Framework for understanding the patterns of student difficulties in quantum mechanics

Learning advanced physics, in general, is challenging not only due to the\nincreased mathematical sophistication but also because one must continue to\nbuild on all of the prior knowledge acquired at the introductory and\nintermediate levels. In addition, learning quantum mechanics can be especially\nchallenging because the paradigms of classical mechanics and quantum mechanics\nare very different. Here, we review research on student reasoning difficulties\nin learning upper-level quantum mechanics and research on students'\nproblem-solving and metacognitive skills in these courses. Some of these\nstudies were multi-university investigations. The investigations suggest that\nthere is large diversity in student performance in upper-level quantum\nmechanics regardless of the university, textbook, or instructor and many\nstudents in these courses have not acquired a functional understanding of the\nfundamental concepts. The nature of reasoning difficulties in learning quantum\nmechanics is analogous to reasoning difficulties found via research in\nintroductory physics courses. The reasoning difficulties were often due to\nover-generalizations of concepts learned in one context to another context\nwhere they are not directly applicable. Reasoning difficulties in\ndistinguishing between closely related concepts and in making sense of the\nformalism of quantum mechanics were common. We conclude with a brief summary of\nthe research-based approached that take advantage of research on student\ndifficulties in order to improve teaching and learning of quantum mechanics.\n

At large research universities, physics graduate teaching assistants (TAs) are often responsible for grading in courses at all levels. However, few studies have focused on TAs’ grading practices in introductory and advanced physics courses. This study was designed to investigate whether physics graduate TAs grade students in introductory physics and quantum mechanics using different criteria and if so, why they may be inclined to do so. To investigate possible discrepancies in TAs’ grading approaches in courses at different levels, we implemented a sequence of instructional activities in a TA professional development course that asked TAs to grade student solutions of introductory physics and upper-level quantum mechanics problems and explain why, if at all, their grading approaches were different or similar in the two contexts. We analyzed the differences in TAs’ grading approaches in the two contexts and discuss the reasons they provided for the differences in their grading approaches in introductory physics and quantum mechanics in individual interviews, class discussions, and written responses. We find that a majority of the TAs graded solutions to quantum mechanics problems differently than solutions to introductory physics problems. In quantum mechanics, the TAs focused more on physics concepts and reasoning and penalized students for not showing evidence of understanding. The findings of the study have implications for TA professional development programs, e.g., the importance of helping TAs think about the difficulty of a problem from an introductory students’ perspective and reflecting on the benefits of formative assessment.

Solving problems presented in multiple representations is an important skill for future physicists and engineers. However, such a task is not easy for most students taking introductory physics courses. We conducted teaching/learning interviews with 20 students in a first‐semester calculus‐based physics course on several topics in introductory mechanics. These interviews helped identify the common difficulties students encountered when solving physics problems posed in multiple representations as well as the hints that help students overcome those difficulties. We found that most representational difficulties arise due to the lack of students’ ability to associate physics knowledge with corresponding mathematical knowledge. Based on those findings, we developed, tested and refined a set of problem‐solving exercises to help students learn to solve problems in graphical and equational representations. We present our findings on students’ common difficulties with graphical and equational representations, the problem‐solving exercises and their impact on students’ problem solving abilities.

When learning physics, students need more than just an understanding of mathematical and physical concepts. Integrating the two fields is crucial, as research indicates that students often struggle even when they have a strong grasp of both. In this paper, we use the heat equation as an example from higher education. Given the importance of the gradient and the Laplacian in the heat equation, as well as fluency and flexibility in using multiple external representations, this study investigates students’ reasoning about the gradient and the Laplacian of a function within different function representations in mathematics and physics contexts. To do so, we designed isomorphic open-response questions, administered as paper-and-pencil tests to 190 first-year students who were taking calculus-based introductory physics courses and multivariable calculus courses during their second semester. These students were enrolled in physics, mathematics, engineering, and bioengineering programs. We found that students face conceptual challenges with both the gradient and the Laplacian. Student difficulties with interpreting the gradient may be related to an incomplete understanding of steepness in graphical representations of functions of two independent variables. Moreover, their understanding of the Laplacian is limited. While context did not significantly affect students’ reasoning in terms of the types of difficulties they encountered, the proportion of students facing these difficulties, and their tendency to link or discuss more than one aspect of a concept, we did find context-specific prevalence of graphical, algebraic-symbolic, or linguistic descriptions. Students used linguistic descriptions more frequently in a physics context. We observed varying abilities to express the meaning of concepts in different answer representations. Linguistic descriptions of concepts proved particularly challenging for both concepts and contexts. Our findings provide insight into students’ understanding of two concepts related to the 2D heat equation.

