Normalization is a particularly important concept within quantum mechanics due to the probabilistic nature of quantum systems.However, students' understanding of normalization has not been an explicit focus in past studies.In this paper, I will present a preliminary framework for students' understanding of mathematical norms and normalization of vectors, using interviews with quantum mechanics students to illustrate how the framework can be used to model and make sense of students' reasoning about the normalization of vectors.
Normalization is a particularly important concept within quantum mechanics due to the probabilistic nature of quantum systems.However, students' understanding of normalization has not been an explicit focus in past studies.In this paper, I will present a preliminary framework for students' understanding of mathematical norms and normalization of vectors, using interviews with quantum mechanics students to illustrate how the framework can be used to model and make sense of students' reasoning about the normalization of vectors.
The perspectives of introductory classical physics students can often negatively influence how those students later interpret quantum phenomena when taking an introductory course in modern physics. A detailed exploration of student perspectives on the interpretation of quantum physics is needed, both to characterize student understanding of physics concepts, and to inform how we might teach traditional content. Our previous investigations of student perspectives on quantum physics have indicated they can be highly nuanced, and may vary both within and across contexts. In order to better understand the contextual and often seemingly contradictory stances of students on matters of interpretation, we interviewed 19 students from four introductory modern physics courses taught at the University of Colorado. We find that students have attitudes and opinions that often parallel the stances of expert physicists when arguing for their favored interpretations of quantum mechanics, allowing for more nuanced characterizations of student perspectives in terms of three key interpretive themes. We present a framework for characterizing student perspectives on quantum mechanics, and demonstrate its utility in interpreting the sometimes-contradictory nature of student responses to previous surveys. We further find that students most often vacillate in their responses when what makes intuitive sense to them is not in agreement with what they consider to be a correct response, underscoring the need to distinguish between the personal and the public perspectives of introductory modern physics students.
The perspectives of introductory classical physics students can often negatively influence how those students later interpret quantum phenomena when taking an introductory course in modern physics. A detailed exploration of student perspectives on the interpretation of quantum physics is needed, both to characterize student understanding of physics concepts, and to inform how we might teach traditional content. Our previous investigations of student perspectives on quantum physics have indicated they can be highly nuanced, and may vary both within and across contexts. In order to better understand the contextual and often seemingly contradictory stances of students on matters of interpretation, we interviewed 19 students from four introductory modern physics courses taught at the University of Colorado. We find that students have attitudes and opinions that often parallel the stances of expert physicists when arguing for their favored interpretations of quantum mechanics, allowing for more nuanced characterizations of student perspectives in terms of three key interpretive themes. We present a framework for characterizing student perspectives on quantum mechanics, and demonstrate its utility in interpreting the sometimes contradictory nature of student responses to previous surveys. We further find that students most often vacillate in their responses when what makes intuitive sense to them is not in agreement with what they consider to be a correct response, underscoring the need to distinguish between the personal and the public perspectives of introductory modern physics students.
Most engineers need a basic understanding of quantum mechanics, and for this the typical college introduction is enough. Such an introduction tends to be axiomatic but physically unenlightening. The rules of quantum mechanics (QM) are a fait accompli; justified because they work. This is a good beginning, but those needing to learn more face a special quantum barrier: more advanced texts often continue in the axiomatic tradition. Infinite-dimensional spaces, amplitude vectors, matrices, and operators in lieu of momentum and energy-these and other concepts may be introduced with little rationalization or intuition. To an inquiring student, QM can quickly become bizarre; something to be manipulated but not understood. This need not be so. Here the authors present one teaching solution. They show that many of QM's key ideas can be reasoned out from just a few physical assumptions with mathematics that is low in dimension, linear and accessible to many. Some results include a simple reason for believing probability amplitudes are more natural to use than probabilities, the duality of functions as vectors and vice versa, an example of how a matrix, an operator and a physical value might be related, the Heisenberg uncertainty principle and Schrodinger's equation.
Operators play a central role in the formalism of quantum mechanics. In particular, operators corresponding to observables encode important information about the results of quantum measurements. We interviewed upper-level undergraduate physics majors about their understanding of the role of operators in quantum measurements. Previous studies have shown that many students think of measurements on quantum systems as being deterministic and that measurements mathematically correspond to operators acting on the initial quantum state. This study is consistent with and expands on those results. We report on how two students make sense of a quantum measurement problem involving sequential measurements and the role that the eigenvalue equation plays in this sense-making.
Abstract In a cultural, history-based, sense-making approach to Quantum Mechanics multi-representations are fundamental to improve conceptual learning of abstract, mathematical aspects. In fact, they can be integrated among each other allowing students to actively manipulate concepts and instruments which would otherwise be difficult and little accessible. In this presentation we will show different (and interconnected) representations of vectors in complex Hilbert spaces and of eigenvalues and eigenvectors of Hermitian matrices together with an evaluation of their efficacy for students conceptual understanding.
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.