As part of an effort to examine student understanding of expressions for probability in an upper-division spins-first quantum mechanics (QM) context, clinical think-aloud interviews were conducted with students following relevant instruction.Students were given various tasks to showcase their conceptual understanding of the mathematics and physics underpinning these expressions.The symbolic forms framework was used as a lens through which to analyze their understanding.Various symbol templates and conceptual schemata were identified, in Dirac and function notations, with multiple schemata paired with different templates.The overlapping linking suggests that defining strict template-schema pairs may not be feasible or productive for studying student interpretations of expressions for probability in upper-division QM courses.
As part of an effort to examine student understanding of expressions for probability in an upper-division spins-first quantum mechanics (QM) context, clinical think-aloud interviews were conducted with students following relevant instruction.Students were given various tasks to showcase their conceptual understanding of the mathematics and physics underpinning these expressions.The symbolic forms framework was used as a lens through which to analyze their understanding.Various symbol templates and conceptual schemata were identified, in Dirac and function notations, with multiple schemata paired with different templates.The overlapping linking suggests that defining strict template-schema pairs may not be feasible or productive for studying student interpretations of expressions for probability in upper-division QM courses.
The Born rule, which describes the formalism for determining probabilities, is one of the most fundamental postulates in quantum mechanics.This paper presents results from an investigation into how students apply the Born rule to determine probabilities for energy and position measurements.The investigation includes two stages with independent methods: a quantitative analysis of student written work and a qualitative analysis of student individual interviews.The data from written tasks suggest that after instruction many students have not developed a coherent model for determining probabilities that they can apply to observables regardless of whether the eigenvalues are discrete or continuous.Moreover, many students seem to lack a functional understanding of quantum states and inner products that allows them to translate between Dirac notation and wave function representation.These results motivate student interviews, which allow us to probe student reasoning in depth.Prior research suggests that various features of each notation used in quantum mechanics may have an impact on how students perform computations.We postulate that the features of quantum notations may also interact with student sensemaking.Therefore, we analyze student interviews through the lens of the structural features of quantum notations framework.In particular, we discuss how different structural features may facilitate or hinder student sensemaking about concepts relevant to determining probabilities.The results from both quantitative and qualitative data suggest that unsuccessfully differentiating between a wave function and its associated state vector in Dirac notation may be a primary barrier for students to develop a model for determining probabilities for discrete and continuous cases.
Dirac notation is often used in upper level quantum mechanics courses, but students struggle with this representation.To investigate the difficulties that advanced students (i.e., upper-level undergraduate and graduate students) have while translating state vectors in Dirac notation to wave functions in position and momentum representations, we administered free-response and multiple-choice questions and conducted individual interviews with students.We find that students display common difficulties with these topics.
The physics education research (PER) literature on quantum mechanics (QM) often acknowledges and minimizes what we refer to as the enigma of QM, viewing these aspects of QM as a challenge to overcome, rather than an inherent feature to be leveraged for learning. Using the overlapping theories of knowledge-in-pieces and resource theory as connected to an epistemological framing perspective, we focus on students’ descriptions of the enigma of QM and the epistemological frames that students view as relevant when approaching QM. Data comes from interviews with 10 undergraduate physics students who had learned quantum physics topics across different courses in the curriculum, ranging from sophomore-level modern physics to an advanced undergraduate-level QM course. To elicit students’ ideas on QM, the interview protocol emphasized their perspectives and empowerment when possible. Iterative rounds of qualitative analysis coded the data for how students described enigmatic aspects of QM, along with three epistemological frames students identified as their reasoning strategies: reasoning about scale, reasoning about the connections between QM and mathematics, and reasoning about the connections between QM and classical mechanics. The results document that students are aware that something about QM is unexplainable, an enigma to acknowledge. We argue that acknowledgment of the enigma of QM is a key part of what we refer to as thinking quantumly. Additionally, the epistemological frame of reasoning about scale has not been widely discussed in the QM-PER literature, which students use to distinguish QM from classical mechanics and to explain why mathematical models are needed. These results are important for amplifying students’ voices in how they approach enigmatic aspects of QM. Thus, our results begin to move the QM-PER literature beyond viewing enigmatic aspects of QM as obstacles to be corrected through refined instruction, toward an asset-based orientation as a source of meaningful insights for the learning process.
The Paradigms team at Oregon State University has developed a quantum mechanics curriculum aimed at middle division students that begins with a strong emphasis on using operators, matrices and Dirac notation to describe quantum systems. The curriculum begins with spin systems, and this content ordering relies on students being able to understand quantum mechanical operators, eigenstates and quantum measurement without prior instruction on wave functions. We have analyzed classroom and an interview video to identify resources students use when considering these quantum ideas. Identification of such resources will inform introductory curricula that are prerequisite to the quantum Paradigms and inform the development of Paradigms materials that will guide students to use these resources productively.
As part of a broad investigation of student understanding in physics, we have examined student ability with superposition throughout introductory and upper-division courses in physics.This research has focused on examining student ability to add and subtract vector quantities and the wave functions associated with quantum physics.We present results from a series of research tasks designed to probe student understanding of superposition in each of these contexts at various points in undergraduate instruction.In addition, we describe and discuss certain patterns in student reasoning that have been identified across the different tasks, contexts, and courses.
Not only is Quantum Mechanics (QM) conceptually rich, it is also a theory that physics students have found abstract and technically formidable. Nevertheless, compared to other classical topics of physics, university students’ understanding of QM has received minimal attention in the physics education literature. The principal purpose of this study was to characterize the variation in the ways that undergraduate physics students depict the basic concepts of QM and to extrapolate the results to scaffold possible changes to instructional practices at the university that provided the context for the study. In so doing, an adaptation of a developmental phenomenographic perspective was chosen. Empirically, the study was approached through in-depth interviews with 35 physics students from two Ethiopian governmental universities after they had been exposed to the traditional QM course for one-third of a semester. Interview responses were analyzed using phenomenographic approach where a picture of students’ depictions was established for each quantum concept by expounding the given responses. For each basic quantum concept addressed, the structure of the description categories was separately constructed, and overall, it was found that naive, quasi-classical ontology and/or variants of classical ways of visualization are dominant in students’ responses. For example, it was found that students’ depictions of the photon concept could be described with three distinct categories of description, which are (a) classical intuitive description, (b) mixed model description and (c) quasi-quantum model description. Similarly, the findings revealed that it is possible to establish three qualitatively different categories of description to picture students’ depictions of matter waves, namely, (a) classical and trajectory-based description, (b) an intricate blend of classical and quantum description and (c) incipient quantum model description. Likewise, it was found that students’ depictions of uncertainty principle can be described as: (a) uncertainty as classical ignorance, (b) uncertainty as measurement disturbance and (c) uncertainty as a quasi-quantum principle. \nWith regard to learning QM, the categories of description made clear several issues: most students did not have enough knowledge to depict the basic concepts of QM properly; they were influenced by the perspective of classical physics and their perceptions in making explanations about QM; and they also applied mixed ideas, one based on their classical model and the other from newly introduced QM. These results are also supported by the findings of previous studies in similar domains. Findings from the study were used to guide the design of multiple representations-based instructions and interactive learning tutorials on the conceptual aspects of QM that has been shown to address specific difficulties identified in the study. Theoretical and practical implications of the study, as well as potential future considerations are drawn.