Creating equitable learning environments has been an area of significant focus for physics education researchers in recent years. Here we introduce a framework that posits that grade penalty is a measure of academic self-concept and investigate if there are gender differences in grade penalties in physics courses for students majoring in physics. In order to quantify grade penalty, we define grade anomaly as the difference between a student’s grade in a course under consideration and their grade point average (GPA) in all other classes thus far. A grade anomaly lower than students expected grade based on their GPA is a grade penalty and higher than expected average grade is a grade bonus. Our framework posits that since women have traditionally been marginalized in physics, female physics majors are more likely to be negatively impacted by a grade penalty in their courses since their academic self-concept as a physics major hinges on them securing a certain grade. In the study presented here, we examine the average grade anomalies across a number of courses for female and male physics majors. We find that these students received grade penalties in almost all physics courses studied, though there were grade bonuses in a few laboratory courses. We also find that in physics courses, on average, women often had larger grade penalties than men, especially in introductory courses. We hypothesize that, because their grade penalties are often larger than men’s, women’s decisions to pursue a physics major and career may be particularly affected by grade penalties received in their various courses. Furthermore, the grade penalty measure can be easily computed by the physics programs concerned with equity.Received 19 January 2022Accepted 11 July 2022DOI:https://doi.org/10.1103/PhysRevPhysEducRes.18.020127Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDiversity & inclusionLearning environmentLearning theoryProfessional TopicsLower undergraduate studentsUpper undergraduate studentsPhysics Education Research

We present a method based on the scattering T operator, and conservation of net real and reactive power, to provide physical bounds on any electromagnetic design objective that can be framed as a net radiative emission, scattering or absorption process. Application of this approach to plane-wave scattering from an arbitrarily shaped, compact body of homogeneous electric susceptibility χ is found to predictively quantify and differentiate the relative performance of dielectric and metallic materials across all optical length scales. When the size of a device is restricted to be much smaller than the wavelength (a subwavelength cavity, antenna, nanoparticle, etc.), the maximum cross-section enhancement that may be achieved via material structuring is found to be much weaker than prior predictions: the response of strong metals (Reχ≪0) exhibits a diluted (homogenized) effective medium scaling ∝χ/Imχ; below a threshold size inversely proportional to the index of refraction (consistent with the half-wavelength resonance condition), the maximum cross-section enhancement possible with dielectrics (Reχ>0) shows the same material dependence as Rayleigh scattering. In the limit of a bounding volume much larger than the wavelength in all dimensions, achievable scattering interactions asymptote to the geometric area, as predicted by ray optics. For representative metal and dielectric materials, geometries capable of scattering power from an incident plane wave within an order of magnitude (typically a factor of two) of the bound are discovered by inverse design. The basis of the method rests entirely on scattering theory and can thus likely be applied to acoustics, quantum mechanics, and other wave physics.Received 27 March 2020Revised 17 June 2020Accepted 10 July 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.033172Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasElectromagnetismGeometrical & wave opticsLight propagation, transmission & absorptionMetamaterialsMie scatteringPhotonicsPlasmonicsScattering theoryAtomic, Molecular & OpticalGeneral Physics

