function and be able to compare and contrast them from the case when the particles can be treated as distinguishable. We discuss how explicit scaffolding is designed via guided teaching-learning sequences for two- or three-particle bosonic and fermionic systems to help students develop intuition about how to construct completely symmetric and antisymmetric wave function, both when spin part of the wave function is ignored and when both spatial and spin degrees of freedom are included. Published by the American Physical Society2025 

We studied the challenges students face with time dependence in quantum mechanics, specifically in the context of Larmor precession of spin. This research informed the creation and evaluation of a learning tutorial aimed at helping students grasp these concepts using a two-state system. The tutorial utilizes visualization tools to enhance students' intuition and emphasizes the integration of qualitative and quantitative understanding. We also examine how students applied semi-classical or quantum mechanical reasoning in their answers and assess the improvement they showed after working with the tutorial. 

Traditional approaches to undergraduate-level quantum mechanics require extensive mathematical preparation, preventing most students from enrolling in a quantum mechanics course until the third year of a physics major. Here we describe an approach to teaching quantum formalism and postulates that can be used with first-year undergraduate students and even high school students. The only pre-requisite is a familiarity with vector dot products. This approach enables students to learn Dirac notation and core postulates of quantum mechanics at a much earlier stage in their academic career, which can help students prepare for careers in quantum science and engineering and advance the Second Quantum Revolution. 

[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Research-based multiple-choice questions implemented in class with peer instruction have been shown to be an effective tool for improving students’ engagement and learning outcomes. Moreover, multiple-choice questions that are carefully sequenced to build on each other can be particularly helpful for students to develop a systematic understanding of concepts pertaining to a theme. Here, we discuss the development, validation, and implementation of a multiple-choice question sequence (MQS) on the topic of quantum measurement in the context of wave functions in the infinite-dimensional Hilbert space. This MQS was developed using students’ common difficulties with quantum measurements as a guide and was implemented in a junior-/senior-level quantum mechanics course at a large research university in the U.S. We compare student performance on assessment tasks focusing on quantum measurement before and after the implementation of the MQS and discuss how different difficulties were reduced and how to further improve students’ conceptual understanding of quantum measurement in infinite-dimensional Hilbert space. Published by the American Physical Society2025 

We examine students’ challenges in determining the number of distinct many-particle stationary states for a system of noninteracting identical particles, focusing on how these insights guided the design, validation, and evaluation of a quantum interactive learning tutorial (QuILT) to aid students’ understanding. Specifically, we focus on systems with a fixed number of available single-particle states and particles, where the total energy is not fixed. The QuILT is designed to provide scaffolding support to help students learn these complex concepts more effectively. This study was conducted in advanced quantum mechanics courses, where written questions were administered to students in class following traditional instruction on the relevant concepts. Additionally, individual interviews were conducted with students to gain deeper insights. Our findings reveal that both upper-level undergraduate and graduate students face similar challenges in understanding these concepts. Additionally, difficulty with basic concepts in combinatorics that are necessary to answer the questions correctly was also found. The QuILT offers scaffolding support to help undergraduate and graduate students systematically reason through these concepts. Published by the American Physical Society2024 

[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] One hallmark of expertise in physics is the ability to translate between different representations of knowledge and use the representations that make the problem-solving process easier. In quantum mechanics, students learn about several ways to represent quantum states, e.g., as state vectors in Dirac notation and as wave functions in position and momentum representation. Many advanced students in upper-level undergraduate and graduate quantum mechanics courses have difficulty translating state vectors in Dirac notation to wave functions in the position or momentum representation and vice versa. They also struggle when translating the wave function between the position and momentum representations. The research presented here describes the difficulties that students have with these concepts and how the research was used as a guide in the development, validation, and evaluation of a Quantum Interactive Learning Tutorial (QuILT) to help students develop a functional understanding of these concepts. The QuILT strives to help students with different representations of quantum states as state vectors in Dirac notation and as wave functions in position and momentum representation and with translating between these representations. We discuss the effectiveness of the QuILT from in-class implementation and evaluation. Published by the American Physical Society2024