Conceptual change from a classical to a quantum perspective is a challenging task for students. Basic concepts such as state and time evolution, already posing problematic issues in classical physics, require a different interpretation because of the redefinition of the concept of state, inextricably linked to fundamental features of new system behavior such as incompatibility between observables. Within an empirical research aimed at investigating connections between conceptual aspects and formalism emerging from student reasoning, we designed questionnaire items to explore how student describe time evolution of states, how they identify stationary states and how they use related concepts of eigenstate of an observable and incompatibility between observables in this process. Data gathering instruments were developed through a case study to collect student points of view on the relevant issue. For each aspect, at least two items were designed in order to cross-examine from different perspectives. Data analysis was organized following qualitative research methodology. The questionnaire was administered to a small group of students together with follow-up interviews. The study shows that most students link energy eigenstates to stationary states, but inverse implication doesn’t emerge with the same strength. The relation between energy eigenstates and those of different observables is clearly stated only when a commutator expression is identified by students. Energy eigenstates and their linear combinations are related to time invariance and time evolution in quantum mechanics in familiar-abstract context of infinite well, while uncertainties emerge in the crucial situation of scattering. Students answer as expected to questions concerning familiar tasks and contexts, but this doesn’t correspond to conceptual understanding of crucial elements under investigation. There is a need to build competence on critical reasoning.
Conceptual change from a classical to a quantum perspective is a challenging task for students. Basic concepts such as state and time evolution, already posing problematic issues in classical physics, require a different interpretation because of the redefinition of the concept of state, inextricably linked to fundamental features of new system behavior such as incompatibility between observables. Within an empirical research aimed at investigating connections between conceptual aspects and formalism emerging from student reasoning, we designed questionnaire items to explore how student describe time evolution of states, how they identify stationary states and how they use related concepts of eigenstate of an observable and incompatibility between observables in this process. Data gathering instruments were developed through a case study to collect student points of view on the relevant issue. For each aspect, at least two items were designed in order to cross-examine from different perspectives. Data analysis was organized following qualitative research methodology. The questionnaire was administered to a small group of students together with follow-up interviews. The study shows that most students link energy eigenstates to stationary states, but inverse implication doesn’t emerge with the same strength. The relation between energy eigenstates and those of different observables is clearly stated only when a commutator expression is identified by students. Energy eigenstates and their linear combinations are related to time invariance and time evolution in quantum mechanics in familiar-abstract context of infinite well, while uncertainties emerge in the crucial situation of scattering. Students answer as expected to questions concerning familiar tasks and contexts, but this doesn’t correspond to conceptual understanding of crucial elements under investigation. There is a need to build competence on critical reasoning.
Learning quantum mechanics entails adopting a new reference frame for the physical interpretation of the world.The quantum perspective is intrinsically connected with math, which becomes a sort of referent for physical meaning, requiring the employment of new formal structures and a new interpretation of familiar ones.Research evidences that students have difficulty both with concepts and with the use of formalism in qualitative tasks.We administered a 15-item questionnaire focused on incompatibility of observables and related formal structures to 40 physics students of three Italian universities.Semi-structured interviews were scheduled on a subset of students.Results concerning translation processes between math and physical meaning show that most students only look at the square modulus in order to reason on physical information encoded in quantum state, thus neglecting phase relations and their connection with incompatibility.
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.
A common task when problem solving in quantum mechanics, including in a spins-first curriculum, involves changing the basis of a given state.Our research in undergraduate quantum mechanics courses at three institutions explores student thinking about basis, basis expansion coefficients, and change of basis in the context of spin-½ systems.Our investigation is based on conceptual and computational written questions as well as student reasoning interviews.We identify student ideas about whether and how changing basis affects the state, examine how students perceive notation as indicative of choice of basis, explore students' interpretations of the structure and meaning of a basis expansion, and identify the range of methods students employ when changing basis.For instance, we find a recurring idea that changing basis alters the physical system, and observe that some students chose to relabel the ket representing a quantum state vector after changing basis.Together, these results paint a broad, qualitative picture of a variety of ways that students grapple with basis and change of basis, with potential implications for instruction.
We investigate the difficulties of advanced undergraduate students toward the end of a full year upper-level quantum mechanics course with concepts related to quantum measurements and time development. Our analysis is based upon a test administered to 89 students from six universities and interviews with 9 students. Strikingly, most students shared the same difficulties despite variations in background, teaching styles, and textbooks. Concepts related to stationary states, eigenstates, and time dependence of expectation values were found to be particularly difficult. An analysis of written tests and interviews suggests that widespread misconceptions originate from an inability to discriminate between related concepts and a tendency to overgeneralize.
Abstract Learning quantum mechanics (QM) is widely recognized as a challenging task, as it involves highly abstract concepts and requires a significant shift from classical physics reasoning. Developing expertise in QM requires students to represent physical concepts using multiple formalisms, such as Dirac notation and algebraic representations, and to move flexibly between them. This study investigates two main categories of student difficulties related to Dirac notation. The first concerns difficulties in expressing a generic state vector in the position and momentum representations. The second relates to challenges in writing the eigenvectors of the position and momentum operators in Dirac notation. To explore these issues, we administered questionnaires to third-year undergraduate physics students enrolled in a QM course at the Lyon 1 University. We present a detailed analysis of student responses, identify the principal difficulties encountered, and compare our findings with results reported in prior American studies. Our results show that many students struggle to establish a clear connection between Dirac notation and its functional algebraic representation. In particular, students find it easier to recognize state vectors and eigenvectors than to construct them explicitly. Additionally, common difficulties include confusion between bras (e.g. x | and p |) and the corresponding operators, as well as confusion between the Dirac delta distribution and the Kronecker delta. Finally, we propose some interpretations of our results in light of those reported in previous studies.
In order to support students in the development of expertise in quantum mechanics, we asked which concepts and structures can act as organizing principles of non-relativistic theory. The research question has been addressed in a multi-step process based on the analysis of categorization studies, on a content analysis of a sample of undergraduate textbooks, and on the results of existing research on learning difficulties. The answer consists of seven concept maps, intended as models of the organizing principles of quantum knowledge needed to account for the results of measurement and time evolution. By means of these instruments, it is possible to visualize and explore the different facets of the interplay between the vector structure of the quantum states and the operator structure of the observables, and to highlight the educational significance of the relations between observables. The maps can be used by instructors as a support for helping students build a well-organized knowledge structure and by researchers as a basis for the design of investigations into student understanding. While this framework may be adapted to different approaches and interpretive stances, it provides indications in favor of a spin-first approach over a waves-first one.