Randomness is a defining element of mixing processes in nature and an essential ingredient to many protocols in quantum information. In this work, we investigate how much randomness is required to transform a given quantum state into another one. Specifically, we ask whether there is a gap between the power of a classical source of randomness compared to that of a quantum one. We provide a complete answer to these questions, by identifying provably optimal protocols for both classical and quantum sources of randomness, based on a dephasing construction. We find that in order to implement any noisy transition on a d-dimensional quantum system it is necessary and sufficient to have a quantum source of randomness of dimension d or a classical one of dimension d. Interestingly, coherences provided by quantum states in a source of randomness offer a quadratic advantage. The process we construct has the additional features to be robust and catalytic; i.e., the source of randomness can be reused. Building upon this formal framework, we illustrate that this dephasing construction can serve as a useful primitive in both equilibration and quantum information theory: We discuss applications describing the smallest measurement device, capturing the smallest equilibrating environment allowed by quantum mechanics, or forming the basis for a cryptographic private quantum channel. We complement the exact analysis with a discussion of approximate protocols based on quantum expanders deriving from discrete Weyl systems. This gives rise to equilibrating environments of remarkably small dimension. Our results highlight the curious feature of randomness that residual correlations and dimension can be traded against each other.Received 13 June 2018Revised 21 August 2018DOI:https://doi.org/10.1103/PhysRevX.8.041016Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum channelsQuantum correlations in quantum informationQuantum cryptographyQuantum quenchThermodynamicsQuantum InformationInterdisciplinary Physics

Analog quantum simulators, which efficiently represent model systems, have the potential to provide new insight toward naturally occurring phenomena beyond the capabilities of classical computers. Incorporating dissipation as a resource unlocks a wider range of out-of-equilibrium processes such as chemical reactions. Here, we operate a hybrid qubit-oscillator circuit quantum electrodynamics simulator and model nonadiabatic molecular dynamics through a conical intersection. We identify dephasing of the electronic qubit as the mechanism that drives wave-packet branching when the corresponding oscillator undergoes large amplitude motion. Furthermore, we directly observe enhanced branching when the wave-packet passes through the conical intersection. Thus, the forces that influence a chemical reaction can be viewed from the perspective of measurement backaction in quantum mechanics—there is an effective measurement-induced dephasing rate that depends on the position of the wave packet relative to the conical intersection. Our results set the groundwork for more complex simulations of chemical dynamics using quantum simulators, offering deeper insight into the role of dissipation in determining macroscopic quantities of interest such as the quantum yield of a chemical reaction.5 MoreReceived 16 May 2022Revised 1 November 2022Accepted 19 December 2022DOI:https://doi.org/10.1103/PhysRevX.13.011008Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum information processing with continuous variablesQuantum simulationQuantum Information

The evaluation of hypotheses, and the ability to learn from critical reflection on experimental and theoretical tests of those hypotheses, is central to an authentic practice of physics. A large part of physics education therefore seeks to help students understand the significance of this kind of reflective practice and to develop the strategies required to accurately update their belief in the utility of various hypotheses and models. Prior work has introduced Bayesian updating activities as one potential means for cultivating such reflective practice within the context of introductory physics courses. These activities are not fixed pieces of curricular matter, but are better thought of as codified practices that are incorporated within lectures, labs, homework, and exams, and which are adaptable to a broad range of course formats. The Bayesian updating activities engage students in the use of hypothetico-deductive reasoning to test a hypothesis or model, followed by Bayesian updating at the conclusion of this test to update their subjective confidence in the hypothesis or model that was tested. Prior work has identified significant gains in pre- and post comparison of student scores on the Epistemological Beliefs Assessment for Physical Science (EBAPS) in introductory algebra-based courses. Here, we conduct a quasi-experimental study of the impact of Bayesian updating activities on student EBAPS scores in introductory calculus-based courses. Our analysis examines the impact to the overall EBAPS score, the subscores for each of the five original axes identified by the authors of the EBAPS, and the subscores for five alternative axes that were recently identified in other work via factor analysis of student responses [Johnson and Willoughby, Phys. Rev. Phys. Educ. Res. 14, 010135 (2018)]. The results of our analysis show meaningful and credible gains on the overall EBAPS scores as well as for a multitude of the subscores. These gains are noteworthy due to their strength and their ability to be achieved with activities that were implemented as relatively minor alterations to a traditional course structure.Received 13 August 2019DOI:https://doi.org/10.1103/PhysRevPhysEducRes.16.010101Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEpistemology, attitudes, & beliefsInstructional materials developmentInstructional strategiesPhysics Education Research

Written quiz responses of 653 students in three separate courses are analyzed in detail. There has been relatively little research on student learning of thermodynamics in physics courses at the university level. A recent study by Loverude et al. has made it evident that students at the introductory level (and beyond) face many significant difficulties in learning fundamental thermodynamic concepts such as the first law of thermodynamics. I have been engaged in an ongoing project with T. J. Greenbowe to investigate student learning of thermodynamics in both physics and chemistry courses. As part of that investigation, a short diagnostic quiz has been administered over the past two years in the calculus-based introductory physics course at Iowa State University (ISU). This quiz focuses on heat, work, and the first law of thermodynamics. At ISU, thermodynamics is studied at the end of the second semester of the twosemester sequence in calculus-based introductory general physics. This course is taught in a traditional manner, with large lecture classes (up to 250 students), weekly recitation sections (about 25 students), and weekly labs taught by graduate students. Homework is assigned and graded every week. Thermal physics comprises 18-20% of the course coverage, and includes a wide variety of topics such as calorimetry, heat conduction, kinetic theory, laws of thermodynamics, heat engines, entropy, etc. The diagnostic quiz used in this study is shown below; it has been administered in three separate classes. The version shown here was administered in May 2001; the other two versions (December 1999 and December 2000) had very minor variations from the one shown here. (There were one or two additional questions on these quizzes which are not discussed here.) The 1999 and 2000 classes were taught by the same instructor, using a different textbook in each course. The 2001 course was taught by a different instructor, using the same text that was employed in the 1999 course. Both instructors are very experienced and have taught introductory physics at ISU for many years. The quiz was administered in two different ways: in 1999 and 2001, it was given as a practice quiz in the final recitation session (last week of class). In almost all cases it was ungraded; one instructor used it as a graded quiz. In 2000 the quiz was administered as an ungraded practice quiz in the very last lecture class of the year. This p-V diagram represents a system consisting of a fixed amount of ideal gas that undergoes two different processes in going from state A to state B:

Research has been conducted relating to the use of representation in physics learning. The aim of the research is to identify the difficulties of students in using vector diagrams in solving the problems of various free body systems, then to determine the effect of the students’ prior knowledge on the implementation of representation in physics learning. The study was conducted on two groups of students, who only learned Introductory Physics and students who had studied Introductory Physics and Mechanics. The findings of the study is almost no difference between groups of students who only have Introductory Physics and groups of students who have learned the knowledge of Introductory Physics and Mechanics. More than 75% of students cannot correctly draws vector diagrams of weight and weight projections in inclined plane, normal force, friction forces and tension. The implication of the research is the need to organize the physics education curriculum for freshman students at the university which gives sufficient prior knowledge on Vector Analysis, especially students in physics departement and engineering departement.

The lack of diversity and the under-performance of underrepresented students in STEM courses have been the focus of researchers in the last decade. In particular, many hypotheses have been put forth for the reasons for the under-representation and under-performance of women in physics. Here, we present a framework for helping all students learn in science courses that takes into account four factors: (1) the characteristics of instruction and learning tools, (2) student characteristics, (3) implementation of instruction and learning tools, and (4) the students’ environments. While there has been much research on factor 1 (characteristics of instruction and learning tools), there has been less focus on factor 2 (students’ characteristics, and in particular, motivational factors). Here, we focus on the baseline characteristics of introductory physics students obtained from survey data to inform factor 2 of the framework. A longitudinal analysis of students’ motivational characteristics in two-semester introductory physics courses was performed by administering pre- and post-surveys that evaluated students’ self-efficacy, grit, fascination with physics, value associated with physics, intelligence mindset, and physics epistemology. We found that female students reported lower levels of self-efficacy, fascination, and value associated with physics, and held a more “fixed” view of intelligence in the context of physics compared to male students. Female students’ fascination and value associated with physics decreased significantly more than males’ after an introductory physics course sequence. In addition, females’ view of physics intelligence became more “fixed” compared to males’ by the end of an introductory physics course sequence. Grit was the only factor on which females reported averages that were equal to or higher than males throughout introductory physics courses. The findings inform the framework and have implications for the development and implementation of effective pedagogies and learning tools to help all students learn.

Quantum mechanics is generally an abstract theory that challenges intuition since it cannot be understood using ordinary experiences. Quantum tunneling is one of the topics which is more counterintuitive and challenges students' conceptual understanding. The study was aimed at investigating if final year undergraduate quantum mechanics students have alternative conceptions regarding quantum tunneling phenomena after a full course of instruction. In this study, students’ alternative conceptions are referred to as any student ideas related to the incorrect belief of scientific principles. Using the deep structured interviews involving 36 students who were registered in a quantum physics course from the University of Malawi, we found that some students hold on to the idea that the energy of a quantum tunneling electron is lost, the wave function graph depicts the physical wave, and the “potential step,” and “barrier” means a physical object/barrier. The study further established that the possible sources of these alternative conceptions emanated from students’ experience with classical tunneling / classical mechanics, quantum mechanics lecturer notes, and previous quantum mechanics lessons. The findings of this study further portray that traditional pedagogy has its limitation in facilitating students' comprehension of quantum tunneling. We recommend that further research into the design of instructional technology for quantum physics should consider these students' alternative conceptions when designing visual, interactive, and mental models, which may help establish students' conceptual understanding of quantum mechanics.

Finding the best ways to introduce quantum physics to undergraduate students in all scientific areas, in particular for chemistry students, is a pressing, but hardly a simple task. In this paper, we discuss the relevance of taking into account lessons from the history of the discipline and the ongoing controversy over its interpretations and foundations in the search for new ways of improving the teaching of quantum physics. We also review and discuss the recent research in science education literature that proposes new ways of introducing quantum mechanics for undergraduate students. From these discussions, we suggest some possibilities – the inclusion of philosophical interpretations and their defense; the emphasis on strictly quantum features of the systems; an emphasis on formalism, without worrying about the ultimate ontological status of mathematics; the incorporation of quantum mechanics applications into real problems; and the need to introduce complementarity when using images – which can be taken into account when devising more effective ways of teaching introductory quantum mechanics for chemistry students.

Computational thinking in physics has many different forms, definitions, and implementations depending on the level of physics or the institution it is presented in. To better integrate computational thinking in introductory physics, we need to understand what physicists find important about computational thinking in introductory physics. We present a qualitative analysis of 26 interviews asking academic (N_a=18) and industrial (N_i=8) physicists about the teaching and learning of computational thinking in introductory physics courses. These interviews are part of a long-term project toward developing an assessment protocol for computational thinking in introductory physics. We find that academic and industrial physicists value students’ ability to read code and that (or ) and spreadsheets were the preferred computational language or environment used. Additionally, the interviewees mentioned that identifying the core physics concepts within a program, explaining code to others, and good program hygiene (i.e., commenting and using meaningful variable names) are important skills for introductory students to acquire. We also find that while a handful of interviewees note that the experience and skills gained from computation are quite useful for student’s future careers, they also describe multiple limiting factors of teaching computation in introductory physics, such as curricular overhaul, not having “space” for computation’, and student rejection. The interviews show that while adding computational thinking to physics students’ repertoire is important, the importance really comes from using computational thinking to learn and understand physics better. This informs us that the assessment we develop should only include the basics of computational thinking needed to assess introductory physics knowledge. Published by the American Physical Society 2024